From 94cf769cd51831b0516369160566e806669d894c Mon Sep 17 00:00:00 2001
From: Mehdi Ataei <mehdi.ataei@autodesk.com>
Date: Thu, 28 Nov 2024 11:40:12 -0500
Subject: [PATCH] Added abstraction layer for boundary condition application
 and stepper initialization, and the capability to add profiles to boundary
 conditions

---
 CHANGELOG.md                                  |   2 +
 examples/cfd/flow_past_sphere_3d.py           |  98 +++++++-----
 examples/cfd/lid_driven_cavity_2d.py          |  41 +++---
 .../cfd/lid_driven_cavity_2d_distributed.py   |  19 ++-
 examples/cfd/turbulent_channel_3d.py          |  57 +++----
 examples/cfd/windtunnel_3d.py                 | 118 ++++++++-------
 examples/performance/mlups_3d.py              |  42 ++----
 .../bc_extrapolation_outflow.py               |  15 +-
 .../boundary_condition/bc_regularized.py      |  24 ++-
 xlb/operator/boundary_condition/bc_zouhe.py   | 104 ++++++++-----
 .../boundary_condition/boundary_condition.py  |  71 ++++++++-
 .../indices_boundary_masker.py                |  13 +-
 .../boundary_masker/mesh_boundary_masker.py   |  66 ++++-----
 xlb/operator/stepper/nse_stepper.py           | 139 +++++++++++++++---
 xlb/operator/stepper/stepper.py               |  37 ++---
 15 files changed, 538 insertions(+), 308 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 1d036de..69ad1f4 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -18,3 +18,5 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 - XLB is now installable via pip
 - Complete rewrite of the codebase for better modularity and extensibility based on "Operators" design pattern
 - Added NVIDIA's Warp backend for state-of-the-art performance
+- Added abstraction layer for boundary condition efficient encoding/decoding of auxiliary data
+- Added the capability to add profiles to boundary conditions
\ No newline at end of file
diff --git a/examples/cfd/flow_past_sphere_3d.py b/examples/cfd/flow_past_sphere_3d.py
index 1b5905e..de74b5a 100644
--- a/examples/cfd/flow_past_sphere_3d.py
+++ b/examples/cfd/flow_past_sphere_3d.py
@@ -1,20 +1,16 @@
 import xlb
 from xlb.compute_backend import ComputeBackend
 from xlb.precision_policy import PrecisionPolicy
-from xlb.helper import create_nse_fields, initialize_eq, check_bc_overlaps
+from xlb.grid import grid_factory
 from xlb.operator.stepper import IncompressibleNavierStokesStepper
 from xlb.operator.boundary_condition import (
     FullwayBounceBackBC,
     HalfwayBounceBackBC,
-    ZouHeBC,
     RegularizedBC,
-    EquilibriumBC,
-    DoNothingBC,
     ExtrapolationOutflowBC,
 )
 from xlb.operator.macroscopic import Macroscopic
-from xlb.operator.boundary_masker import IndicesBoundaryMasker
-from xlb.utils import save_fields_vtk, save_image
+from xlb.utils import save_image
 import warp as wp
 import numpy as np
 import jax.numpy as jnp
@@ -34,18 +30,19 @@ def __init__(self, omega, grid_shape, velocity_set, backend, precision_policy):
         self.velocity_set = velocity_set
         self.backend = backend
         self.precision_policy = precision_policy
-        self.grid, self.f_0, self.f_1, self.missing_mask, self.bc_mask = create_nse_fields(grid_shape)
-        self.stepper = None
+        self.omega = omega
         self.boundary_conditions = []
+        self.u_max = 0.04
 
-        # Setup the simulation BC, its initial conditions, and the stepper
-        self._setup(omega)
+        # Create grid using factory
+        self.grid = grid_factory(grid_shape, compute_backend=backend)
 
-    def _setup(self, omega):
+        # Setup the simulation BC and stepper
+        self._setup()
+
+    def _setup(self):
         self.setup_boundary_conditions()
-        self.setup_boundary_masker()
-        self.initialize_fields()
-        self.setup_stepper(omega)
+        self.setup_stepper()
 
     def define_boundary_indices(self):
         box = self.grid.bounding_box_indices()
@@ -69,31 +66,62 @@ def define_boundary_indices(self):
 
     def setup_boundary_conditions(self):
         inlet, outlet, walls, sphere = self.define_boundary_indices()
-        bc_left = RegularizedBC("velocity", (0.04, 0.0, 0.0), indices=inlet)
-        # bc_left = EquilibriumBC(rho = 1, u=(0.04, 0.0, 0.0), indices=inlet)
+        bc_left = RegularizedBC("velocity", profile=self.bc_profile(), indices=inlet)
         bc_walls = FullwayBounceBackBC(indices=walls)
-        # bc_outlet = RegularizedBC("pressure", 1.0, indices=outlet)
-        # bc_outlet = DoNothingBC(indices=outlet)
         bc_outlet = ExtrapolationOutflowBC(indices=outlet)
         bc_sphere = HalfwayBounceBackBC(indices=sphere)
         self.boundary_conditions = [bc_walls, bc_left, bc_outlet, bc_sphere]
 
-    def setup_boundary_masker(self):
-        # check boundary condition list for duplicate indices before creating bc mask
-        check_bc_overlaps(self.boundary_conditions, self.velocity_set.d, self.backend)
-
-        indices_boundary_masker = IndicesBoundaryMasker(
-            velocity_set=self.velocity_set,
-            precision_policy=self.precision_policy,
-            compute_backend=self.backend,
+    def setup_stepper(self):
+        self.stepper = IncompressibleNavierStokesStepper(
+            omega=self.omega,
+            grid=self.grid,
+            boundary_conditions=self.boundary_conditions,
+            collision_type="BGK",
         )
-        self.bc_mask, self.missing_mask = indices_boundary_masker(self.boundary_conditions, self.bc_mask, self.missing_mask, (0, 0, 0))
-
-    def initialize_fields(self):
-        self.f_0 = initialize_eq(self.f_0, self.grid, self.velocity_set, self.precision_policy, self.backend)
-
-    def setup_stepper(self, omega):
-        self.stepper = IncompressibleNavierStokesStepper(omega, boundary_conditions=self.boundary_conditions, collision_type="BGK")
+        self.f_0, self.f_1, self.bc_mask, self.missing_mask = self.stepper.init_fields()
+
+    def bc_profile(self):
+        u_max = self.u_max  # u_max = 0.04
+        # Get the grid dimensions for the y and z directions
+        H_y = float(self.grid_shape[1] - 1)  # Height in y direction
+        H_z = float(self.grid_shape[2] - 1)  # Height in z direction
+
+        @wp.func
+        def bc_profile_warp(index: wp.vec3i):
+            # Poiseuille flow profile: parabolic velocity distribution
+            y = self.precision_policy.store_precision.wp_dtype(index[1])
+            z = self.precision_policy.store_precision.wp_dtype(index[2])
+
+            # Calculate normalized distance from center
+            y_center = y - (H_y / 2.0)
+            z_center = z - (H_z / 2.0)
+            r_squared = (2.0 * y_center / H_y) ** 2.0 + (2.0 * z_center / H_z) ** 2.0
+
+            # Parabolic profile: u = u_max * (1 - r²)
+            return wp.vec(u_max * wp.max(0.0, 1.0 - r_squared), 0.0, 0.0, 0.0, 0.0, length=5)
+
+        def bc_profile_jax():
+            y = jnp.arange(self.grid_shape[1])
+            z = jnp.arange(self.grid_shape[2])
+            Y, Z = jnp.meshgrid(y, z, indexing="ij")
+
+            # Calculate normalized distance from center
+            y_center = Y - (H_y / 2.0)
+            z_center = Z - (H_z / 2.0)
+            r_squared = (2.0 * y_center / H_y) ** 2.0 + (2.0 * z_center / H_z) ** 2.0
+
+            # Parabolic profile for x velocity, zero for y and z
+            u_x = u_max * jnp.maximum(0.0, 1.0 - r_squared)
+            u_y = jnp.zeros_like(u_x)
+            u_z = jnp.zeros_like(u_x)
+
+            return jnp.stack([u_x, u_y, u_z])
+
+        if self.backend == ComputeBackend.JAX:
+            return bc_profile_jax
+        elif self.backend == ComputeBackend.WARP:
+            return bc_profile_warp
 
     def run(self, num_steps, post_process_interval=100):
         start_time = time.time()
@@ -134,8 +162,8 @@ def post_process(self, i):
 
 if __name__ == "__main__":
     # Running the simulation
-    grid_shape = (512 // 2, 128 // 2, 128 // 2)
-    backend = ComputeBackend.WARP
+    grid_shape = (256 // 2, 128 // 2, 128 // 2)
+    backend = ComputeBackend.JAX
     precision_policy = PrecisionPolicy.FP32FP32
 
     velocity_set = xlb.velocity_set.D3Q19(precision_policy=precision_policy, backend=backend)
diff --git a/examples/cfd/lid_driven_cavity_2d.py b/examples/cfd/lid_driven_cavity_2d.py
index f8bd65a..515a39a 100644
--- a/examples/cfd/lid_driven_cavity_2d.py
+++ b/examples/cfd/lid_driven_cavity_2d.py
@@ -1,8 +1,7 @@
 import xlb
 from xlb.compute_backend import ComputeBackend
 from xlb.precision_policy import PrecisionPolicy
-from xlb.helper import create_nse_fields, initialize_eq, check_bc_overlaps
-from xlb.operator.boundary_masker import IndicesBoundaryMasker
+from xlb.grid import grid_factory
 from xlb.operator.stepper import IncompressibleNavierStokesStepper
 from xlb.operator.boundary_condition import HalfwayBounceBackBC, EquilibriumBC
 from xlb.operator.macroscopic import Macroscopic
@@ -26,19 +25,21 @@ def __init__(self, omega, prescribed_vel, grid_shape, velocity_set, backend, pre
         self.velocity_set = velocity_set
         self.backend = backend
         self.precision_policy = precision_policy
-        self.grid, self.f_0, self.f_1, self.missing_mask, self.bc_mask = create_nse_fields(grid_shape)
-        self.stepper = None
+        self.omega = omega
         self.boundary_conditions = []
         self.prescribed_vel = prescribed_vel
 
-        # Setup the simulation BC, its initial conditions, and the stepper
-        self._setup(omega)
+        # Create grid using factory
+        self.grid = grid_factory(grid_shape, compute_backend=backend)
 
-    def _setup(self, omega):
+        # Setup the simulation BC and stepper
+        self._setup()
+
+    def _setup(self):
         self.setup_boundary_conditions()
-        self.setup_boundary_masker()
-        self.initialize_fields()
-        self.setup_stepper(omega)
+        self.setup_stepper()
+        # Initialize fields using the stepper
+        self.f_0, self.f_1, self.bc_mask, self.missing_mask = self.stepper.init_fields()
 
     def define_boundary_indices(self):
         box = self.grid.bounding_box_indices()
@@ -54,21 +55,13 @@ def setup_boundary_conditions(self):
         bc_walls = HalfwayBounceBackBC(indices=walls)
         self.boundary_conditions = [bc_walls, bc_top]
 
-    def setup_boundary_masker(self):
-        # check boundary condition list for duplicate indices before creating bc mask
-        check_bc_overlaps(self.boundary_conditions, self.velocity_set.d, self.backend)
-        indices_boundary_masker = IndicesBoundaryMasker(
-            velocity_set=self.velocity_set,
-            precision_policy=self.precision_policy,
-            compute_backend=self.backend,
+    def setup_stepper(self):
+        self.stepper = IncompressibleNavierStokesStepper(
+            omega=self.omega,
+            grid=self.grid,
+            boundary_conditions=self.boundary_conditions,
+            collision_type="BGK",
         )
-        self.bc_mask, self.missing_mask = indices_boundary_masker(self.boundary_conditions, self.bc_mask, self.missing_mask)
-
-    def initialize_fields(self):
-        self.f_0 = initialize_eq(self.f_0, self.grid, self.velocity_set, self.precision_policy, self.backend)
-
-    def setup_stepper(self, omega):
-        self.stepper = IncompressibleNavierStokesStepper(omega, boundary_conditions=self.boundary_conditions)
 
     def run(self, num_steps, post_process_interval=100):
         for i in range(num_steps):
diff --git a/examples/cfd/lid_driven_cavity_2d_distributed.py b/examples/cfd/lid_driven_cavity_2d_distributed.py
index cdc9027..c990c77 100644
--- a/examples/cfd/lid_driven_cavity_2d_distributed.py
+++ b/examples/cfd/lid_driven_cavity_2d_distributed.py
@@ -10,15 +10,24 @@ class LidDrivenCavity2D_distributed(LidDrivenCavity2D):
     def __init__(self, omega, prescribed_vel, grid_shape, velocity_set, backend, precision_policy):
         super().__init__(omega, prescribed_vel, grid_shape, velocity_set, backend, precision_policy)
 
-    def setup_stepper(self, omega):
-        stepper = IncompressibleNavierStokesStepper(omega, boundary_conditions=self.boundary_conditions)
-        distributed_stepper = distribute(
+    def setup_stepper(self):
+        # Create the base stepper
+        stepper = IncompressibleNavierStokesStepper(
+            omega=self.omega,
+            grid=self.grid,
+            boundary_conditions=self.boundary_conditions,
+            collision_type="BGK",
+        )
+
+        # Distribute the stepper
+        self.stepper = distribute(
             stepper,
             self.grid,
             self.velocity_set,
         )
-        self.stepper = distributed_stepper
-        return
+
+        # Initialize fields using the distributed stepper
+        self.f_0, self.f_1, self.bc_mask, self.missing_mask = self.stepper.init_fields()
 
 
 if __name__ == "__main__":
diff --git a/examples/cfd/turbulent_channel_3d.py b/examples/cfd/turbulent_channel_3d.py
index eb73fdc..dd95db2 100644
--- a/examples/cfd/turbulent_channel_3d.py
+++ b/examples/cfd/turbulent_channel_3d.py
@@ -2,12 +2,12 @@
 import time
 from xlb.compute_backend import ComputeBackend
 from xlb.precision_policy import PrecisionPolicy
-from xlb.helper import create_nse_fields, initialize_eq
+from xlb.grid import grid_factory
 from xlb.operator.stepper import IncompressibleNavierStokesStepper
 from xlb.operator.boundary_condition import RegularizedBC
 from xlb.operator.macroscopic import Macroscopic
-from xlb.operator.boundary_masker import IndicesBoundaryMasker
 from xlb.utils import save_fields_vtk, save_image
+from xlb.helper import initialize_eq
 import warp as wp
 import numpy as np
 import jax.numpy as jnp
@@ -48,18 +48,16 @@ def __init__(self, channel_half_width, Re_tau, u_tau, grid_shape, velocity_set,
         self.u_tau = u_tau
         self.visc = u_tau * channel_half_width / Re_tau
         self.omega = 1.0 / (3.0 * self.visc + 0.5)
-        # DeltaPlus = Re_tau / channel_half_width
-        # DeltaPlus = u_tau / nu * Delta where u_tau / nu = Re_tau / channel_half_width
-
         self.grid_shape = grid_shape
         self.velocity_set = velocity_set
         self.backend = backend
         self.precision_policy = precision_policy
-        self.grid, self.f_0, self.f_1, self.missing_mask, self.bc_mask = create_nse_fields(grid_shape)
-        self.stepper = None
         self.boundary_conditions = []
 
-        # Setup the simulation BC, its initial conditions, and the stepper
+        # Create grid using factory
+        self.grid = grid_factory(grid_shape, compute_backend=backend)
+
+        # Setup the simulation BC and stepper
         self._setup()
 
     def get_force(self):
@@ -71,9 +69,10 @@ def get_force(self):
 
     def _setup(self):
         self.setup_boundary_conditions()
-        self.setup_boundary_masker()
-        self.initialize_fields()
         self.setup_stepper()
+        # Initialize fields using the stepper
+        self.f_0, self.f_1, self.bc_mask, self.missing_mask = self.stepper.init_fields()
+        self.initialize_fields()
 
     def define_boundary_indices(self):
         # top and bottom sides of the channel are no-slip and the other directions are periodic
@@ -81,21 +80,27 @@ def define_boundary_indices(self):
         walls = [box["bottom"][i] + box["top"][i] for i in range(self.velocity_set.d)]
         return walls
 
+    def bc_profile(self):
+        @wp.func
+        def bc_profile_warp(index: wp.vec3i):
+            return wp.vec(0.0, length=1)
+
+        def bc_profile_jax():
+            return jnp.zeros(1)
+
+        if self.backend == ComputeBackend.JAX:
+            return bc_profile_jax
+        elif self.backend == ComputeBackend.WARP:
+            return bc_profile_warp
+
     def setup_boundary_conditions(self):
         walls = self.define_boundary_indices()
-        bc_walls = RegularizedBC("velocity", (0.0, 0.0, 0.0), indices=walls)
+        bc_walls = RegularizedBC("velocity", profile=self.bc_profile(), indices=walls)
         self.boundary_conditions = [bc_walls]
 
-    def setup_boundary_masker(self):
-        indices_boundary_masker = IndicesBoundaryMasker(
-            velocity_set=self.velocity_set,
-            precision_policy=self.precision_policy,
-            compute_backend=self.backend,
-        )
-        self.bc_mask, self.missing_mask = indices_boundary_masker(self.boundary_conditions, self.bc_mask, self.missing_mask)
-
     def initialize_fields(self):
-        shape = (self.velocity_set.d,) + (self.grid_shape)
+        # Initialize with random velocity field
+        shape = (self.velocity_set.d,) + self.grid_shape
         np.random.seed(0)
         u_init = np.random.random(shape)
         if self.backend == ComputeBackend.JAX:
@@ -105,9 +110,12 @@ def initialize_fields(self):
         self.f_0 = initialize_eq(self.f_0, self.grid, self.velocity_set, self.precision_policy, self.backend, u=u_init)
 
     def setup_stepper(self):
-        force = self.get_force()
         self.stepper = IncompressibleNavierStokesStepper(
-            self.omega, boundary_conditions=self.boundary_conditions, collision_type="KBC", forcing_scheme="exact_difference", force_vector=force
+            omega=self.omega,
+            grid=self.grid,
+            boundary_conditions=self.boundary_conditions,
+            collision_type="BGK",
+            force_vector=self.get_force(),
         )
 
     def run(self, num_steps, print_interval, post_process_interval=100):
@@ -142,14 +150,12 @@ def post_process(self, i):
         u_magnitude = (u[0] ** 2 + u[1] ** 2 + u[2] ** 2) ** 0.5
         fields = {"rho": rho[0], "u_x": u[0], "u_y": u[1], "u_z": u[2], "u_magnitude": u_magnitude}
         save_fields_vtk(fields, timestep=i)
-        save_image(fields["u_magnitude"][:, grid_size_y // 2, :], timestep=i)
+        save_image(fields["u_magnitude"][:, self.grid_shape[1] // 2, :], timestep=i)
 
         # Save monitor plot
         self.plot_uplus(u, i)
-        return
 
     def plot_uplus(self, u, timestep):
-        # Compute moving average of drag coefficient, 100, 1000, 10000
         # mean streamwise velocity in wall units u^+(z)
         # Wall distance in wall units to be used inside output_data
         zz = np.arange(self.grid_shape[-1])
@@ -165,6 +171,7 @@ def plot_uplus(self, u, timestep):
         ax.set_ylim([0, 20])
         fname = "uplus_" + str(timestep // 10000).zfill(5) + ".png"
         plt.savefig(fname, format="png")
+        plt.close()
 
 
 if __name__ == "__main__":
diff --git a/examples/cfd/windtunnel_3d.py b/examples/cfd/windtunnel_3d.py
index 96c79f7..b6ec4f8 100644
--- a/examples/cfd/windtunnel_3d.py
+++ b/examples/cfd/windtunnel_3d.py
@@ -3,20 +3,16 @@
 import time
 from xlb.compute_backend import ComputeBackend
 from xlb.precision_policy import PrecisionPolicy
-from xlb.helper import create_nse_fields, initialize_eq, check_bc_overlaps
+from xlb.grid import grid_factory
 from xlb.operator.stepper import IncompressibleNavierStokesStepper
 from xlb.operator.boundary_condition import (
+    HalfwayBounceBackBC,
     FullwayBounceBackBC,
-    EquilibriumBC,
-    DoNothingBC,
     RegularizedBC,
-    HalfwayBounceBackBC,
     ExtrapolationOutflowBC,
-    GradsApproximationBC,
 )
 from xlb.operator.force.momentum_transfer import MomentumTransfer
 from xlb.operator.macroscopic import Macroscopic
-from xlb.operator.boundary_masker import IndicesBoundaryMasker, MeshBoundaryMasker, MeshDistanceBoundaryMasker
 from xlb.utils import save_fields_vtk, save_image
 import warp as wp
 import numpy as np
@@ -37,13 +33,14 @@ def __init__(self, omega, wind_speed, grid_shape, velocity_set, backend, precisi
         self.velocity_set = velocity_set
         self.backend = backend
         self.precision_policy = precision_policy
-        self.grid, self.f_0, self.f_1, self.missing_mask, self.bc_mask = create_nse_fields(grid_shape)
-        self.stepper = None
+        self.omega = omega
         self.boundary_conditions = []
-
-        # Setup the simulation BC, its initial conditions, and the stepper
         self.wind_speed = wind_speed
-        self.omega = omega
+
+        # Create grid using factory
+        self.grid = grid_factory(grid_shape, compute_backend=backend)
+
+        # Setup the simulation BC and stepper
         self._setup()
 
         # Make list to store drag coefficients
@@ -52,11 +49,10 @@ def __init__(self, omega, wind_speed, grid_shape, velocity_set, backend, precisi
         self.lift_coefficients = []
 
     def _setup(self):
-        # NOTE: it is important to initialize fields before setup_boundary_masker is called because f_0 or f_1 might be used to store BC information
-        self.initialize_fields()
         self.setup_boundary_conditions()
-        self.setup_boundary_masker()
         self.setup_stepper()
+        # Initialize fields using the stepper
+        self.f_0, self.f_1, self.bc_mask, self.missing_mask = self.stepper.init_fields()
 
     def voxelize_stl(self, stl_filename, length_lbm_unit):
         mesh = trimesh.load_mesh(stl_filename, process=False)
@@ -85,57 +81,70 @@ def define_boundary_indices(self):
         length_phys_unit = mesh_extents.max()
         length_lbm_unit = self.grid_shape[0] / 4
         dx = length_phys_unit / length_lbm_unit
-        shift = np.array([self.grid_shape[0] * dx / 4, (self.grid_shape[1] * dx - mesh_extents[1]) / 2, 0.0])
+        mesh_vertices = mesh_vertices / dx
+        shift = np.array([self.grid_shape[0] / 4, (self.grid_shape[1] - mesh_extents[1] / dx) / 2, 0.0])
         car = mesh_vertices + shift
-        self.grid_spacing = dx
         self.car_cross_section = np.prod(mesh_extents[1:]) / dx**2
 
         return inlet, outlet, walls, car
 
     def setup_boundary_conditions(self):
         inlet, outlet, walls, car = self.define_boundary_indices()
-        bc_left = EquilibriumBC(rho=1.0, u=(self.wind_speed, 0.0, 0.0), indices=inlet)
-        # bc_left = RegularizedBC('velocity', (self.wind_speed, 0.0, 0.0), indices=inlet)
+        bc_left = RegularizedBC("velocity", profile=self.bc_profile(), indices=inlet)
         bc_walls = FullwayBounceBackBC(indices=walls)
         bc_do_nothing = ExtrapolationOutflowBC(indices=outlet)
-        # bc_car = HalfwayBounceBackBC(mesh_vertices=car)
-        bc_car = GradsApproximationBC(mesh_vertices=car)
-        # bc_car = FullwayBounceBackBC(mesh_vertices=car)
+        bc_car = HalfwayBounceBackBC(mesh_vertices=car)
         self.boundary_conditions = [bc_walls, bc_left, bc_do_nothing, bc_car]
 
-    def setup_boundary_masker(self):
-        # check boundary condition list for duplicate indices before creating bc mask
-        check_bc_overlaps(self.boundary_conditions, self.velocity_set.d, self.backend)
+    def bc_profile(self):
+        u_max = self.wind_speed
+        # Get the grid dimensions for the y and z directions
+        H_y = float(self.grid_shape[1] - 1)  # Height in y direction
+        H_z = float(self.grid_shape[2] - 1)  # Height in z direction
 
-        indices_boundary_masker = IndicesBoundaryMasker(
-            velocity_set=self.velocity_set,
-            precision_policy=self.precision_policy,
-            compute_backend=self.backend,
-        )
-        # mesh_boundary_masker = MeshBoundaryMasker(
-        #     velocity_set=self.velocity_set,
-        #     precision_policy=self.precision_policy,
-        #     compute_backend=self.backend,
-        # )
-        mesh_distance_boundary_masker = MeshDistanceBoundaryMasker(
-            velocity_set=self.velocity_set,
-            precision_policy=self.precision_policy,
-            compute_backend=self.backend,
-        )
-        bclist_other = self.boundary_conditions[:-1]
-        bc_mesh = self.boundary_conditions[-1]
-        dx = self.grid_spacing
-        origin, spacing = (0, 0, 0), (dx, dx, dx)
-        self.bc_mask, self.missing_mask = indices_boundary_masker(bclist_other, self.bc_mask, self.missing_mask)
-        # self.bc_mask, self.missing_mask = mesh_boundary_masker(bc_mesh, origin, spacing, self.bc_mask, self.missing_mask)
-        self.bc_mask, self.missing_mask, self.f_1 = mesh_distance_boundary_masker(bc_mesh, origin, spacing, self.bc_mask, self.missing_mask, self.f_1)
-
-    def initialize_fields(self):
-        self.f_0 = initialize_eq(self.f_0, self.grid, self.velocity_set, self.precision_policy, self.backend)
-        self.f_1 = initialize_eq(self.f_1, self.grid, self.velocity_set, self.precision_policy, self.backend)
+        @wp.func
+        def bc_profile_warp(index: wp.vec3i):
+            # Poiseuille flow profile: parabolic velocity distribution
+            y = self.precision_policy.store_precision.wp_dtype(index[1])
+            z = self.precision_policy.store_precision.wp_dtype(index[2])
+
+            # Calculate normalized distance from center
+            y_center = y - (H_y / 2.0)
+            z_center = z - (H_z / 2.0)
+            r_squared = (2.0 * y_center / H_y) ** 2.0 + (2.0 * z_center / H_z) ** 2.0
+
+            # Parabolic profile: u = u_max * (1 - r²)
+            return wp.vec(u_max * wp.max(0.0, 1.0 - r_squared), 0.0, 0.0, 0.0, 0.0, length=5)
+
+        def bc_profile_jax():
+            y = jnp.arange(self.grid_shape[1])
+            z = jnp.arange(self.grid_shape[2])
+            Y, Z = jnp.meshgrid(y, z, indexing="ij")
+
+            # Calculate normalized distance from center
+            y_center = Y - (H_y / 2.0)
+            z_center = Z - (H_z / 2.0)
+            r_squared = (2.0 * y_center / H_y) ** 2.0 + (2.0 * z_center / H_z) ** 2.0
+
+            # Parabolic profile for x velocity, zero for y and z
+            u_x = u_max * jnp.maximum(0.0, 1.0 - r_squared)
+            u_y = jnp.zeros_like(u_x)
+            u_z = jnp.zeros_like(u_x)
+
+            return jnp.stack([u_x, u_y, u_z])
+
+        if self.backend == ComputeBackend.JAX:
+            return bc_profile_jax
+        elif self.backend == ComputeBackend.WARP:
+            return bc_profile_warp
 
     def setup_stepper(self):
-        self.stepper = IncompressibleNavierStokesStepper(self.omega, boundary_conditions=self.boundary_conditions, collision_type="KBC")
+        self.stepper = IncompressibleNavierStokesStepper(
+            omega=self.omega,
+            grid=self.grid,
+            boundary_conditions=self.boundary_conditions,
+            collision_type="BGK",
+        )
 
     def run(self, num_steps, print_interval, post_process_interval=100):
         # Setup the operator for computing surface forces at the interface of the specified BC
@@ -176,7 +185,7 @@ def post_process(self, i):
         fields = {"u_magnitude": u_magnitude}
 
         save_fields_vtk(fields, timestep=i)
-        save_image(fields["u_magnitude"][:, grid_size_y // 2, :], timestep=i)
+        save_image(fields["u_magnitude"][:, self.grid_shape[1] // 2, :], timestep=i)
 
         # Compute lift and drag
         boundary_force = self.momentum_transfer(self.f_0, self.f_1, self.bc_mask, self.missing_mask)
@@ -190,7 +199,6 @@ def post_process(self, i):
 
         # Save monitor plot
         self.plot_drag_coefficient()
-        return
 
     def plot_drag_coefficient(self):
         # Compute moving average of drag coefficient, 100, 1000, 10000
@@ -230,14 +238,14 @@ def plot_drag_coefficient(self):
     # Configuration
     backend = ComputeBackend.WARP
     precision_policy = PrecisionPolicy.FP32FP32
+
     velocity_set = xlb.velocity_set.D3Q27(precision_policy=precision_policy, backend=backend)
     wind_speed = 0.02
     num_steps = 100000
     print_interval = 1000
 
     # Set up Reynolds number and deduce relaxation time (omega)
-    # Re = 50000.0
-    Re = 500000000000.0
+    Re = 5000.0
     clength = grid_size_x - 1
     visc = wind_speed * clength / Re
     omega = 1.0 / (3.0 * visc + 0.5)
diff --git a/examples/performance/mlups_3d.py b/examples/performance/mlups_3d.py
index 2001fb2..d684614 100644
--- a/examples/performance/mlups_3d.py
+++ b/examples/performance/mlups_3d.py
@@ -2,9 +2,10 @@
 import argparse
 import time
 import warp as wp
+import numpy as np
 from xlb.compute_backend import ComputeBackend
 from xlb.precision_policy import PrecisionPolicy
-from xlb.helper import create_nse_fields, initialize_eq
+from xlb.grid import grid_factory
 from xlb.operator.stepper import IncompressibleNavierStokesStepper
 from xlb.operator.boundary_condition import FullwayBounceBackBC, EquilibriumBC
 from xlb.distribute import distribute
@@ -40,32 +41,21 @@ def setup_simulation(args):
     return backend, precision_policy
 
 
-def create_grid_and_fields(cube_edge):
-    grid_shape = (cube_edge, cube_edge, cube_edge)
-    grid, f_0, f_1, missing_mask, bc_mask = create_nse_fields(grid_shape)
-
-    return grid, f_0, f_1, missing_mask, bc_mask
-
-
-def define_boundary_indices(grid):
+def run(backend, precision_policy, grid_shape, num_steps):
+    # Create grid and setup boundary conditions
+    grid = grid_factory(grid_shape)
     box = grid.bounding_box_indices()
     box_no_edge = grid.bounding_box_indices(remove_edges=True)
     lid = box_no_edge["top"]
     walls = [box["bottom"][i] + box["left"][i] + box["right"][i] + box["front"][i] + box["back"][i] for i in range(len(grid.shape))]
-    return lid, walls
+    walls = np.unique(np.array(walls), axis=-1).tolist()
 
+    boundary_conditions = [EquilibriumBC(rho=1.0, u=(0.02, 0.0, 0.0), indices=lid), FullwayBounceBackBC(indices=walls)]
 
-def setup_boundary_conditions(grid):
-    lid, walls = define_boundary_indices(grid)
-    bc_top = EquilibriumBC(rho=1.0, u=(0.02, 0.0, 0.0), indices=lid)
-    bc_walls = FullwayBounceBackBC(indices=walls)
-    return [bc_top, bc_walls]
-
-
-def run(f_0, f_1, backend, precision_policy, grid, bc_mask, missing_mask, num_steps):
-    omega = 1.0
-    stepper = IncompressibleNavierStokesStepper(omega, boundary_conditions=setup_boundary_conditions(grid))
+    # Create stepper
+    stepper = IncompressibleNavierStokesStepper(omega=1.0, grid=grid, boundary_conditions=boundary_conditions, collision_type="BGK")
 
+    # Distribute if using JAX backend
     if backend == ComputeBackend.JAX:
         stepper = distribute(
             stepper,
@@ -73,6 +63,8 @@ def run(f_0, f_1, backend, precision_policy, grid, bc_mask, missing_mask, num_st
             xlb.velocity_set.D3Q19(precision_policy=precision_policy, backend=backend),
         )
 
+    # Initialize fields and run simulation
+    f_0, f_1, bc_mask, missing_mask = stepper.init_fields()
     start_time = time.time()
 
     for i in range(num_steps):
@@ -80,8 +72,7 @@ def run(f_0, f_1, backend, precision_policy, grid, bc_mask, missing_mask, num_st
         f_0, f_1 = f_1, f_0
     wp.synchronize()
 
-    end_time = time.time()
-    return end_time - start_time
+    return time.time() - start_time
 
 
 def calculate_mlups(cube_edge, num_steps, elapsed_time):
@@ -93,11 +84,8 @@ def calculate_mlups(cube_edge, num_steps, elapsed_time):
 def main():
     args = parse_arguments()
     backend, precision_policy = setup_simulation(args)
-    velocity_set = xlb.velocity_set.D3Q19(precision_policy=precision_policy, backend=backend)
-    grid, f_0, f_1, missing_mask, bc_mask = create_grid_and_fields(args.cube_edge)
-    f_0 = initialize_eq(f_0, grid, velocity_set, precision_policy, backend)
-
-    elapsed_time = run(f_0, f_1, backend, precision_policy, grid, bc_mask, missing_mask, args.num_steps)
+    grid_shape = (args.cube_edge, args.cube_edge, args.cube_edge)
+    elapsed_time = run(backend, precision_policy, grid_shape, args.num_steps)
     mlups = calculate_mlups(args.cube_edge, args.num_steps, elapsed_time)
 
     print(f"Simulation completed in {elapsed_time:.2f} seconds")
diff --git a/xlb/operator/boundary_condition/bc_extrapolation_outflow.py b/xlb/operator/boundary_condition/bc_extrapolation_outflow.py
index 8a2a482..fa75490 100644
--- a/xlb/operator/boundary_condition/bc_extrapolation_outflow.py
+++ b/xlb/operator/boundary_condition/bc_extrapolation_outflow.py
@@ -56,12 +56,15 @@ def __init__(
             mesh_vertices,
         )
 
+        # Set the flag for auxilary data recovery
+        self.needs_aux_recovery = True
+
         # find and store the normal vector using indices
         self._get_normal_vec(indices)
 
         # Unpack the two warp functionals needed for this BC!
         if self.compute_backend == ComputeBackend.WARP:
-            self.warp_functional, self.prepare_bc_auxilary_data = self.warp_functional
+            self.warp_functional, self.update_bc_auxilary_data = self.warp_functional
 
     def _get_normal_vec(self, indices):
         # Get the frequency count and most common element directly
@@ -92,9 +95,9 @@ def _roll(self, fld, vec):
             return jnp.roll(fld, (vec[0], vec[1], vec[2]), axis=(1, 2, 3))
 
     @partial(jit, static_argnums=(0,), inline=True)
-    def prepare_bc_auxilary_data(self, f_pre, f_post, bc_mask, missing_mask):
+    def update_bc_auxilary_data(self, f_pre, f_post, bc_mask, missing_mask):
         """
-        Prepare the auxilary distribution functions for the boundary condition.
+        Update the auxilary distribution functions for the boundary condition.
         Since this function is called post-collisiotn: f_pre = f_post_stream and f_post = f_post_collision
         """
         sound_speed = 1.0 / jnp.sqrt(3.0)
@@ -171,7 +174,7 @@ def functional(
             return _f
 
         @wp.func
-        def prepare_bc_auxilary_data(
+        def update_bc_auxilary_data(
             index: Any,
             timestep: Any,
             missing_mask: Any,
@@ -180,7 +183,7 @@ def prepare_bc_auxilary_data(
             f_pre: Any,
             f_post: Any,
         ):
-            # Preparing the formulation for this BC using the neighbour's populations stored in f_aux and
+            # Update the auxilary data for this BC using the neighbour's populations stored in f_aux and
             # f_pre (post-streaming values of the current voxel). We use directions that leave the domain
             # for storing this prepared data.
             _f = f_post
@@ -199,7 +202,7 @@ def prepare_bc_auxilary_data(
 
         kernel = self._construct_kernel(functional)
 
-        return (functional, prepare_bc_auxilary_data), kernel
+        return (functional, update_bc_auxilary_data), kernel
 
     @Operator.register_backend(ComputeBackend.WARP)
     def warp_implementation(self, f_pre, f_post, bc_mask, missing_mask):
diff --git a/xlb/operator/boundary_condition/bc_regularized.py b/xlb/operator/boundary_condition/bc_regularized.py
index d1abd4e..f36439e 100644
--- a/xlb/operator/boundary_condition/bc_regularized.py
+++ b/xlb/operator/boundary_condition/bc_regularized.py
@@ -44,7 +44,7 @@ class RegularizedBC(ZouHeBC):
     def __init__(
         self,
         bc_type,
-        prescribed_value,
+        profile,
         velocity_set: VelocitySet = None,
         precision_policy: PrecisionPolicy = None,
         compute_backend: ComputeBackend = None,
@@ -54,7 +54,7 @@ def __init__(
         # Call the parent constructor
         super().__init__(
             bc_type,
-            prescribed_value,
+            profile,
             velocity_set,
             precision_policy,
             compute_backend,
@@ -127,15 +127,11 @@ def _construct_warp(self):
         # assign placeholders for both u and rho based on prescribed_value
         _d = self.velocity_set.d
         _q = self.velocity_set.q
-        u = self.prescribed_value if self.bc_type == "velocity" else (0,) * _d
-        rho = self.prescribed_value if self.bc_type == "pressure" else 0.0
 
         # Set local constants TODO: This is a hack and should be fixed with warp update
         # _u_vec = wp.vec(_d, dtype=self.compute_dtype)
         # compute Qi tensor and store it in self
         _u_vec = wp.vec(self.velocity_set.d, dtype=self.compute_dtype)
-        _rho = self.compute_dtype(rho)
-        _u = _u_vec(u[0], u[1], u[2]) if _d == 3 else _u_vec(u[0], u[1])
         _opp_indices = self.velocity_set.opp_indices
         _w = self.velocity_set.w
         _c = self.velocity_set.c
@@ -222,6 +218,15 @@ def functional_velocity(
             # Find normal vector
             normals = get_normal_vectors(missing_mask)
 
+            # Find the value of u from the missing directions
+            for l in range(wp.static(_q)):
+                # Since we are only considering normal velocity, we only need to find one value
+                if missing_mask[l] == wp.uint8(1):
+                    # Create velocity vector by multiplying the prescribed value with the normal vector
+                    prescribed_value = f_1[_opp_indices[l], index[0], index[1], index[2]]
+                    _u = -prescribed_value * normals
+                    break
+
             # calculate rho
             fsum = _get_fsum(_f, missing_mask)
             unormal = self.compute_dtype(0.0)
@@ -253,6 +258,13 @@ def functional_pressure(
             # Find normal vector
             normals = get_normal_vectors(missing_mask)
 
+            # Find the value of rho from the missing directions
+            for q in range(wp.static(_q)):
+                # Since we need only one scalar value, we only need to find one value
+                if missing_mask[q] == wp.uint8(1):
+                    _rho = f_1[_opp_indices[q], index[0], index[1], index[2]]
+                    break
+
             # calculate velocity
             fsum = _get_fsum(_f, missing_mask)
             unormal = -self.compute_dtype(1.0) + fsum / _rho
diff --git a/xlb/operator/boundary_condition/bc_zouhe.py b/xlb/operator/boundary_condition/bc_zouhe.py
index 3e89992..95643f9 100644
--- a/xlb/operator/boundary_condition/bc_zouhe.py
+++ b/xlb/operator/boundary_condition/bc_zouhe.py
@@ -21,6 +21,7 @@
     boundary_condition_registry,
 )
 from xlb.operator.equilibrium import QuadraticEquilibrium
+import jax
 
 
 class ZouHeBC(BoundaryCondition):
@@ -38,7 +39,7 @@ class ZouHeBC(BoundaryCondition):
     def __init__(
         self,
         bc_type,
-        prescribed_value,
+        profile,
         velocity_set: VelocitySet = None,
         precision_policy: PrecisionPolicy = None,
         compute_backend: ComputeBackend = None,
@@ -50,7 +51,7 @@ def __init__(
         assert bc_type in ["velocity", "pressure"], f"type = {bc_type} not supported! Use 'pressure' or 'velocity'."
         self.bc_type = bc_type
         self.equilibrium_operator = QuadraticEquilibrium()
-        self.prescribed_value = prescribed_value
+        self.profile = profile
 
         # Call the parent constructor
         super().__init__(
@@ -62,11 +63,14 @@ def __init__(
             mesh_vertices,
         )
 
-        # Set the prescribed value for pressure or velocity
-        dim = self.velocity_set.d
-        if self.compute_backend == ComputeBackend.JAX:
-            self.prescribed_value = jnp.atleast_1d(prescribed_value)[(slice(None),) + (None,) * dim]
-            # TODO: this won't work if the prescribed values are a profile with the length of bdry indices!
+        # This BC needs auxilary data initialization before streaming
+        self.needs_aux_init = True
+
+        # This BC needs auxilary data recovery after streaming
+        self.needs_aux_recovery = True
+
+        # This BC needs one auxilary data for the density or normal velocity
+        self.num_of_aux_data = 1
 
         # This BC needs padding for finding missing directions when imposed on a geometry that is in the domain interior
         self.needs_padding = True
@@ -84,13 +88,25 @@ def _get_normal_vec(self, missing_mask):
         normals = -jnp.tensordot(main_c, m, axes=(-1, 0))
         return normals
 
+    @partial(jit, static_argnums=(0, 2, 3), inline=True)
+    def _broadcast_prescribed_values(self, prescribed_values, prescribed_values_shape, target_shape):
+        broadcast_dims = [0]  # Always include the leading dimension
+        p_idx = 1  # Start from the second dimension for prescribed_values
+        for t_idx, t_dim in enumerate(target_shape):
+            if p_idx < len(prescribed_values_shape) + 1 and prescribed_values_shape[p_idx] == t_dim:
+                broadcast_dims.append(t_idx)
+                p_idx += 1
+        broadcast_dims = tuple(broadcast_dims)
+        return jax.lax.broadcast_in_dim(prescribed_values, target_shape, broadcast_dims)
+
     @partial(jit, static_argnums=(0,), inline=True)
     def get_rho(self, fpop, missing_mask):
         if self.bc_type == "velocity":
-            vel = self.prescribed_value
+            target_shape = (self.velocity_set.d,) + fpop.shape[1:]
+            vel = self._broadcast_prescribed_values(self.prescribed_values, self.prescribed_values.shape, target_shape)
             rho = self.calculate_rho(fpop, vel, missing_mask)
         elif self.bc_type == "pressure":
-            rho = self.prescribed_value
+            rho = self.prescribed_values
         else:
             raise ValueError(f"type = {self.bc_type} not supported! Use 'pressure' or 'velocity'.")
         return rho
@@ -98,9 +114,10 @@ def get_rho(self, fpop, missing_mask):
     @partial(jit, static_argnums=(0,), inline=True)
     def get_vel(self, fpop, missing_mask):
         if self.bc_type == "velocity":
-            vel = self.prescribed_value
+            target_shape = (self.velocity_set.d,) + fpop.shape[1:]
+            vel = self._broadcast_prescribed_values(self.prescribed_values, self.prescribed_values.shape, target_shape)
         elif self.bc_type == "pressure":
-            rho = self.prescribed_value
+            rho = self.prescribed_values
             vel = self.calculate_vel(fpop, rho, missing_mask)
         else:
             raise ValueError(f"type = {self.bc_type} not supported! Use 'pressure' or 'velocity'.")
@@ -134,14 +151,13 @@ def calculate_rho(self, fpop, vel, missing_mask):
         return rho
 
     @partial(jit, static_argnums=(0,), inline=True)
-    def calculate_equilibrium(self, fpop, missing_mask):
+    def calculate_equilibrium(self, f_post, missing_mask):
         """
         This is the ZouHe method of calculating the missing macroscopic variables at the boundary.
         """
-        rho = self.get_rho(fpop, missing_mask)
-        vel = self.get_vel(fpop, missing_mask)
+        rho = self.get_rho(f_post, missing_mask)
+        vel = self.get_vel(f_post, missing_mask)
 
-        # compute feq at the boundary
         feq = self.equilibrium_operator(rho, vel)
         return feq
 
@@ -176,14 +192,11 @@ def _construct_warp(self):
         # assign placeholders for both u and rho based on prescribed_value
         _d = self.velocity_set.d
         _q = self.velocity_set.q
-        u = self.prescribed_value if self.bc_type == "velocity" else (0,) * _d
-        rho = self.prescribed_value if self.bc_type == "pressure" else 0.0
 
         # Set local constants TODO: This is a hack and should be fixed with warp update
         # _u_vec = wp.vec(_d, dtype=self.compute_dtype)
         _u_vec = wp.vec(self.velocity_set.d, dtype=self.compute_dtype)
-        _rho = self.compute_dtype(rho)
-        _u = _u_vec(u[0], u[1], u[2]) if _d == 3 else _u_vec(u[0], u[1])
+        _rho = self.compute_dtype(0.0)
         _opp_indices = self.velocity_set.opp_indices
         _c = self.velocity_set.c
         _c_float = self.velocity_set.c_float
@@ -207,7 +220,7 @@ def _get_fsum(
         def get_normal_vectors(
             missing_mask: Any,
         ):
-            if wp.static(_d == 3):
+            if _d == 3:
                 for l in range(_q):
                     if missing_mask[l] == wp.uint8(1) and wp.abs(_c[0, l]) + wp.abs(_c[1, l]) + wp.abs(_c[2, l]) == 1:
                         return -_u_vec(_c_float[0, l], _c_float[1, l], _c_float[2, l])
@@ -231,62 +244,79 @@ def bounceback_nonequilibrium(
         def functional_velocity(
             index: Any,
             timestep: Any,
-            missing_mask: Any,
-            f_0: Any,
-            f_1: Any,
+            _missing_mask: Any,
             f_pre: Any,
             f_post: Any,
+            _f_pre: Any,
+            _f_post: Any,
         ):
             # Post-streaming values are only modified at missing direction
-            _f = f_post
+            _f = _f_post
 
             # Find normal vector
-            normals = get_normal_vectors(missing_mask)
+            normals = get_normal_vectors(_missing_mask)
 
             # calculate rho
-            fsum = _get_fsum(_f, missing_mask)
+            fsum = _get_fsum(_f, _missing_mask)
             unormal = self.compute_dtype(0.0)
+
+            # Find the value of u from the missing directions
+            for l in range(_q):
+                # Since we are only considering normal velocity, we only need to find one value (all values are the same in the missing directions)
+                if _missing_mask[l] == wp.uint8(1):
+                    # Create velocity vector by multiplying the prescribed value with the normal vector
+                    # TODO: This can be optimized by saving _missing_mask[l] in the bc class later since it is the same for all boundary cells
+                    prescribed_value = f_post[_opp_indices[l], index[0], index[1], index[2]]
+                    _u = -prescribed_value * normals
+                    break
+
             for d in range(_d):
                 unormal += _u[d] * normals[d]
+
             _rho = fsum / (self.compute_dtype(1.0) + unormal)
 
             # impose non-equilibrium bounceback
-            feq = self.equilibrium_operator.warp_functional(_rho, _u)
-            _f = bounceback_nonequilibrium(_f, feq, missing_mask)
+            _feq = self.equilibrium_operator.warp_functional(_rho, _u)
+            _f = bounceback_nonequilibrium(_f, _feq, _missing_mask)
             return _f
 
         @wp.func
         def functional_pressure(
             index: Any,
             timestep: Any,
-            missing_mask: Any,
-            f_0: Any,
-            f_1: Any,
+            _missing_mask: Any,
             f_pre: Any,
             f_post: Any,
+            _f_pre: Any,
+            _f_post: Any,
         ):
             # Post-streaming values are only modified at missing direction
-            _f = f_post
+            _f = _f_post
 
             # Find normal vector
-            normals = get_normal_vectors(missing_mask)
+            normals = get_normal_vectors(_missing_mask)
+
+            # Find the value of rho from the missing directions
+            for q in range(_q):
+                # Since we need only one scalar value, we only need to find one value (all values are the same in the missing directions)
+                if _missing_mask[q] == wp.uint8(1):
+                    _rho = f_post[_opp_indices[q], index[0], index[1], index[2]]
+                    break
 
             # calculate velocity
-            fsum = _get_fsum(_f, missing_mask)
+            fsum = _get_fsum(_f, _missing_mask)
             unormal = -self.compute_dtype(1.0) + fsum / _rho
             _u = unormal * normals
 
             # impose non-equilibrium bounceback
             feq = self.equilibrium_operator.warp_functional(_rho, _u)
-            _f = bounceback_nonequilibrium(_f, feq, missing_mask)
+            _f = bounceback_nonequilibrium(_f, feq, _missing_mask)
             return _f
 
         if self.bc_type == "velocity":
             functional = functional_velocity
         elif self.bc_type == "pressure":
             functional = functional_pressure
-        elif self.bc_type == "velocity":
-            functional = functional_pressure
 
         kernel = self._construct_kernel(functional)
 
diff --git a/xlb/operator/boundary_condition/boundary_condition.py b/xlb/operator/boundary_condition/boundary_condition.py
index bf1eef2..ae99a04 100644
--- a/xlb/operator/boundary_condition/boundary_condition.py
+++ b/xlb/operator/boundary_condition/boundary_condition.py
@@ -7,6 +7,8 @@
 from typing import Any
 from jax import jit
 from functools import partial
+import jax
+import jax.numpy as jnp
 
 from xlb.velocity_set.velocity_set import VelocitySet
 from xlb.precision_policy import PrecisionPolicy
@@ -57,13 +59,25 @@ def __init__(
         # A flag for BCs that need implicit boundary distance between the grid and a mesh (to be set to True if applicable inside each BC)
         self.needs_mesh_distance = False
 
+        # A flag for BCs that need auxilary data initialization before stepper
+        self.needs_aux_init = False
+
+        # A flag to track if the BC is initialized with auxilary data
+        self.is_initialized_with_aux_data = False
+
+        # Number of auxilary data needed for the BC (for prescribed values)
+        self.num_of_aux_data = 0
+
+        # A flag for BCs that need auxilary data recovery after streaming
+        self.needs_aux_recovery = False
+
         if self.compute_backend == ComputeBackend.WARP:
             # Set local constants TODO: This is a hack and should be fixed with warp update
             _f_vec = wp.vec(self.velocity_set.q, dtype=self.compute_dtype)
             _missing_mask_vec = wp.vec(self.velocity_set.q, dtype=wp.uint8)  # TODO fix vec bool
 
         @wp.func
-        def prepare_bc_auxilary_data(
+        def update_bc_auxilary_data(
             index: Any,
             timestep: Any,
             missing_mask: Any,
@@ -102,10 +116,10 @@ def _get_thread_data(
         # Construct some helper warp functions for getting tid data
         if self.compute_backend == ComputeBackend.WARP:
             self._get_thread_data = _get_thread_data
-            self.prepare_bc_auxilary_data = prepare_bc_auxilary_data
+            self.update_bc_auxilary_data = update_bc_auxilary_data
 
     @partial(jit, static_argnums=(0,), inline=True)
-    def prepare_bc_auxilary_data(self, f_pre, f_post, bc_mask, missing_mask):
+    def update_bc_auxilary_data(self, f_pre, f_post, bc_mask, missing_mask):
         """
         A placeholder function for prepare the auxilary distribution functions for the boundary condition.
         currently being called after collision only.
@@ -146,3 +160,54 @@ def kernel(
                 f_post[l, index[0], index[1], index[2]] = self.store_dtype(_f[l])
 
         return kernel
+
+    def _construct_aux_data_init_kernel(self, functional):
+        """
+        Constructs the warp kernel for the auxilary data recovery.
+        """
+        _id = wp.uint8(self.id)
+        _opp_indices = self.velocity_set.opp_indices
+        _num_of_aux_data = self.num_of_aux_data
+
+        # Construct the warp kernel
+        @wp.kernel
+        def aux_data_init_kernel(
+            f_0: wp.array4d(dtype=Any),
+            f_1: wp.array4d(dtype=Any),
+            bc_mask: wp.array4d(dtype=wp.uint8),
+            missing_mask: wp.array4d(dtype=wp.bool),
+        ):
+            # Get the global index
+            i, j, k = wp.tid()
+            index = wp.vec3i(i, j, k)
+
+            # read tid data
+            _f_0, _f_1, _boundary_id, _missing_mask = self._get_thread_data(f_0, f_1, bc_mask, missing_mask, index)
+
+            # Apply the functional
+            if _boundary_id == _id:
+                # prescribed_values is a q-sized vector of type wp.vec
+                prescribed_values = functional(index)
+                # Write the result for all q directions, but only store up to num_of_aux_data
+                # TODO: Somehow raise an error if the number of prescribed values does not match the number of missing directions
+                counter = wp.int32(0)
+                for l in range(wp.static(self.velocity_set.q)):
+                    if _missing_mask[l] == wp.uint8(1) and counter < _num_of_aux_data:
+                        f_1[_opp_indices[l], index[0], index[1], index[2]] = self.store_dtype(prescribed_values[counter])
+                        counter += 1
+
+        return aux_data_init_kernel
+
+    def aux_data_init(self, f_0, f_1, bc_mask, missing_mask):
+        if self.compute_backend == ComputeBackend.WARP:
+            # Launch the warp kernel
+            wp.launch(
+                self._construct_aux_data_init_kernel(self.profile),
+                inputs=[f_0, f_1, bc_mask, missing_mask],
+                dim=f_0.shape[1:],
+            )
+        elif self.compute_backend == ComputeBackend.JAX:
+            # We don't use boundary aux encoding/decoding in JAX
+            self.prescribed_values = self.profile()
+        self.is_initialized_with_aux_data = True
+        return f_0, f_1
diff --git a/xlb/operator/boundary_masker/indices_boundary_masker.py b/xlb/operator/boundary_masker/indices_boundary_masker.py
index 0c1f7e1..d165952 100644
--- a/xlb/operator/boundary_masker/indices_boundary_masker.py
+++ b/xlb/operator/boundary_masker/indices_boundary_masker.py
@@ -111,16 +111,15 @@ def kernel(
             is_interior: wp.array1d(dtype=wp.bool),
             bc_mask: wp.array4d(dtype=wp.uint8),
             missing_mask: wp.array4d(dtype=wp.bool),
-            start_index: wp.vec3i,
         ):
             # Get the index of indices
             ii = wp.tid()
 
             # Get local indices
             index = wp.vec3i()
-            index[0] = indices[0, ii] - start_index[0]
-            index[1] = indices[1, ii] - start_index[1]
-            index[2] = indices[2, ii] - start_index[2]
+            index[0] = indices[0, ii]
+            index[1] = indices[1, ii]
+            index[2] = indices[2, ii]
 
             # Check if index is in bounds
             shape = wp.vec3i(missing_mask.shape[1], missing_mask.shape[2], missing_mask.shape[3])
@@ -198,11 +197,6 @@ def warp_implementation(self, bclist, bc_mask, missing_mask, start_index=None):
         wp_id_numbers = wp.array(id_numbers, dtype=wp.uint8)
         wp_is_interior = wp.array(is_interior, dtype=wp.bool)
 
-        if start_index is None:
-            start_index = wp.vec3i(0, 0, 0)
-        else:
-            start_index = wp.vec3i(*start_index)
-
         # Launch the warp kernel
         wp.launch(
             self.warp_kernel,
@@ -213,7 +207,6 @@ def warp_implementation(self, bclist, bc_mask, missing_mask, start_index=None):
                 wp_is_interior,
                 bc_mask,
                 missing_mask,
-                start_index,
             ],
         )
 
diff --git a/xlb/operator/boundary_masker/mesh_boundary_masker.py b/xlb/operator/boundary_masker/mesh_boundary_masker.py
index edfdb42..9b8edc4 100644
--- a/xlb/operator/boundary_masker/mesh_boundary_masker.py
+++ b/xlb/operator/boundary_masker/mesh_boundary_masker.py
@@ -36,7 +36,6 @@ def jax_implementation(
         self,
         bc,
         origin,
-        spacing,
         id_number,
         bc_mask,
         missing_mask,
@@ -47,7 +46,7 @@ def jax_implementation(
         wp.init()
         bc_mask = wp.from_jax(bc_mask)
         missing_mask = wp.from_jax(missing_mask)
-        bc_mask, missing_mask = self.warp_implementation(bc, origin, spacing, bc_mask, missing_mask, start_index)
+        bc_mask, missing_mask = self.warp_implementation(bc, origin, bc_mask, missing_mask, start_index)
         return wp.to_jax(bc_mask), wp.to_jax(missing_mask)
 
     def _construct_warp(self):
@@ -59,49 +58,36 @@ def _construct_warp(self):
         @wp.kernel
         def kernel(
             mesh_id: wp.uint64,
-            origin: wp.vec3,
-            spacing: wp.vec3,
             id_number: wp.int32,
             bc_mask: wp.array4d(dtype=wp.uint8),
             missing_mask: wp.array4d(dtype=wp.bool),
-            start_index: wp.vec3i,
         ):
             # get index
             i, j, k = wp.tid()
 
             # Get local indices
-            index = wp.vec3i()
-            index[0] = i - start_index[0]
-            index[1] = j - start_index[1]
-            index[2] = k - start_index[2]
+            index = wp.vec3i(i, j, k)
 
             # position of the point
             ijk = wp.vec3(wp.float32(index[0]), wp.float32(index[1]), wp.float32(index[2]))
-            ijk = ijk + wp.vec3(0.5, 0.5, 0.5)  # cell center
-            pos = wp.cw_mul(ijk, spacing) + origin
-
+            pos = ijk + wp.vec3(0.5, 0.5, 0.5)  # cell center
             # Compute the maximum length
-            max_length = wp.sqrt(
-                (spacing[0] * wp.float32(bc_mask.shape[1])) ** 2.0
-                + (spacing[1] * wp.float32(bc_mask.shape[2])) ** 2.0
-                + (spacing[2] * wp.float32(bc_mask.shape[3])) ** 2.0
-            )
+            max_length = wp.sqrt(2.0) / 2.0  # half of unit cell diagonal
 
             # evaluate if point is inside mesh
-            query = wp.mesh_query_point_sign_winding_number(mesh_id, pos, max_length)
+            query = wp.mesh_query_point_no_sign(mesh_id, pos, max_length)
             if query.result:
                 # set point to be solid
-                if query.sign <= 0:  # TODO: fix this
-                    # Stream indices
-                    for l in range(1, _q):
-                        # Get the index of the streaming direction
-                        push_index = wp.vec3i()
-                        for d in range(self.velocity_set.d):
-                            push_index[d] = index[d] + _c[d, l]
-
-                        # Set the boundary id and missing_mask
-                        bc_mask[0, push_index[0], push_index[1], push_index[2]] = wp.uint8(id_number)
-                        missing_mask[l, push_index[0], push_index[1], push_index[2]] = True
+                # Stream indices
+                for l in range(1, _q):
+                    # Get the index of the streaming direction
+                    push_index = wp.vec3i()
+                    for d in range(self.velocity_set.d):
+                        push_index[d] = index[d] + _c[d, l]
+
+                    # Set the boundary id and missing_mask
+                    bc_mask[0, push_index[0], push_index[1], push_index[2]] = wp.uint8(id_number)
+                    missing_mask[l, push_index[0], push_index[1], push_index[2]] = True
 
         return None, kernel
 
@@ -109,20 +95,27 @@ def kernel(
     def warp_implementation(
         self,
         bc,
-        origin,
-        spacing,
         bc_mask,
         missing_mask,
-        start_index=(0, 0, 0),
     ):
         assert bc.mesh_vertices is not None, f'Please provide the mesh vertices for {bc.__class__.__name__} BC using keyword "mesh_vertices"!'
         assert bc.indices is None, f"Please use IndicesBoundaryMasker operator if {bc.__class__.__name__} is imposed on known indices of the grid!"
-        assert (
-            bc.mesh_vertices.shape[1] == self.velocity_set.d
-        ), "Mesh points must be reshaped into an array (N, 3) where N indicates number of points!"
+        assert bc.mesh_vertices.shape[1] == self.velocity_set.d, (
+            "Mesh points must be reshaped into an array (N, 3) where N indicates number of points!"
+        )
         mesh_vertices = bc.mesh_vertices
         id_number = bc.id
 
+        # Check mesh extents against domain dimensions
+        domain_shape = bc_mask.shape[1:]  # (nx, ny, nz)
+        mesh_min = np.min(mesh_vertices, axis=0)
+        mesh_max = np.max(mesh_vertices, axis=0)
+
+        if any(mesh_min < 0) or any(mesh_max >= domain_shape):
+            raise ValueError(
+                f"Mesh extents ({mesh_min}, {mesh_max}) exceed domain dimensions {domain_shape}. The mesh must be fully contained within the domain."
+            )
+
         # We are done with bc.mesh_vertices. Remove them from BC objects
         bc.__dict__.pop("mesh_vertices", None)
 
@@ -140,12 +133,9 @@ def warp_implementation(
             self.warp_kernel,
             inputs=[
                 mesh.id,
-                origin,
-                spacing,
                 id_number,
                 bc_mask,
                 missing_mask,
-                start_index,
             ],
             dim=bc_mask.shape[1:],
         )
diff --git a/xlb/operator/stepper/nse_stepper.py b/xlb/operator/stepper/nse_stepper.py
index e08e95c..fad6644 100644
--- a/xlb/operator/stepper/nse_stepper.py
+++ b/xlb/operator/stepper/nse_stepper.py
@@ -7,6 +7,7 @@
 
 from xlb import DefaultConfig
 from xlb.compute_backend import ComputeBackend
+from xlb.precision_policy import Precision
 from xlb.operator import Operator
 from xlb.operator.stream import Stream
 from xlb.operator.collision import BGK, KBC
@@ -16,31 +17,112 @@
 from xlb.operator.boundary_condition.boundary_condition import ImplementationStep
 from xlb.operator.boundary_condition.boundary_condition_registry import boundary_condition_registry
 from xlb.operator.collision import ForcedCollision
+from xlb.operator.boundary_masker import IndicesBoundaryMasker, MeshBoundaryMasker
+from xlb.helper import check_bc_overlaps
 
 
 class IncompressibleNavierStokesStepper(Stepper):
-    def __init__(self, omega, boundary_conditions=[], collision_type="BGK", forcing_scheme="exact_difference", force_vector=None):
-        velocity_set = DefaultConfig.velocity_set
-        precision_policy = DefaultConfig.default_precision_policy
-        compute_backend = DefaultConfig.default_backend
+    def __init__(
+        self,
+        omega,
+        grid,
+        boundary_conditions=[],
+        collision_type="BGK",
+        forcing_scheme="exact_difference",
+        force_vector=None,
+    ):
+        super().__init__(grid, boundary_conditions)
 
         # Construct the collision operator
         if collision_type == "BGK":
-            self.collision = BGK(omega, velocity_set, precision_policy, compute_backend)
+            self.collision = BGK(omega, self.velocity_set, self.precision_policy, self.compute_backend)
         elif collision_type == "KBC":
-            self.collision = KBC(omega, velocity_set, precision_policy, compute_backend)
+            self.collision = KBC(omega, self.velocity_set, self.precision_policy, self.compute_backend)
 
         if force_vector is not None:
             self.collision = ForcedCollision(collision_operator=self.collision, forcing_scheme=forcing_scheme, force_vector=force_vector)
 
         # Construct the operators
-        self.stream = Stream(velocity_set, precision_policy, compute_backend)
-        self.equilibrium = QuadraticEquilibrium(velocity_set, precision_policy, compute_backend)
-        self.macroscopic = Macroscopic(velocity_set, precision_policy, compute_backend)
-
-        operators = [self.macroscopic, self.equilibrium, self.collision, self.stream]
-
-        super().__init__(operators, boundary_conditions)
+        self.stream = Stream(self.velocity_set, self.precision_policy, self.compute_backend)
+        self.equilibrium = QuadraticEquilibrium(self.velocity_set, self.precision_policy, self.compute_backend)
+        self.macroscopic = Macroscopic(self.velocity_set, self.precision_policy, self.compute_backend)
+
+    def init_fields(self, initializer=None):
+        """Initialize the fields required for the stepper.
+
+        Args:
+            initializer: Optional operator to initialize the distribution functions.
+                        If provided, it should be a callable that takes (grid, velocity_set,
+                        precision_policy, compute_backend) as arguments and returns initialized f_0.
+                        If None, default equilibrium initialization is used with rho=1 and u=0.
+
+        Returns:
+            Tuple of (f_0, f_1, bc_mask, missing_mask):
+                - f_0: Initial distribution functions
+                - f_1: Copy of f_0 for double-buffering
+                - bc_mask: Boundary condition mask indicating which BC applies to each node
+                - missing_mask: Mask indicating which populations are missing at boundary nodes
+        """
+        # Initialize distribution functions
+        f_0 = self.grid.create_field(cardinality=self.velocity_set.q, dtype=self.precision_policy.store_precision)
+
+        if initializer is not None:
+            f_0 = initializer(self.grid, self.velocity_set, self.precision_policy, self.compute_backend)
+        else:
+            from xlb.helper.initializers import initialize_eq
+
+            f_0 = initialize_eq(f_0, self.grid, self.velocity_set, self.precision_policy, self.compute_backend)
+
+        # Create f_1 and copy f_0 using backend-specific copy
+        f_1 = f_0.copy() if self.compute_backend == ComputeBackend.JAX else wp.clone(f_0)
+
+        # Initialize masks with correct precision types
+        bc_mask = self.grid.create_field(cardinality=1, dtype=Precision.UINT8)
+        missing_mask = self.grid.create_field(cardinality=self.velocity_set.q, dtype=Precision.BOOL)
+
+        # Process boundary conditions and update masks
+        bc_mask, missing_mask = self._process_boundary_conditions(self.boundary_conditions, bc_mask, missing_mask)
+        # Initialize auxiliary data if needed
+        f_0, f_1 = self._initialize_auxiliary_data(self.boundary_conditions, f_0, f_1, bc_mask, missing_mask)
+
+        return f_0, f_1, bc_mask, missing_mask
+
+    @classmethod
+    def _process_boundary_conditions(cls, boundary_conditions, bc_mask, missing_mask):
+        """Process boundary conditions and update boundary masks."""
+        # Check for boundary condition overlaps
+        check_bc_overlaps(boundary_conditions, DefaultConfig.velocity_set.d, DefaultConfig.default_backend)
+        # Create boundary maskers
+        indices_masker = IndicesBoundaryMasker(
+            velocity_set=DefaultConfig.velocity_set,
+            precision_policy=DefaultConfig.default_precision_policy,
+            compute_backend=DefaultConfig.default_backend,
+        )
+        # Split boundary conditions by type
+        bc_with_vertices = [bc for bc in boundary_conditions if bc.mesh_vertices is not None]
+        bc_with_indices = [bc for bc in boundary_conditions if bc.indices is not None]
+        # Process indices-based boundary conditions
+        if bc_with_indices:
+            bc_mask, missing_mask = indices_masker(bc_with_indices, bc_mask, missing_mask)
+        # Process mesh-based boundary conditions for 3D
+        if DefaultConfig.velocity_set.d == 3 and bc_with_vertices:
+            mesh_masker = MeshBoundaryMasker(
+                velocity_set=DefaultConfig.velocity_set,
+                precision_policy=DefaultConfig.default_precision_policy,
+                compute_backend=DefaultConfig.default_backend,
+            )
+            for bc in bc_with_vertices:
+                bc_mask, missing_mask = mesh_masker(bc, bc_mask, missing_mask)
+
+        return bc_mask, missing_mask
+
+    @staticmethod
+    def _initialize_auxiliary_data(boundary_conditions, f_0, f_1, bc_mask, missing_mask):
+        """Initialize auxiliary data for boundary conditions that require it."""
+        for bc in boundary_conditions:
+            if bc.needs_aux_init and not bc.is_initialized_with_aux_data:
+                f_0, f_1 = bc.aux_data_init(f_0, f_1, bc_mask, missing_mask)
+        return f_0, f_1
 
     @Operator.register_backend(ComputeBackend.JAX)
     @partial(jit, static_argnums=(0))
@@ -76,7 +158,7 @@ def jax_implementation(self, f_0, f_1, bc_mask, missing_mask, timestep):
 
         # Apply collision type boundary conditions
         for bc in self.boundary_conditions:
-            f_post_collision = bc.prepare_bc_auxilary_data(f_post_stream, f_post_collision, bc_mask, missing_mask)
+            f_post_collision = bc.update_bc_auxilary_data(f_post_stream, f_post_collision, bc_mask, missing_mask)
             if bc.implementation_step == ImplementationStep.COLLISION:
                 f_post_collision = bc(
                     f_post_stream,
@@ -108,6 +190,8 @@ def _construct_warp(self):
         # Group active boundary conditions
         active_bcs = set(boundary_condition_registry.id_to_bc[bc.id] for bc in self.boundary_conditions)
 
+        _opp_indices = self.velocity_set.opp_indices
+
         @wp.func
         def apply_bc(
             index: Any,
@@ -134,7 +218,7 @@ def apply_bc(
                             f_result = wp.static(self.boundary_conditions[i].warp_functional)(index, timestep, missing_mask, f_0, f_1, f_pre, f_post)
                     if wp.static(self.boundary_conditions[i].id in extrapolation_outflow_bc_ids):
                         if _boundary_id == wp.static(self.boundary_conditions[i].id):
-                            f_result = wp.static(self.boundary_conditions[i].prepare_bc_auxilary_data)(
+                            f_result = wp.static(self.boundary_conditions[i].update_bc_auxilary_data)(
                                 index, timestep, missing_mask, f_0, f_1, f_pre, f_post
                             )
             return f_result
@@ -161,6 +245,23 @@ def get_thread_data(
 
             return _f0_thread, _f1_thread, _missing_mask
 
+        @wp.func
+        def apply_aux_recovery_bc(
+            index: Any,
+            _boundary_id: Any,
+            _missing_mask: Any,
+            f_0: Any,
+            _f1_thread: Any,
+        ):
+            # Unroll the loop over boundary conditions
+            for i in range(wp.static(len(self.boundary_conditions))):
+                if wp.static(self.boundary_conditions[i].needs_aux_recovery):
+                    if _boundary_id == wp.static(self.boundary_conditions[i].id):
+                        # Perform the swapping of data
+                        for l in range(self.velocity_set.q):
+                            if _missing_mask[l] == wp.uint8(1):
+                                f_0[_opp_indices[l], index[0], index[1], index[2]] = self.store_dtype(_f1_thread[_opp_indices[l]])
+
         @wp.kernel
         def kernel(
             f_0: wp.array4d(dtype=Any),
@@ -192,13 +293,11 @@ def kernel(
             # Apply post-collision boundary conditions
             _f_post_collision = apply_bc(index, timestep, _boundary_id, _missing_mask, f_0, f_1, _f_post_stream, _f_post_collision, False)
 
+            # Apply auxiliary recovery for boundary conditions (swapping)
+            apply_aux_recovery_bc(index, _boundary_id, _missing_mask, f_0, _f1_thread)
+
             # Store the result in f_1
             for l in range(self.velocity_set.q):
-                # TODO: Improve this later
-                if wp.static("GradsApproximationBC" in active_bcs):
-                    if _boundary_id == wp.static(boundary_condition_registry.bc_to_id["GradsApproximationBC"]):
-                        if _missing_mask[l] == wp.uint8(1):
-                            f_0[_opp_indices[l], index[0], index[1], index[2]] = self.store_dtype(_f1_thread[_opp_indices[l]])
                 f_1[l, index[0], index[1], index[2]] = self.store_dtype(_f_post_collision[l])
 
         return None, kernel
diff --git a/xlb/operator/stepper/stepper.py b/xlb/operator/stepper/stepper.py
index 44aab5f..667c399 100644
--- a/xlb/operator/stepper/stepper.py
+++ b/xlb/operator/stepper/stepper.py
@@ -8,25 +8,28 @@ class Stepper(Operator):
     Class that handles the construction of lattice boltzmann stepping operator
     """
 
-    def __init__(self, operators, boundary_conditions):
-        # Get the boundary condition ids
-        from xlb.operator.boundary_condition.boundary_condition_registry import boundary_condition_registry
-
-        self.operators = operators
+    def __init__(self, grid, boundary_conditions):
+        self.grid = grid
         self.boundary_conditions = boundary_conditions
-
         # Get velocity set, precision policy, and compute backend
-        velocity_sets = set([op.velocity_set for op in self.operators if op is not None])
-        assert len(velocity_sets) < 2, "All velocity sets must be the same. Got {}".format(velocity_sets)
-        velocity_set = DefaultConfig.velocity_set if not velocity_sets else velocity_sets.pop()
-
-        precision_policies = set([op.precision_policy for op in self.operators if op is not None])
-        assert len(precision_policies) < 2, "All precision policies must be the same. Got {}".format(precision_policies)
-        precision_policy = DefaultConfig.default_precision_policy if not precision_policies else precision_policies.pop()
-
-        compute_backends = set([op.compute_backend for op in self.operators if op is not None])
-        assert len(compute_backends) < 2, "All compute backends must be the same. Got {}".format(compute_backends)
-        compute_backend = DefaultConfig.default_backend if not compute_backends else compute_backends.pop()
+        velocity_set = DefaultConfig.velocity_set
+        precision_policy = DefaultConfig.default_precision_policy
+        compute_backend = DefaultConfig.default_backend
 
         # Initialize operator
         super().__init__(velocity_set, precision_policy, compute_backend)
+
+    def init_fields(self, initializer=None, init_kernel=None):
+        """Initialize the fields required for the stepper.
+
+        Args:
+            initializer: Optional initializer operator to initialize the fields.
+                        If None, equilibrium initialization is used.
+            init_kernel: Optional custom kernel to initialize the populations.
+                        This kernel should accept (grid, f_0) as arguments and return the initialized f_0.
+                        If provided, this takes precedence over initializer.
+
+        Returns:
+            Tuple of (f_0, f_1, bc_mask, missing_mask)
+        """
+        raise NotImplementedError("Subclasses must implement init_fields()")