Add isotropic thermal conduction and RKL2 supertimestepping (#1)

* Add simple anisotropic step function test

* Separate FillDerived and EstimateTimestep in driver in prep for STS list

* Add diffflux parameter

* Add RKL2 STS task list

* Add calc of RKL2 stages

* Remove unncessary register for rkl2

* Adopt STS RKL2 variable naming

* Move calc of dt_diff into PreStep

* Make tlim an argument for diff step test

* Adjust RKL2 conv test to gaussian profile

* Add conv panel to conv plot

* auto-format

* rename diffusion integrator parameter

* Add isotropic thermal conduction

* Add isotropic cond to conv test

* Add RKL2 conv test

* Add new dt max ratio for rkl2 param

* Add prolongation and fluxcorrect to RKL2 task list

* Use base container as active STS container (workaround some AMR bug for using prolong/restric with non-base containers)

* Add isotropic Spitzer thermal conduction timestep

* Calc isotropic, non-const thermal diff

* Fix calc of saturated heat flux

* Add LimO3 limiter

* Add limo3 convergence

* Fix LimO3 recon

* Fix saturated conduction prefactor

* Remove calc of saturated conduction from cond coeff

* Add upwinded saturated conduction in x-dir

* Add saturated conduction prefactor

* Add x2 and x3 sat cond fluxes

* Increase default rkl2 ratio to 400 and allow flux correction for all integrators

* Remove parabolic timestep constraint for saturated conduction limit regime

* Add perturb to cloud pgen

* Add perturb to B (knowing this is not great...)

* Revert "Add perturb to B (knowing this is not great...)"

This reverts commit 1d0cb19.

* Revert "Add perturb to cloud pgen"

This reverts commit 6df018f.

* Limit cooling to upper bound of TFloor and cooling table cutoff

* Add oblique B field

* Update coords and driver

* Fix test cases and add success check

* Add changelog

* Address comments

.github/workflows/ci.yml

@@ -53,6 +53,7 @@ jobs:
             build/tst/regression/outputs/cluster_hse/analytic_comparison.png
             build/tst/regression/outputs/cluster_tabular_cooling/convergence.png
             build/tst/regression/outputs/aniso_therm_cond_ring_conv/ring_convergence.png
+            build/tst/regression/outputs/aniso_therm_cond_gauss_conv/cond.png
             build/tst/regression/outputs/field_loop/field_loop.png
             build/tst/regression/outputs/riemann_hydro/shock_tube.png
             build/tst/regression/outputs/turbulence/parthenon.hst

CHANGELOG.md

@@ -3,6 +3,7 @@
 ## Current develop (i.e., `main` branch)
 
 ### Added (new features/APIs/variables/...)
+- [[PR 1]](https://github.com/parthenon-hpc-lab/athenapk/pull/1) Add isotropic thermal conduction and RKL2 supertimestepping
 
 ### Changed (changing behavior/API/variables/...)
 - [[PR 84]](https://github.com/parthenon-hpc-lab/athenapk/pull/84) Bump Parthenon to latest develop (2024-02-15)

README.md

@@ -14,7 +14,8 @@ Current features include
   - HLLE (hydro and MHD), HLLC (hydro), and HLLD (MHD) Riemann solvers
   - adiabatic equation of state
   - MHD based on hyperbolic divergence cleaning following Dedner+ 2002
-  - anisotropic thermal conduction
+  - isotropic and anisotropic thermal conduction
+  - operator-split, second-order RKL2 supertimestepping for diffusive terms
   - optically thin cooling based on tabulated cooling tables with either Townsend 2009 exact integration or operator-split subcycling
 - static and adaptive mesh refinement
 - problem generators for

docs/input.md

@@ -69,15 +69,22 @@ conserved to primitive conversion if both are defined.
 
 #### Diffusive processes
 
-##### Anisotropic thermal conduction (required MHD)
+##### Isotropic (hydro and MHD) and anisotropic thermal conduction (only MHD)
 In the presence of magnetic fields thermal conduction is becoming anisotropic with the flux along
 the local magnetic field direction typically being much stronger than the flux perpendicular to the magnetic field.
 
 From a theoretical point of view, thermal conduction is included in the system of MHD equations by an additional
 term in the total energy equation:
 ```math
-\delta_t E + \nabla \cdot (... + \mathbf{F}) \quad \mathrm{with}\\
-\mathbf{F} = - \kappa \mathbf{\hat b} (\mathbf{\hat b \cdot \nabla T})
+\delta_t E + \nabla \cdot (... + \mathbf{F}_\mathrm{c})
+```
+where the full thermal conduction flux $`\mathbf{F}_\mathrm{c}`$ contains both the classic thermal conduction
+```math
+\mathbf{F}_\mathrm{classic} = - \kappa \mathbf{\hat b} (\mathbf{\hat b \cdot \nabla T})
+```
+as well as the saturated flux (as introduced by [^CM77])
+```math
+\mathbf{F}_\mathrm{sat} = - 5 \phi \rho^{-1/2} p^{3/2} \mathrm{sgn}(\mathbf{\hat b \cdot \nabla T}) \mathbf{\hat b}
 ```
 
 From an implementation point of view, two options implemented and can be configured within a `<diffusion>` block in the input file.
@@ -86,23 +93,52 @@ the integration step (before flux correction in case of AMR, and calculating the
 Moreover, they are implemented explicitly, i.e., they add a (potentially very restrictive) constraint to the timestep due to the scaling with $`\propto \Delta_x^2`$.
 Finally, we employ limiters for calculating the temperature gradients following Sharma & Hammett (2007)[^SH07].
 This prevents unphysical conduction against the gradient, which may be introduced because the off-axis gradients are not centered on the interfaces.
+Similarly, to account for the different nature of classic and saturated fluxes (parabolic and hyperbolic, respectively),
+we follow [^M+12] and use a smooth transition
+```math
+\mathbf{F}_\mathrm{c} = \frac{q}{q + F_\mathrm{classic}} \mathbf{F}_\mathrm{classic} \quad \mathrm{with} \quad q = 5 \phi \rho^{-1/2} p^{3/2}
+```
+and upwinding of the hyperbolic, saturated fluxes.
 
-To enable conduction, set
+To enable thermal conduction, set
 
 Parameter: `conduction` (string)
 - `none` : No thermal conduction
+- `isotropic` : Isotropic thermal conduction
+- `anisotropic` : Anisotropic thermal conduction
+
+In addition the coefficient (or diffusivity) needs to be set
+
+Parameter: `conduction_coeff` (string)
 - `spitzer` : Anisotropic thermal conduction with a temperature dependent classic Spitzer thermal conductivity
   $`\kappa (T) = c_\kappa T^{5/2} \mathrm{erg/s/K/cm}`$ and
-  $`c_\kappa`$ being constant prefactor (set via `diffusion/spitzer_cond_in_erg_by_s_K_cm` with a default value of $`4.6\times10^{-7}`$). Note, as indicated by the units in the input parameter name, this kind of thermal conductivity requires a full set of units
+  $`c_\kappa`$ being constant prefactor (set via the additional `diffusion/spitzer_cond_in_erg_by_s_K_cm` parameter with a default value of $`4.6\times10^{-7}`$ which assumes a fully ionized hydrogen plasma [^CM77] with $`\ln \lambda = 40`$ approximating ICM conditions). Note, as indicated by the units in the input parameter name, this kind of thermal conductivity requires a full set of units
   to be defined for the simulation.
-- `thermal_diff` : Contrary to a temperature dependent conductivity, a simple thermal diffusivity can be used instead for which
+- `fixed` : Contrary to a temperature dependent conductivity, a simple thermal diffusivity can be used instead for which
 the conduction flux is $`\mathbf{F} = - \chi \rho \mathbf{\hat b} (\mathbf{\hat b \cdot \nabla \frac{p_\mathrm{th}}{\rho}})`$
-Here, the strength, $`\chi`$, is controlled via the `thermal_diff_coeff_code` parameter in code units.
+Here, the strength, $`\chi`$, is controlled via the additional `thermal_diff_coeff_code` parameter in code units.
 Given the dimensions of $`L^2/T`$ it is referred to a thermal diffusivity rather than thermal conductivity.
 
+Parameter: `conduction_sat_phi` (float)
+- Default value 0.3\
+Factor to account for the uncertainty in the estimated of saturated fluxes, see [^CM77].
+Default value corresponds to the typical value used in literature and goes back to [^MMM80] and [^BM82].
+
+
 [^SH07]:
     P. Sharma and G. W. Hammett, "Preserving monotonicity in anisotropic diffusion," Journal of Computational Physics, vol. 227, no. 1, Art. no. 1, 2007, doi: https://doi.org/10.1016/j.jcp.2007.07.026.
 
+[^M+12]:
+    A. Mignone, C. Zanni, P. Tzeferacos, B. van Straalen, P. Colella, and G. Bodo, “THE PLUTO CODE FOR ADAPTIVE MESH COMPUTATIONS IN ASTROPHYSICAL FLUID DYNAMICS,” The Astrophysical Journal Supplement Series, vol. 198, Art. no. 1, Dec. 2011, doi: https://doi.org/10.1088/0067-0049/198/1/7
+
+[^CM77]:
+    L. Cowie and C. F. McKee, “The evaporation of spherical clouds in a hot gas. I. Classical and saturated mass loss rates.,” , vol. 211, pp. 135–146, Jan. 1977, doi: https://doi.org/10.1086/154911
+
+[^MMM80]:
+    C. E. Max, C. F. McKee, and W. C. Mead, “A model for laser driven ablative implosions,” The Physics of Fluids, vol. 23, Art. no. 8, 1980, doi: https://doi.org/10.1063/1.863183
+
+[^BM82]:
+    S. A. Balbus and C. F. McKee, “The evaporation of spherical clouds in a hot gas. III - Suprathermal evaporation,” , vol. 252, pp. 529–552, Jan. 1982, doi: https://doi.org/10.1086/159581
 
 ### Additional MHD options in `<hydro>` block
 

inputs/diffusion.in

@@ -14,7 +14,8 @@ Bx = 1.0     # Bx for x1 step function (permutated for iprobs in other direction
 By = 0.0     # By for x1 step function (permutated for iprobs in other directions)
 
 #iprob = 10   # Diffusion of Gaussian profile in x1 direction
-sigma = 0.1  # standard deviation of Gaussian for iprob=10
+t0 = 0.5      # Temporal offset for initial Gaussian profile
+amp = 1e-6    # Amplitude of Gaussian profile
 
 iprob = 20   # ring diffusion in x1-x2 plane; 21 for x2-x3 plane; 22 for x3-x1 plane
 
@@ -59,8 +60,11 @@ reconstruction = dc
 gamma = 2.0
 
 <diffusion>
-conduction = thermal_diff
+integrator = unsplit
+conduction = anisotropic
+conduction_coeff = fixed
 thermal_diff_coeff_code = 0.01
+rkl2_max_dt_ratio = 400.0
 
 <parthenon/output0>
 file_type = hdf5

src/CMakeLists.txt

@@ -7,6 +7,7 @@ add_executable(
         eos/adiabatic_glmmhd.cpp
         units.hpp
         eos/adiabatic_hydro.cpp
+        hydro/diffusion/diffusion.cpp
         hydro/diffusion/diffusion.hpp
         hydro/diffusion/conduction.cpp
         hydro/hydro_driver.cpp