Adaptively coupled multiphysics simulations with Trixi.jl

We extended the capabilities of the numerical simulation framework Trixi.jl to be able to simulate adaptively coupled multiphysics systems. Coupling is performed through the boundary values of the systems where the coupling functions can be freely defined, depending on the physical nature of the interface. This allows us to couple any pair of systems, like Navier-Stokes equations with magnetohydrodynamic equations. This is particularly useful for hierarchical systems found in e.g. Astrophysics where we can have a complex model for a small part of the domain and a simplified model on a larger part. This can greatly reduce the computational cost and decrease the computational time. To account for dynamic changes in the physics that need to be solved at any given point in space, we support adaptively coupled domains. The criteria for changing the domain boundaries can be freely defined and tailored to the problem. One application is the propagation of magnetic fields in space where we solve the magnetohydrodynamic equations only for the part of the domain with a significant magnetic field.

1. Adaptively Coupled Multiphysics
Simulations with Trixi.jl
Simon Candelaresi, Michael Schlottke-Lakemper

2. Luca Rüedi
(zuonline.ch)

3. Magnetotail
3
(ESA)
bow shock
magnetic streamlines
(movie)

4. Modeling
4
(ESA)
kinetic model
(expensive)
magnetohydrodynamics (MHD)
(cheap)
interface coupling

5. Coupled Systems
5
MHD:
heat diffusion:

6. Coupled Systems
6
Navier-Stokes
Navier-Stokes
heat diffusionheat diffusion
(Ioannis Karathanasis)

7. Trixi.jl
7
open source:
github.com/trixi-framework/Trixi.jl
Discontinuous Galerkin
(Stack Overflow)
AMR
OrdinaryDiffEq (SciML)
callbacks
using Trixi
equations = LinearScalarAdvectionEquation2D((0.5, -0.3)
solver = DGSEM(polydeg = 3)
cells_per_dimension = (16, 16)
stepsize_callback = StepsizeCallback(cfl = 1.6)
mesh = StructuredMesh(cells_per_dimension, (-1, -1), (1, 1))
semi = SemidiscretizationHyperbolic(mesh, equations,
initial_condition, solver)
ode = semidiscretize(semi, (0.0, 1.0));
callbacks = CallbackSet(stepsize_callback)
sol = solve(ode, CarpenterKennedy2N54(),
dt = 1.0, callback = callbacks);

8. Coupling via Converter Functions
8
Two system with any number of shared variables, including 0:
User can define converter functions.
Any pair of systems can be coupled.
coupling_function12 = (x, u, equations_other, equations_own)
-> SVector(u[1], u[2], u[3], 0.0, u[4], 0.0, 0.0, 0.0, 0.0)
coupling_function21 = (x, u, equations_other, equations_own) -> SVector(u[1], u[2], u[3], u[5])
2.5d MHD 2d Euler

9. Work Flow Coupling
9
equations
initial condition
solver
parent mesh
coupling functions
boundaries
mesh view
semidiscretize (ode)
semidiscretize coupled (ode)
OrdinaryDiffEq
equations
initial condition
solver
coupling functions
boundaries
mesh view
semidiscretize (ode)

10. Isothermal-Polytropic
10
coupling functions:
“periodic coupling”

11. Isothermal-Polytropic
11

12. Adaptive Coupling
12
strong currents and magnetic fields
Euler
Euler MHD
MHD
Use callback functions to remesh.
Use coupling functions to copy data.

13. Adaptive Coupling
13
1. Generate the new grid.
2. Write new u-solution vectors
3. Generate new ODE for OrdinaryDiffEq (integrator).
4. Reinitialize ODE integrator with new problem and new solution vector.

14. Euler and MHD
14

15. Conclusion
Flexible coupling through converter functions.
Free domain definitions.
Adaptive coupling with arbitrary criteria.
[email protected]
Coupled hierarchy of models.