A fast, robust, and simple implicit method for adaptive time-stepping on adaptive mesh-refinement grids

摘要

Implicit solvers present strong limitations when used on supercomputingfacilities and in particular for adaptive mesh-refinement codes. We present anew method for implicit adaptive time-stepping on adaptive meshrefinement-grids. We implement it in the radiation hydrodynamics solver wedesigned for the RAMSES code for astrophysical purposes and, more particularly,for protostellar collapse. We briefly recall the radiation hydrodynamicsequations and the adaptive time-stepping methodology used for hydrodynamicalsolvers. We then introduce the different types of boundary conditions(Dirichlet, Neumann, and Robin) that are used at the interface between levelsand present our implementation of the new method in the RAMSES code. The methodis tested against classical diffusion and radiation hydrodynamics tests, afterwhich we present an application for protostellar collapse. We show that usingDirichlet boundary conditions at level interfaces is a good compromise betweenrobustness and accuracy and that it can be used in structure formationcalculations. The gain in computational time over our former unique time stepmethod ranges from factors of 5 to 50 depending on the level of adaptivetime-stepping and on the problem. We successfully compare the old and newmethods for protostellar collapse calculations that involve highly nonlinearphysics. We have developed a simple but robust method for adaptivetime-stepping of implicit scheme on adaptive mesh-refinement grids. It can beapplied to a wide variety of physical problems that involve diffusionprocesses.