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

¹ École Normale Supérieure de Lyon, CRAL, UMR CNRS 5574, Université Lyon I, 46 Allée d’Italie, 69364 Lyon Cedex 07, France
e-mail: benoit.commercon@ens-lyon.fr
² Laboratoire de radioastronomie (UMR 8112 CNRS), École Normale Supérieure et Observatoire de Paris, 24 rue Lhomond, 75231 Paris Cedex 05, France
³ LESIA−Observatoire de Paris, CNRS, UPMC, Université Paris-Diderot, 5 place Jules Janssen, 92195 Meudon, France
⁴ Universität Zürich, Institut für Theoretische Physik, Winterthurerstrasse 190, 8057 Zürich, Switzerland

Received: 16 October 2013
Accepted: 17 December 2013

Abstract

Context. Implicit solvers present strong limitations when used on supercomputing facilities and in particular for adaptive mesh-refinement codes.

Aims. We present a new method for implicit adaptive time-stepping on adaptive mesh-refinement grids. We implement it in the radiation-hydrodynamics solver we designed for the RAMSES code for astrophysical purposes and, more particularly, for protostellar collapse.

Methods. We briefly recall the radiation-hydrodynamics equations and the adaptive time-stepping methodology used for hydrodynamical solvers. We then introduce the different types of boundary conditions (Dirichlet, Neumann, and Robin) that are used at the interface between levels and present our implementation of the new method in the RAMSES code. The method is tested against classical diffusion and radiation-hydrodynamics tests, after which we present an application for protostellar collapse.

Results. We show that using Dirichlet boundary conditions at level interfaces is a good compromise between robustness and accuracy and that it can be used in structure formation calculations. The gain in computational time over our former unique time step method ranges from factors of 5 to 50 depending on the level of adaptive time-stepping and on the problem. We successfully compare the old and new methods for protostellar collapse calculations that involve highly non linear physics.

Conclusions. We have developed a simple but robust method for adaptive time-stepping of implicit scheme on adaptive mesh-refinement grids. It can be applied to a wide variety of physical problems that involve diffusion processes.