Local time steps for a finite volume scheme
For computational scientists solving time-dependent PDEs on adaptive grids, this work offers a conservative and efficient approach to local time stepping, though it is an incremental improvement over existing domain decomposition and multigrid methods.
The paper proposes a strategy for using different time steps in different spatial regions for time-dependent problems on locally refined grids, presenting two conservative finite volume approximations and an iterative solver that reduces to standard time stepping on coarse and fine grids, with numerical results demonstrating accuracy.
We present a strategy for solving time-dependent problems on grids with local refinements in time using different time steps in different regions of space. We discuss and analyze two conservative approximations based on finite volume with piecewise constant projections and domain decomposition techniques. Next we present an iterative method for solving the composite-grid system that reduces to solution of standard problems with standard time stepping on the coarse and fine grids. At every step of the algorithm, conservativity is ensured. Finally, numerical results illustrate the accuracy of the proposed methods.