A parallel explicit/implicit time stepping scheme on block-adaptive grids

We present a parallel explicit/implicit time integration scheme well suited for block-adaptive grids. The basic idea of the algorithm is that the time stepping scheme can differ in the blocks of the grid for a given time step: an explicit scheme is used in the blocks where the local stability requirement is not violated and an implicit scheme is used in the blocks where the explicit scheme would be unstable. The implicit scheme is second order in time. The non-linear system of equations is linearized with Newton linearization. The linear system is solved with a preconditioned Krylov subspace iterative scheme. The Schwarz type preconditioning is also based on the block structure of the grid. We discuss load balancing for parallel execution and the optimal choice of the time step for speed and robustness. The parallel efficiency of the scheme is demonstrated for the equations of magnetohydrodynamics with a geophysics application in three dimensions. The control of the numerical divergence of the magnetic field in combination with the explicit/implicit time stepping scheme is also discussed.