Across-the-space parallelism still remains the most mature, convenient and natural way to parallelize large scale problems. One of the major problems here is that implicit time stepping is often difficult to parallelize due to the structure of the system. Approximate implicit schemes have been suggested to circumvent the problem [5]. These schemes have attractive stability properties and they a...

This paper constructs strong-stability-preserving general linear time-stepping methods that are well suited for hyperbolic PDEs discretized by the method of lines. These methods generalize both Runge–Kutta (RK) and linear multistep schemes. They have high stage orders and hence are less susceptible than RK methods to order reduction from source terms or nonhomogeneous boundary conditions. A glo...

Due to the catastrophic effects of tsunamis, tsunami simulations are desired to increase the understanding of the phenomena, as well as to serve as mitigation and early warning systems for the affected areas. Even with the advent of supercomputers, the latter one is far out of reach due to the contradicting need for accuracy and speed. A common solution to this problem are adaptively refined gr...

The application of nonlinear schemes like dual time stepping as preconditioners in matrix-free Newton-Krylov-solvers is considered and analyzed. We provide a novel formulation of the left preconditioned operator that says it is in fact linear in the matrix-free sense, but changes the Newton scheme. This allows to get some insight in the convergence properties of these schemes which are demonstr...

We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order 0 < β < 1. From the known structure of the nonsmooth solution and by introducing corresponding correction terms, we can obtain uniformly second-order accuracy from these schemes. We prove the convergence and linear stability of the proposed schemes. Numeri...

An explicit one-step time discretization for discontinuous Galerkin schemes applied to advection-diffusion equations is presented. The main idea is based on a predictor corrector approach which was proposed in [1]. The interesting feature is that the predictor is local and takes only into account the time evolution of the data within each grid cell. In this paper, we focus ourselves to a contin...

This paper constructs strong-stability-preserving general linear time-stepping methods that are well suited for hyperbolic PDEs discretized by the method of lines. These methods generalize both Runge-Kutta (RK) and linear multistep schemes. They have high stage orders and hence are less susceptible than RK methods to order reduction from source terms or nonhomogeneous boundary conditions. A glo...

This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convectiondiffusion problems and the respective transient problems with implicit time integration. Monotonic schemes that combine explicit time stepping with dG space discretization are very common, but the design of such ...