Arbitrary high order accurate time integration schemes for linear problems

Numerical schemes for wave propagation over long distances need good wave propagation properties with low dispersion and low dissipation errors. Suitable numerical methods are methods with high order of accuracy in space and time. For space discretization on structured grids, high order finite difference schemes are efficient, and, if a complicated computational domain requires an unstructured grid, discontinuous Galerkin methods are recently employed with success. The time integration is often performed by a Runge-Kutta scheme. These schemes need for the order of accuracy O > 4 more than O stages, which reduces performance concerning CPU-time as well as storage requirements, because the numerical solution of more than one stage has to be stored. However, it is interesting to use schemes of accuracy order higher than 4, especially to capture wave propagation over long distances or if very accurate computations are needed. In this paper we consider a time integration approach for linear wave problems based on a Taylor expansion. Here we construct and analyze schemes of arbitrary high order accuracy in space and time using this time integration technique within the finite difference as well as the discontinuous Galerkin framework. We present a stability analysis as well as performance comparisons with schemes relying on other time integration methods. A modification for the DG schemes is presented that accentuates the computational performance. Numerical experiments are realized for the system of linearized Euler Equations, but the formulation allows an application of the proposed schemes to any linear hyperbolic system.