Reformulations for general advectiondiffusionreaction equations and locally implicit ADER schemes
Following Cattaneo's original idea, in this article we first present two relaxation formulations for time-dependent, non-linear systems of advectiondiffusionreaction equations. Such formulations yield time-dependent non-linear hyperbolic balance laws with stiff source terms. Then we present a locally implicit version of the ADER method to solve these stiff systems to high accuracy. The new ingredient of the numerical methodology is a locally implicit solution of the generalised Riemann problem. We illustrate the formulations and the resulting numerical approach by solving the compressible NavierStokes equations.
advectiondiffusionreaction equations generalised riemann problems relaxation methods ader high-order schemes cauchykowalewski procedure