Reformulations for general advection–diffusion–reaction 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 advection–diffusion–reaction 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 Navier–Stokes equations.