Compact Approximate Taylor Methods for Systems of Conservation Laws

Abstract

A new family of high order methods for systems of conservation laws is introduced: the Compact Approximate Taylor (CAT) methods. As in the approximate Taylor methods proposed by Zorío et al. (J Sci Comput 71(1):246–273, 2017) the Cauchy–Kovalevskaya procedure is circumvented by using Taylor approximations in time that are computed in a recursive way. The difference is that here this strategy is applied locally to compute the numerical fluxes what leads to methods that have $(2 p +1)$-point stencil and order of accuracy 2p, where p is an arbitrary integer. Moreover we prove that they reduce to the high-order Lax–Wendroff methods for linear problems and hence they are linearly $L^2$-stable under a $CFL-1$ condition. In order to prevent the spurious oscillations that appear close to discontinuities two shock-capturing techniques have been considered: a flux-limiter technique and WENO reconstruction for the first time derivative (WENO-CAT methods). We follow [25] in the second approach. A number of test cases are considered to compare these methods with other WENO-based schemes: the linear transport equation, Burgers equation, the 1D compressible Euler equations, and the ideal Magnetohydrodynamics equations are considered. Although CAT methods present an extra computational cost due to the local character, this extra cost is compensated by the fact that they still give good solutions with CFL values close to 1.