Multi-moment ADER-Taylor methods for systems of conservation laws with source terms in one dimension

Abstract

A new integration method combining the ADER time discretization with a multi-moment finite-volume framework is introduced. ADER runtime is reduced by performing only one Cauchy–Kowalewski (C–K) procedure per cell per time step and by using the Differential Transform Method for high-order derivatives. Three methods are implemented: (1) single-moment WENO (WENO), (2) two-moment Hermite WENO (HWENO), and (3) entirely local multi-moment (MM-Loc). MM-Loc evolves all moments, sharing the locality of Galerkin methods yet with a constant time step during p-refinement.Five 1-D experiments validate the methods: (1) linear advection, (2) Burger’s equation shock, (3) transient shallow-water (SW), (4) steady-state SW simulation, and (5) SW shock. WENO and HWENO methods showed expected polynomial h-refinement convergence and successfully limited oscillations for shock experiments. MM-Loc showed expected polynomial h-refinement and exponential p-refinement convergence for linear advection and showed sub-exponential (yet super-polynomial) convergence with p-refinement in the SW case.HWENO accuracy was generally equal to or better than a five-moment MM-Loc scheme. MM-Loc was less accurate than RKDG at lower refinements, but with greater h- and p-convergence, RKDG accuracy is eventually surpassed. The ADER time integrator of MM-Loc also proved more accurate with p-refinement at a CFL of unity than a semi-discrete RK analog of MM-Loc. Being faster in serial and requiring less frequent inter-node communication than Galerkin methods, the ADER-based MM-Loc and HWENO schemes can be spatially refined and have the same runtime, making them a competitive option for further investigation.