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 one Cauchy-Kowalewski procedure per time step. 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 CFL during p -refinement. Four 1-D experiments validate the methods: (1) linear advection, (2) Burger's equation, (3) transient shallow-water (SW) , and (4) steady-state SW simulation. WENO and HWENO methods showed expected polynomial h -refinement convergence and successfully limited oscillations for Burger's equation's shock. 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. MM-Loc is generally less accurate than a selected Runge-Kutta Discontinuous Galerkin (RKDG) method yet with better h -refinement convergence. MM-Loc, being faster and requiring less frequent communication than RKDG, can be spatially refined and have the same RKDG runtime. Therefore, MM-Loc is a competitive option for further investigation.