Equivalence of ADER and Lax-Wendroff in DG / FR framework for linear problems

TL;DR

This paper addresses whether ADER and Lax-Wendroff schemes are equivalent within a DG/Flux Reconstruction framework for linear problems. It recasts the ADER-DG corrector as a FR-based method and demonstrates that, for linear flux $f(u)=a u$, the ADER-FR time-averaged flux matches the LW-FR flux with D2 dissipation, yielding identical Fourier stability limits. The key contribution is an analytical equivalence proof supplemented by numerical validation on linear advection, where ADER-FR and LW-FR with D2 align up to machine precision while D1 dissipation introduces measurable discrepancies. This work unifies two high-order single-stage approaches, informing stability analyses and enabling cross-method insights for efficient, high-order time stepping in DG/FR schemes.

Abstract

ADER (Arbitrary high order by DERivatives) and Lax-Wendroff (LW) schemes are two high order single stage methods for solving time dependent partial differential equations. ADER is based on solving a locally implicit equation to obtain a space-time predictor solution while LW is based on an explicit Taylor's expansion in time. We cast the corrector step of ADER Discontinuous Galerkin (DG) scheme into an equivalent quadrature free Flux Reconstruction (FR) framework and then show that the obtained ADER-FR scheme is equivalent to the LWFR scheme with D2 dissipation numerical flux for linear problems. This also implies that the two schemes have the same Fourier stability limit for time step size. The equivalence is verified by numerical experiments.

Figures (1)

• Figure 1: $L^2$ error $\| u_h - u_{\operatorname{exact}} \|_2$ versus time for wave packet test for different polynomial degrees with 240 degrees of freedom. Periodic : (a) $N = 1$, (b) $N = 2$, (c) $N = 3$. Non-periodic : (a) $N=1$, (b) $N=2$, (c) $N=3$

Theorems & Definitions (2)

• Remark 1
• Remark 2