Convergence of entropy-stable continuous summation-by-parts discretizations of symmetric hyperbolic conservation laws

TL;DR

This paper proves convergence of entropy-stable split-form discretizations for symmetric hyperbolic PDEs using a continuous summation-by-parts framework, clarifying their stability and accuracy for nonlinear problems.

Contribution

It provides the first convergence proof for entropy-stable split-form discretizations of nonlinear hyperbolic PDEs within the C-SBP framework.

Findings

Convergence is guaranteed for sufficiently small mesh sizes.

Error bounds remain finite and tend to zero as mesh is refined.

Results clarify the link between stability, consistency, and convergence.

Abstract

The Lax equivalence theorem guarantees convergence of stable and consistent discretizations for linear hyperbolic partial differential equations (PDEs). For nonlinear problems, however, stability and consistency alone do not generally guarantee convergence, even for smooth solutions, and existing convergence results typically rely either on projection-based error decompositions or on linearization arguments that do not directly extend to entropy-stable split-form discretizations. In particular, general convergence results for entropy-stable discretizations of hyperbolic PDEs are currently lacking, despite their widespread use. In this work, we prove convergence under smoothness assumptions on the exact solution and fluxes for entropy-stable split-form discretizations of scalar and symmetric hyperbolic systems with homogeneous flux functions within the continuous summation-by-parts (C-SBP) framework. The scalar inviscid Burgers equation is presented as a canonical example. The analysis is based on a stability-consistency argument that yields a nonlinear error evolution inequality whose solution provides an explicit upper bound on the numerical error. We show that, for sufficiently small mesh spacing, and for degree-$p$ C-SBP discretizations in $d$ spatial dimensions with $p>1+d/2$, this bound remains finite on any finite time interval and tends to zero as the mesh is refined, implying convergence despite the presence of local linear instabilities. The results help clarify the relationship between consistency, entropy stability, nonlinear error growth, and convergence for discretizations of nonlinear hyperbolic problems.