Entropy-stable, high-order summation-by-parts discretizations without interface penalties

Abstract

The paper presents high-order accurate, energy-, and entropy-stable discretizations constructed from summation-by-parts (SBP) operators. Notably, the discretizations assemble global SBP operators and use continuous solutions, unlike previous efforts that use discontinuous SBP discretizations. Derivative-based dissipation and local-projection stabilization (LPS) are investigated as options for stabilizing the baseline discretization. These stabilizations are equal up to a multiplicative constant in one dimension, but only LPS remains well conditioned for general, multidimensional SBP operators. Furthermore, LPS is able to take advantage of the additional nodes required by degree $2p$ diagonal-norms, resulting in an element-local stabilization with a bounded spectral radius. An entropy-stable version of LPS is easily obtained by applying the projection on the entropy variables. Numerical experiments with the linear-advection and Euler equations demonstrate the accuracy, efficiency, and robustness of the stabilized discretizations, and the continuous approach compares favorably with the more common discontinuous SBP methods.

To appear in the Journal of Scientific Computing

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• On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws
• Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts
• On the Rate of Error Growth in Time for Numerical Solutions of Nonlinear Dispersive Wave Equations
• Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations
• Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions