Mark H. Carpenter

TL;DR: In this article, the Lax-Wendroff theorem states that conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal-norm skew-symmetric summation-by-parts spatial operator yield discrete operators that are conservative.

TL;DR: A systematic approach is presented that enables ''energy stable'' modifications for existing WENO schemes of any order and develops new weight functions and derive constraints on their parameters, which provide consistency, much faster convergence of the high-order ESWenO schemes to their underlying linear schemes for smooth solutions with arbitrary number of vanishing derivatives, and better resolution near strong discontinuities than the conventional counterparts.

TL;DR: Stable and spectrally accurate numerical methods are constructed on arbitrary grids for partial differential equations that are equivalent to conventional spectral methods but do not rely on specific grid distributions.

TL;DR: The extent to which a high-order accurate shock-capturing method can be relied upon for aeroacoustics applications that involve the interaction of shocks with other waves has not been previously quantified and is initiated in this work.

TL;DR: In this paper, it was shown that two-dimensional shocks are asymptotically first-order regardless of the design accuracy of the numerical method, and the practical implications of this finding are discussed in the context of the efficacy of high-order numerical methods for discontinuous flows.