Exact Total Variation Minimizers as Non-Oscillatory Limiters in High-Order Methods for Conservation Laws

Abstract

High-order numerical methods for conservation laws can generate spurious oscillations near discontinuities. We propose to suppress these oscillations by post-processing the numerical solution with a total variation (TV) denoising step that reduces the total variation of the nodal values. For a general analysis operator, the resulting discrete minimization problem need not satisfy the submodularity property that existing exact max-flow algorithms require, so we instead compute the TV minimizer exactly using a differential inclusion algorithm. In one dimension, the differential inclusion algorithm computes the TV minimizer in finitely many steps, requires no tuning of algorithmic parameters, and preserves the total mass. In two dimensions, we apply the 1D algorithm dimension by dimension. We test the method as a post-processing limiter for Fourier pseudospectral methods and a fifth-order finite difference scheme applied to scalar conservation laws, compressible Euler equations, and the two-dimensional incompressible Euler equations.

Disclosure

“Conceptualization, Methodology, Funding acquisition. Xiangxiong Zhang: Conceptualization, Methodology, Software, Investigation, Writing – original draft, Writing – review & editing, Supervision, Funding acquisition. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work, the authors used Claude Code (An- thropic) in order to assist with mathematical discussions, numerical imple- mentation, and draftin”

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2026-08-23T19:12:25.936079Z

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