Linear high order finite difference methods with essentially non-oscillatory limiters for hyperbolic conservation laws

Document Type

Article

Publication Date

4-15-2026

Abstract

For high order finite difference and finite volume methods solving hyperbolic conservation laws, the major challenge is to achieve nonlinear stability in the presence of discontinuous solutions. Total variation diminishing or total variation bounded flux limiters are normally set up to achieve the nonlinear stability. High order essentially non-oscillatory methods (ENO or weighted ENO) were designed to avoid constructing high order polynomials across discontinuities to ensure nonlinear stability. However, adaptively reconstructing high order polynomials and doing so in the characteristic space often contributes significantly to the overall computational cost. Alternatives were proposed as hybrid approaches: simply put, applying limiters in the discontinuous regions of the solution while using linear high order methods in the smooth regions. The key to the success of the hybrid approach lies in the differentiation between smooth and nonsmooth regions, which is highly nontrivial given discrete data sets. In this paper, an irregularity detecting mechanism is provided along the discrete profile of the solution to determine when the nonlinear ENO or WENO methods are needed. The irregularity detector does not depend on manually adjusted parameters when problems change. Such an irregularity detector is easy to implement in the dimensional splitting setting of the finite difference methods. The numerical evidences demonstrate the performance of the newly defined irregularity detector. When applied to high order finite difference methods, the numerical results are accurate and non-oscillatory with improved efficiency.

Publication Title

Journal of Computational Physics

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