A Globally Conservative Compact Framework for Conservation Laws: Fourth-Order Schemes with Enhanced Resolution and Stability

TL;DR

This paper introduces a new framework for designing globally conservative compact finite difference schemes that achieve high resolution, stability, and optimal algebraic conservation for solving conservation laws.

Contribution

It develops a rigorous framework for globally conservative compact schemes, including an algorithm for fourth-order schemes with validated high performance.

Findings

Schemes maintain strict conservation for polynomial flux functions.

Proposed schemes achieve optimal algebraic order and high resolution.

Numerical experiments confirm excellent stability and accuracy.

Abstract

The compact finite difference method is a powerful tool for discretizing conservation laws, owing to its inherent flexibility in developing high-resolution and highly stable schemes. In this paper, we propose a framework for the design of genuine globally conservative compact finite difference schemes, which addresses a critical requirement in conservation laws. Within our framework, we rigorously establish that the discrete conservation law maintains strict conservation for flux functions in polynomial spaces with optimal algebraic order, i.e., the discrete scheme achieves an optimal algebraic precision.Our work advances the existing conservative compact finite difference schemes, which rely on approaches to maintaining global conservation that are fundamentally consistent with the method proposed by Lele [Lele, J. Comput. Phys., 1992]. As an application, we propose an algorithm for designing globally conservative fourth-order schemes, aimed at optimizing resolution and asymptotic stability. Three schemes are generated using the algorithm, with their excellent performance across multiple aspects validated through numerical experiments.