A Globally Conservative Compact Framework for Conservation Laws: Fourth-Order Schemes with Enhanced Resolution and Stability
This work addresses a critical requirement for high-resolution and stable numerical simulations in computational fluid dynamics and related fields, representing an incremental advancement over existing conservative compact schemes.
The authors tackled the problem of designing globally conservative compact finite difference schemes for conservation laws, resulting in a framework that achieves strict conservation with optimal algebraic precision and an algorithm that generates fourth-order schemes with enhanced resolution and stability, validated through numerical experiments.
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.