基于修正数值通量的HWENO高精度有限体积方法

High-Order HWENO Finite Volume Method Based on Modified Numerical Flux

摘要: 本文为双曲守恒律方程构造了一种改进的通用数值通量，并发展了一类新型高阶有限体积格式。传统Godunov格式基于分段常数近似与黎曼求解器计算界面通量，虽具备守恒性与间断捕捉能力，但仅为一阶精度，数值耗散显著，难以兼顾光滑区域的分辨率要求。为此，本文在Godunov格式框架基础上，采用分段二次多项式重构单元解，并通过守恒约束与插值条件确定多项式系数，突破低阶格式的精度局限。同时，引入Hermite Weighted Essentially Non-Oscillatory (HWENO)方法对守恒变量进行高阶重构，结合HWENO限制器实现导数近似，辅以SSP Runge-Kutta方法完成时间离散，构建了完整的高阶格式体系。对经典基准算例的数值精度测试结果表明，所提格式适用于标量守恒律方程与欧拉方程的求解，在光滑区域可稳定达到理论五阶收敛精度，展现出优异的高阶近似性能。

Abstract: This paper constructs an improved general numerical flux and develops a new class of high-order finite volume schemes for hyperbolic conservation laws. The traditional Godunov scheme computes the interface flux based on piecewise constant approximation and Riemann solvers. Although it satisfies conservation and possesses good shock-capturing capability, it is only first-order accurate with requirements in smooth regions. To address this limitation, based on the Godunov framework, this paper reconstructs the cell solution using piecewise quadratic: polynomials, with coefficients determined by conservation constraints and interpolation conditions, thereby overcoming the accuracy limitations of low-order schemes. Meanwhile, the Hermite Weighted Essentially Non-Oscillatory (HWENO) method is introduced for high-order reconstruction of conservative variables. Combined with HWENO limiters for derivative approximation and the SSP Runge-Kutta method for time discretization, a complete high-order scheme system is established. Numerical accuracy tests on classical benchmark problems show that the proposed scheme is applicable to solving scalar conservation laws and Euler equations, stably achieving the theoretical fifth-order convergence accuracy in smooth regions, and demonstrating excellent high-order approximation performance.