对流扩散方程在任意细分上的高阶完全通量格式
High-order complete flux schemes for convection-diffusion equations on arbitrary subdivisions
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中文总结 AI 辅助
研究对流扩散方程在任意网格的高阶格式，通过推导精确法向通量构建完全通量有限体积法，对任意多项式次数离散空间有效，在不同空间有良好表现，数值实验验证了方法的稳健性与最优精度。
中文摘要 AI 辅助
我们为对流扩散方程开发了一种完全通量有限体积法，该方法适用于二维和三维的任意网格以及任意多项式次数的离散空间。与标准有限体积离散化不同，我们从基础偏微分方程推导出穿过每个控制体积边/面的精确法向通量。此精确通量自然分为齐次部分（经典的沙夫特 - 古梅尔通量）和通过包含切向通量与源项的格林函数表示的非齐次部分。所得公式与连续方程完全等效，选定离散空间后可产生无任何修正或稳定化的高阶格式。对于分段线性空间，该格式在对流主导区域达到最优二阶精度并能在适度粗糙网格上保持正值。对于二次空间，基于拉格朗日元素或B样条的标准有限体积法除非专门设计控制体积网格否则无法达到最优\(L^2\)收敛，而本文提出的完全通量格式总能独立于控制体积网格实现最优\(L^2\)收敛。二维和三维数值实验证实了该方法的稳健性和最优精度。
英文摘要
We develop a novel complete flux finite volume method for convection-diffusion equations on arbitrary subdivisions in two and three dimensions. Unlike standard finite volume discretizations, where the numerical flux is directly approximated from the flux definition, we derive the exact normal flux across each control volume edge/face from the underlying PDE. This exact flux splits naturally into a homogeneous part (the classical Scharfetter--Gummel flux) and an inhomogeneous part based on a Green's function that incorporates the tangential flux and the source term. The resulting formulation is exactly equivalent to the continuous equation and, once the discrete space is chosen, yields high-order schemes without using correction or stabilization strategies. From this framework, we develop concrete numerical schemes on arbitrary grids using Lagrange finite element spaces and B-spline spaces, together with their companion dual meshes (control volume partitions). Numerical experiments in two and three dimensions confirm the optimal convergence and positivity preservation of the proposed schemes.
