多边形网格上扩散方程新的单调格式

NEW MONOTONE FINITE VOLUME SCHEMES FOR DIFFUSION EQUATIONS ON POLYGONAL MESHES

本文在星形多边形网格上,构造了扩散方程新的单调有限体积格式.该格式与现有的基于非线性两点流的单调格式的主要区别是,在网格边的法向流离散模板中包含当前边上的点,在推导离散法向流的表达式时采用了定义于当前边上的辅助未知量,这样既可适应网格几何大变形,同时又兼顾了当前网格边上物理量的变化.在光滑解情形证明了离散法向流的相容性对于具有强各向异性、非均匀张量扩散系数的扩散方程,证明了新格式是单调的,即格式可以保持解析解的正性.数值结果表明在扭曲网格上,所构造的格式是局部守恒和保正的,对光滑解有高于一阶的精度,并且,针对非平衡辐射限流扩散问题,数值结果验证了新格式在计算效率和守恒精度上优于九点格式.

We construct a new monotone finite volume method for diffusion equations on starshaped polygonal meshes. A distinct feature of the new scheme is that the discrete stencil of normal flux on a ceU-edge can contain the cell-edge, which is different from the existing monotone schemes based on a nonlinear two-point flux approximation. That is, in the construction of discrete normal flux on each cell-edge, both the geometric character of distorted cells and the physical variables defined on that cell-edge are taken into account. The consistence of discrete flux is discussed, and the new scheme is proved to be monotone, i.e. it preserves positivity of analytical solutions for diffusion equations with strongly anisotropic and heterogeneous full tensor coefficients. Moreover a design principle of general monotone schemes is proposed. Numerical results are presented to demonstrate the numerical performance of our new monotone scheme such as positivity-preserving, conservation, accuracy and efficiency on distorted meshes.