基于非结构自适应网格的复合有限体积法

利用文献[1]中将Lax-Wendroff格式和Lax-Friedrichs格式整体复合作用构成二维无结构网格上的复合型有限体积法,同时利用Delaunay方法,根据流场流动特性变化的梯度值为指示器对网格进行加密和粗化,实现自适应,并将此方法应用到二维浅水波方程的求解上,进行了二维部分溃坝,倾斜水跃的数值实验.结果表明,该方法是一个计算稳定、能适应复杂的求解域、能很好地捕捉激波、且计算速度快的算法.

The Composite Finite Volume Method on Unstructured Adaptive Meshes

A composite finite volume method (FVM) for hyperbolic conservation laws is introduced on unstructured (triangular) meshes and tested for the 2D free-surface flow equations. The methodology is based on the theory of the (remainder)-effect of finite difference schemes and the property that the numerical dissipation and dispersion of the schemes are compensated each other in a composite scheme. The composite FVM is formed by global composition of several Lax-Wendroff type steps followed by a diffusive Lax-Friedrichs type step, which filters out the oscillations around shocks typical for the Lax-Wendroff scheme. Meanwhile a method for generating adaptived grid on planar domains by Delaunay method is presented, in which the refinement and unrefinement adapt to inherent flow parameters, such as magnitude of the free (surface) gradient and depth-averaged vorticity. To test the efficiency and reliability of the present method, two typical (problems) of discontinuous solutions of two-dimensional shallow water are solved. The numerical results show that the (proposed) method is easy to implement, is accurate, robust and is highly stable.