A Meshless Local Discontinuous Galerkin Method with Satellite Slope Limiter for Convective-dominated problems
This paper derives a new class of stabilized meshless method for the simulations of the Convective-dominated problems, which are based on the general Discontinuous Galerkin (DG) framework and the local symmetric weak form (LSWF). In contrast to the usual meshbased DG methods which approximate the partial differential equations depend on the non-overlapping mesh, the presented method represents the solutions locally on a set of independent overlapping sub-domains. In addition, by using the similar idea of meshbased DG method, the standard numerical flux functions are employed to determine the information exchanged between each pairs of overlapped sub-domains, and a compatible meshless Slope Limiter for the stabilization of high-order Meshless Local Discontinuous Galerkin (MLDG) schemes is developed in one dimensional cases based on the theoretical analysis of the numerical stability of MLDG schemes. Because of the presented method does not need of any additional meshes, either for approximation purposes or for Galerkin integration purposes, it is a truly meshless method which inherit both the advantages of DG method and meshless method. Finally, the resulting first and second order MLDG schemes are verified by several classical one-dimensional numerical tests with the existence of discontinues, including the transport equation of square wave, the non-viscous Burgers equation, the one dimensional Lax shock-tube problem, the Shu-Osher problem and the Woodward-Colella interacting shock wave problem. All the numerical results are fairly agreed with the exact solutions whatever solving problems with the equidistant nodes of random distribution nodes. In essence, the MLDG methods is found to be a meshless, stable, high-order accurate, and highly parallelizable schemes that can easily handle the complicated convective-dominated problems.
meshless methods, local symmetric weak form, Discontinuous Galerkin methods, slope limiters,convection-dominated problems, shock waves
Wei-ran Gao Hongfu Qiang
201 Staff room, Xi’an Hi-Tech Institute, Xi’an, Shaanxi, PRC, PC 710025
国际会议
南京
英文
25-41
2009-10-12（万方平台首次上网日期，不代表论文的发表时间）