An Adaptive Moving Mesh Method With Local Time Stepping For Conservation Laws And Convection-Dominated Equations

This paper consists of two chapters. On one hand, A variety of physical and engineering problems develop dynamically singular or nearly singular so-lutions in fairly localized regions. For these problems, we are only interested in high resolutions in fairly small solution domain. With uniform meshes, the amount of computational time is too large to enable us to obtain useful numerical approximations, particular in multi-dimensions. Therefore, devel-oping effective and robust adaptive mesh methods for these problems becomes necessary. Main object of the first chapter is to investigate adaptive mesh re-distribution methods. Successful implementation of the adaptive strategy can increase the accuracy of the numerical approximations and also decrease the computational cost.On the other hand, in order tomore efficiently resolve these practical problems, fine meshes over a small portion of the physical domain is often required. However this local refinement reduces the allowable time step very small which is determined by a global CFL time step for the explicit time discrtizations typically employed. Based on the first chapter, in the second chapter, we will develop an efficient adaptive moving mesh algorithm with local time stepping. The principle idea will be demonstrated by investigat-ing the conservation laws and the convection-dominated problems. The local time steps are determined by a local CFL condition. Serval test problems are carried out by using the proposed algorithm. Compared with non-local time stepping method, we can increase the efficiency of the time stepping signifi-cantly in certain situations. Numerical experiments demonstrate the efficiency and robustness of the proposed method.