Mapped WENO Method And Globally Adaptive Mesh Method

This paper studies the following two problems: the mapped WENO method and the global adaptive mesh method for1D hyperbolic conservation laws.In Chapter1, we review the WENO scheme and two present mapped WENO methods, that is, WENO-HAP and WENO-PM6.In Chapter2, we propose a smoothness indicator of sums of squares. Numerical experiments show that, the smoothness indicator of sums of squares can improve the accuracy of some WENO methods, specially in the long time simulation.In Chapter3, we propose a new mapped WENO method, that is WENO-IM(2,0.1). For the fifth order methods, the WENO-IM(2,0.1) can provide more accurate numerical solutions than WENO-HAP and WENO-PM6no matter in short time or long time simulation. However, the higher order WENO-IM(2,0.1) does not perform as well as WENO-RM(260) which is proposed in Chapter4.In Chapter4, we propose another new mapped WENO method, that is WENO-RM(260). For the fifth and seventh order methods, WENO-RM(260) performs compara-bly with WENO-PM6, and much better than WENO-HAP in the long time simulation. For the ninth order methods, WENO-RM(260) can provide more accurate numerical so-lutions than WENO-HAP and WENO-PM6, and can avoid the non-physical oscillation well, however, WENO-HAP and WENO-PM6will generate the non-physical oscillation in the long time simulation.In Chapter6, we propose a globally adaptive mesh method for1D conservation laws. This method firstly obtains the spatial error estimation by using the stagger meshes, then generates the new mesh by using the equidistribution principle to the spatial error estimation. Numerical experiments show that, for some kinds of problems, our globally adaptive mesh method is more efficient than the uniform mesh method, and avoids the non-physical oscillation coming from the structure of mesh near the shock well.