The High Order Semi-lagrangian Methods Of Hyperbolic Conservation Laws

The semi-Lagrangian approach is wildly used in Vlasov equation and weather forecast, this approach is based on the compound mode of the Lagrangian and Eulerian approaches, and thus possesses inherent advantages of the two schemes. On the one hand,the semi-Lagrangian process could be devised as a approach with high order accuracy.On the other hand, the semi-Lagrangian process without restriction about the CFL condition, so the method can save time in the simulations. Moreover, weighted essentially non-oscillatory process is a high-order precision method while maintaining no shock transitions near discontinuities. The designed method can perform high order precision, as well as have the ability to deal with the oscillation, so the paper proposes the high order semi-Lagrangian processes to compute the conservation laws, and test the orders of accuracy and non-oscillation of the presented methods, which further enrich the theory of semi-Lagrangian processes in the computation of conservation laws.Firstly, the paper presents the high order semi-Lagrangian FV approaches for solving the one dimensional hyperbolic conservation laws. The paper adopts the leftward fourth order RK method to compute the initial value problem of the characteristic curves,and the characteristic curves are applied to achieve the equivalent transformation in the different time levels. The updated value then can be reconstructed by WENO schemes to increase their accuracy in space. Along the characteristic curves, the locations between the initial points and terminal points are changing, so the paper proposes the different WENO reconstructions to meet different cases. Furthermore, through tests of precision and non-oscillation, the high order of precision and ability of capture discontinuities of the methods are thusly proved.Secondly, the paper presents the high order semi-Lagrangian FD approaches for solving the two dimensional hyperbolic conservation laws. The paper introduces the new WENO reconstructions based on the Legendre polynomials, which have the same candidate stencils and orders of the accuracy as the conventional WENO method, but have no integral computation to achieve the reconstructions. So the method save time in the simulations, and more suitable for the numerical fluxes, which are not located at the mesh points. Besides, a series of numerical experiments of two-dimensional conservation laws are proposed to prove the abilities of high order precision and handling the oscillations.Finally, the paper explores the fifth-order mapped compact semi-Lagrangian FD method. According to the sign of characteristic velocities, the paper reconstructs the different WENO methods, and make generalization of the methods. The WENO schemes adopt the common nonlinear weights will make the lose accuracy near certain smooth extrema, so the paper presents the mapped weighted to deal with this problem. An accuracy test and non-oscillatory tests are used to demonstrate that the fifth-order mapped compact semi-Lagrangian FD method can reach the fifth order accuracy and maintain the ability of capturing discontinuities.In summary, high order semi-Lagrangian processes are high order accuracy and high resolution numerical approaches for computing hyperbolic conservation laws. The paper focuses on the high order semi-Lagrangian FV approaches to compute the one dimensional scalars, Euler system and shallow equations with source term, high order semiLagrangian FD method for solving two dimensional hyperbolic conservation laws as well as fifth-order mapped compact semi-Lagrangian FD method. Moreover, simulative solutions illustrate the orders of accuracy and non-oscillation, and reflect the superiority of semi-Lagrangian method for solving the hyperbolic cases. Therefore, the designed methods in the thesis are suitable for the hyperbolic conservation laws.