Analysis Of Mathematical Characters On High-Order Schemes

For the sake of seeking an scheme with both accuracy and efficiency, this article compare and analyze some famous schemes. Firstly, this article introduces a permissible weak solution, use classical schemes,first order TVD scheme,Harten scheme,Godunov scheme,second order Godunov scheme(MUSCL method) and third order Godunov scheme(PPM method) to solve model equation. Secondly, this article introduces the regulation of how to judge the type of Riemann solution and the method of how to solve, use Roe scheme,Harten scheme and AUSM scheme to solve the shock tube problem of Euler equation. Finally, this article use Crank-Nicolson scheme,High-Order compact implicit difference method to solve linear Burgers equation and use the cpu-time function provided by Fortran95 to test the efficiency of above schemes. Numerical results prove: When solve the model equation, third order Godunov scheme (PPM method) is more accurate and efficient than other schemes, When solve the Euler equation, although AUSM scheme's computational speed is fast, the scheme only have first order accuracy, When solve the linear Burgers equation, High-Order compact implicit difference method have forth order in space and high efficiency.