| In this thesis, we study some issues in the high order numerical methods for solving nonlinear hyperbolic conservation laws.; The first part focuses on the accuracy problem of the Fourier spectral vanishing viscosity schemes for solving scalar nonlinear conservation law in one spatial dimension. Extensive numerical experiments are carried out to assess the accuracy of the schemes. The asymptotic behavior of the spectral viscosity schemes when viscosity tends to zero has also been studied. Certain entropy violation phenomena have been demonstrated.; In the second part, we study the negative linear weights problem in high order weighted essentially non-oscillatory (WENO) schemes for solving nonlinear hyperbolic conservation laws. A simple and effective strategy, the splitting technique, will be presented to remedy the instability caused by the negative weights. Two 5th order finite volume WENO schemes will also be presented with the help from the splitting technique to treat the negative weights appeared in the schemes.; The last part is numerical simulation of complex flow problems using very high order WENO schemes. The ability and efficiency of high order WENO schemes will be shown. |
