High Order Numerical Methods For Nonlinear Partial Differential Equations With High Order Derivatives

Based on the framework of the Weighted Essentially Non-Oscillatory(WENO)method and several high order time discretization schemes,this paper develops high-order numerical schemes for several types of nonlinear differential equations with high-order spatial derivatives:(1)the high-order strong-stability preserving(SSP)implicit-explicit(IMEX)Runge-Kutta WENO method for solving long wave equations,(2)the fourthorder compact difference exponential time differencing(ETD)Runge-Kutta method for long wave equations,(3)the direct WENO(DWENO)method for nonlinear dispersion equations,(4)the hybrid Hermite WENO(HWENO)method for nonlinear degenerate parabolic equations.This article mainly includes the following four parts:First,for the shock-similarity wave problem of the long wave equation,a high-order numerical method coupled by the third-order finite difference WENO method in space direction and the third-order SSP IMEX Runge-Kutta method in time direction is constructed,so as to realize the numerical scheme reaches the third-order computational accuracy in space and time at the same time.Numerical results show that this method has good non-oscillation property and the ability of loosing CFL(Courant-Friedrich-Lewy)constraint.Second,in order to achieve a more accurate and effective numerical approximation in the time direction,a fourth-order compact finite difference exponential time discretization scheme is proposed for the long wave equation.This scheme uses the fourth-order compact finite difference method in space and the fourth-order ETD Runge-Kutta method in the time direction to achieve the purpose of obtain fourth-order accuracy in space and time at the same time.The divergence caused by the cancellation error in the calculation process of the ETD Runge-Kutta format is avoided by the Cauchy integral formula,thereby achieving the stability of the ETD Runge-Kutta scheme.Numerical experiments verify the effectiveness of this method against loose CFL restrictions,and demonstrate the high-order calculation accuracy of the numerical method.Third,a finite difference DWENO scheme in arbitrary order is constructed for nonlinear dispersion equations,and the fifth-order scheme for solving nonlinear dispersion equations is described specifically.This scheme only needs to be reconstructed once to achieve the purpose of approximating the third derivative with high-order accuracy.Numerical experiments verify the advantages of the non-oscillation,high-order calculation accuracy of the method.Fourth,with the assistance of direct discontinuous Galerkin(DDG)flux,a hybrid HWENO scheme based on zero-order and first-order moments is proposed for nonlinear degenerate parabolic equations.The computational cell is determined by a troubledcell indicator,that is,the cell whose solution may be discontinuous,and the HWENO reconstruction is used on this troubled-cell to avoid oscillation while the linear approximation is used directly on the other cell.This method only uses adjacent cell information to maintain the compactness of the scheme,and can approximate the second derivative more accurately.This method has fifth-order accuracy in one-dimensional test cases,and fourth-order accuracy for two-dimensional problems.Numerical experiments have proved the high-order accuracy and non-oscillation performance of the scheme.