Structure-Preserving Difference Methods For Two Kinds Of Fourth-Order Nonlinear Wave Equations

This thesis focuses on the numerical solutions of two classes of nonlinear fourthorder wave equations,which play an important role in the field of viscoelastic materials.Some numerical methods have been proposed for solving the two kinds of equations,but they only have second-order accuracy in space.In addition,it is usually regard that structure-preserving methods that preserve one or more intrinsic properties of a given dynamical system are better than traditional methods in terms of long-term stability of numerical simulations.However,when applying implicit or explicit approximations to the non-linear wave equations,the nonlinear terms of the model can cause serious damage to the stability of the method.Fortunately,in recent years a method known as scalar auxiliary variables can effectively overcome this problem.In such a context,this paper intends to apply exponential scalar auxiliary variables to two kinds of fourth-order nonlinear wave equations,and construct corresponding high-order linear structure-preserving difference methods.Firstly,the nonlinear fourth-order strain wave equations are transformed into a new equivalent system by introducing exponential scalar auxiliary variables.Based on the obtained equivalent system,a fully discrete high-order difference scheme is obtained by apply the secondorder central difference and the fourth-order compact difference in the time and space directions,respectively,and it is theoretically proved to be structure-preserving.The validity of the numerical scheme is also verified by several numerical examples.Similar technique is then applied to the fourth-order nonlinear strongly damped wave equation.A difference scheme with accuracy of second-order in time and fourth-order in space is proposed.The validity and structural performance of the proposed difference scheme are analyzed and verified by theoretical and numerical simulation.