Sixth-order Explicit Finite Difference Schemes For Solving Nonlinear Wave Equations

Nonlinear wave equations have been widely used in various fields of science and engineering.In this thesis,we investigate high-order methods for numerically solving nonlinear wave equations with variable coefficients and periodic or Dirichlet boundary conditions,and propose two sixth-order explicit finite difference schemes.Firstly,for the one-dimensional nonlinear wave equation with variable coefficients,the correction technique of the central difference method with truncation error remainder is used to discretize the secondorder derivative in time and the second-order derivatives in space using a sixth-order difference operator.In this way,a nonlinear high-order explicit(NHOE)difference scheme is obtained.This scheme achieves fourth-and sixth-order convergence accuracy in time and spatial directions,respectively.To avoid iterations and improve computational efficiency,the other linearized high-order explicit(LHOE)difference scheme is established by linearizing the nonlinear source terms.This scheme also has fourth-order accuracy in time and sixth-order accuracy in space,respectively.In order to match the temporal accuracy with the spatial accuracy and to allow with large time step sizes in the calculation,the Richardson extrapolation technique is used to improve the temporal accuracy of both schemes to sixth order.From this,two sixth-order explicit difference schemes are obtained for solving one-dimensional nonlinear wave equations with variable coefficients.Then,the discrete Fourier method is used to analyze that the stability condition of the high-order difference scheme for the corresponding the one-dimensional linear problem is vmaxτ/h∈(0,1.50].Moreover,we have successfully extended these two established sixth-order explicit difference schemes to solve the one-dimensional coupled sine-Gordon equations.The numerical results verify the accuracy and effectiveness of the two new sixth-order explicit difference schemes.Secondly,for the two-dimensional nonlinear wave equation with variable coefficients,we use the same discretization method to obtain the nonlinear high-order explicit(NHOE)difference scheme by discretizing the second-order derivatives in time and in space using the correction technique of the central difference method with truncation error remainder and the sixth-order difference operator,respectively.Then we linearize the nonlinear source term to obtain the linearized high-order explicit(LHOE)difference scheme.The temporal accuracy of both schemes is improved to sixth order using the Richardson extrapolation formula.The stability condition of the high-order difference scheme for the corresponding twodimensional linear problem is obtained by discrete Fourier analysis,i.e.vmaxτ/h∈(0,1.0607).Then,the two established sixth-order explicit difference schemes are extended to solve the two-dimensional coupled sine-Gordon equations.Numerical experiments are performed to verify the accuracy and effectiveness of the two sixth-order explicit schemes.Finally,the above method is extended to the three-dimensional variable coefficient nonlinear wave equation,and the corresponding NHOE and LHOE schemes are obtained.We use the discrete Fourier analysis to obtain the stability condition of the high-order difference scheme for the corresponding three-dimensional linear problem is vmaxτ/h∈(0,0.8660).Then,two established sixth-order explicit difference schemes are extended to solve the three-dimensional coupled sine-Gordon equations.Finally,we numerically verify the accuracy and effectiveness of these two new sixth-order explicit schemes.