Numerical solutions of the nonlinear wave equations with energy-preserving sixth-order finite difference schemes.

In this paper, novel energy-preserving finite difference schemes are presented and analyzed for one-dimensional and two-dimensional nonlinear wave equations with variable coefficients, respectively. Initially, the original differential equations are reformulated as corresponding Hamiltonian systems. Subsequently, the semi-discrete schemes are constructed using the sixth-order finite difference formula. The discrete energy method is employed to rigorously prove the energy conservation laws, convergence, and stability of the solutions for the spatial semi-discrete scheme. Next, the time derivatives in the semi-discrete schemes are discretized with the third-order strong stability preserving Runge-Kutta method, then the local truncation error of the finite difference scheme is provided. The accuracy and effectiveness of the constructed schemes, along with the theoretical analysis, are demonstrated through numerical experiments. [ABSTRACT FROM AUTHOR]