Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

doi:10.1007/s42967-022-00193-2

Abstract

Abstract: In this paper, two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation. Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense, the new schemes are proved to perfectly preserve the total energy in the discrete sense. By using the standard energy method and the cut-off function technique, the optimal error estimates of the numerical solutions are established, and the convergence rates are of O(h⁴ + τ²) with mesh-size h and time-step τ. In order to improve the computational efficiency, an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step. The convergence of the iterative algorithm is also rigorously analyzed. Several numerical results are carried out to test the error estimates and conservative properties.

Key words: Nonlinear fourth-order wave equation, Compact finite difference scheme, Error estimate, Energy conservation, Iterative algorithm

CLC Number:

Xiaoyi Liu, Tingchun Wang, Shilong Jin, Qiaoqiao Xu. Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1509-1530.

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