Hamiltonian conserved Crank-Nicolson schemes for a semi-linear wave equation based on the exponential scalar auxiliary variables approach

Abstract

The keys to constructing numerical schemes for nonlinear partial differential equations are accuracy, handling of the nonlinear terms, and physical properties (energy dissipation or conservation). In this paper, we employ the exponential scalar auxiliary variable (E-SAV) method to solve a semi-linear wave equation. By defining two different variables and combining the Crank−Nicolson scheme, two semi-discrete schemes are proposed, both of which are second-order and maintain Hamiltonian conservation. Two numerical experiments are presented to verify the reliability of the theory.

Keywords

• semi-linear wave equation
• exponential scalar auxiliary variable
• error analysis
• crank−nicolson scheme