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

Huanhuan Li; Lei Kang; Meng Li; Xianbing Luo; Shuwen Xiang; Huanhuan Li; Lei Kang; Meng Li; Xianbing Luo; Shuwen Xiang

doi:10.3934/era.2024200

Electronic Research Archive

2024, Volume 32, Issue 7: 4433-4453. doi: 10.3934/era.2024200

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

School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China

Received: 17 April 2024 Revised: 29 June 2024 Accepted: 08 July 2024 Published: 15 July 2024

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.
- semi-linear wave equation,
- exponential scalar auxiliary variable,
- error analysis,
- Crank−Nicolson scheme
Citation: Huanhuan Li, Lei Kang, Meng Li, Xianbing Luo, Shuwen Xiang. Hamiltonian conserved Crank-Nicolson schemes for a semi-linear wave equation based on the exponential scalar auxiliary variables approach[J]. Electronic Research Archive, 2024, 32(7): 4433-4453. doi: 10.3934/era.2024200

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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.

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