A linearly implicit energy-preserving exponential integrator for the nonlinear Klein-Gordon equation

TL;DR

This paper introduces a linearly implicit exponential energy-preserving integrator for the nonlinear Klein-Gordon equation, improving computational efficiency by reducing the algebraic system to linear, while maintaining energy conservation.

Contribution

It generalizes a previous energy-preserving integrator to be linearly implicit using the scalar auxiliary variable approach, making it more efficient for conservative systems.

Findings

The new method involves only a linear system with constant coefficients.

Numerical experiments show high efficiency and energy preservation.

The scheme is effective for the nonlinear Klein-Gordon equation.

Abstract

In this paper, we generalize the exponential energy-preserving integrator proposed in the recent paper [SIAM J. Sci. Comput. 38(2016) A1876-A1895] for conservative systems, which now becomes linearly implicit by further utilizing the idea of the scalar auxiliary variable approach. Comparing with the original exponential energy-preserving integrator which usually leads to a nonlinear algebraic system, our new method only involve a linear system with constant coefficient matrix. Taking the nonlinear Klein-Gordon equation for example, we derive the concrete energy-preserving scheme and demonstrate its high efficiency through numerical experiments.