Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach (Advances in Applied Mathematics and Mechanics, 2023)
In this paper, we present a quadratic auxiliary variable (QAV) technique todevelop a novel class of arbitrarily high-order energy-preserving algorithms for theCamassa-Holm equation. The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition. Thenthe reformulated QAV system is discretized by applying the Fourier pseudo-spectralmethod in space and the symplectic Runge-Kutta methods in time, which arrives ata class of fully discrete schemes. Under the consistent initial condition, they can berewritten as a new fully discrete system by eliminating the introduced auxiliary variable, which is rigorously proved to be energy-preserving and symmetric. Ample numerical experiments are conducted to confirm the expected order of accuracy, conservative property and efficiency of the proposed methods. The presented numericalstrategy makes it possible to directly apply a special class of Runge-Kutta methodsto develop energy-preserving algorithms for a general conservative system with anypolynomial energy.