Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation
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In this paper, we develop a novel class of arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. With the aid of the invariant energy quadratization approach, the Camassa-Holm equation is first reformulated into an equivalent system with a quadratic energy functional, which inherits a modified energy conservation law. The new system are then discretized by the standard Fourier pseudo-spectral method, which can exactly preserve the semi-discrete energy conservation law. Subsequently, the Runge-Kutta method and the Gauss collocation method are applied for the resulting semi-discrete system to arrive at some arbitrarily high-order fully discrete schemes. We prove that the schemes provided by the Runge-Kutta method with the certain condition and the Gauss collocation method can conserve the discrete energy conservation law. Numerical results are addressed to confirm accuracy and efficiency of the proposed schemes.
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