A Linearly Implicit Structure-Preserving Scheme for the Camassa–Holm Equation Based on Multiple Scalar Auxiliary Variables Approach

Abstract

In this paper, we present a linearly implicit energy-preserving scheme for the Camassa-Holm equation by using the multiple scalar auxiliary variables approach, which is first developed to construct efficient and robust energy stable schemes for gradient systems. The Camassa-Holm equation is first reformulated into an equivalent system by utilizing the multiple scalar auxiliary variables approach, which inherits a modified energy. Then, the system is discretized in space aided by the standard Fourier pseudo-spectral method and a semi-discrete system is obtained, which is proven to preserve a semi-discrete modified energy. Subsequently, the linearized Crank-Nicolson method is applied for the resulting semi-discrete system to arrive at a fully discrete scheme. The main feature of the new scheme is to form a linear system with a constant coefficient matrix at each time step and produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution. Several numerical results are addressed to confirm accuracy and efficiency of the proposed scheme.

Linearly implicit structure-preserving schemes for Hamiltonian systems

“…Recently, more work has been put into developing linearly implicit energy-preserving schemes for Hamiltonian PDEs, e.g. the partitioned averaged vector field (PAVF) method [25] and schemes based on the invariant energy quadratization (IEQ) approach [26] or the multiple scalar auxiliary variables (MSAV) approach [27]. However, little attention has been given to linearly implicit local energy-preserving methods.…”

Scalar auxiliary variable approach for conservative/dissipative partial differential equations with unbounded energy

“…Due to its attractive properties, the SAV approach has been intensively studied in these years. It has been applied to many PDEs, for example, the two-dimensional sine-Gordon equation [6], the fractional nonlinear Schrödinger equation [12], the Camassa-Holm equation [21], and the imaginary time gradient flow [42]. Shen and Xu [32] and Li, Shen and Rui [22] conducted a convergence analysis of SAV schemes.…”

Efficient energy-preserving exponential integrators for multi-components Hamiltonian systems

“…One systematical methodology to construct linearly implicit schemes for general systems is the energy quadratization technique, which includes the invariant energy quadratization (IEQ) approach [45,46,21] and the scalar auxiliary variable (SAV) approach [40,39]. Although the energy quadratization technique is originally proposed for dissipative gradient flow models, it has also been applied to Hamiltonian systems [8,7,28,30], but no multi-components systems have been concerned so far. One main reason that restricts the further applications may be attributed to the preservation of so-called modified energy, and continuous efforts are still made to improve this shortage [11].…”