Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Abstract

The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg-de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg-de Vries equation.

Linearly Implicit Energy-Preserving Fourier Pseudospectral Schemes for the Complex Modified Korteweg–de Vries Equation

“…which is constructed by the discrete variational derivative method [6]. One can prove that scheme (3.4) precisely conserves the discrete mass and energy (for more details, refer to [20] and the references therein). REMARK 3.6.…”

“…The result shows that our schemes are more efficient than the nonlinear scheme. Secondly, we compare the proposed methods with the existing methods, such as the LIRK4 method, the Petrov-Galerkin method [13], FVEM [20] and the nonlinear scheme (3.4). The computations are conducted on [−30, 30] until T = 20.…”

“…Given the parameters h = 1/2, τ = 0.01 and θ = π/4, we compute the L ∞ errors between the exact solution and the numerical solutions obtained by the above methods, respectively, as shown in Table 5. For t ∈ [0, 20], the numerical results obtained by using the pseudospectral method are much better than those of the Petrov-Galerkin method [13] and the FVEM [20]. The numerical solution of the linear1 and the related exact solution over t ∈ [0, 20] are presented in Figure 2.…”

“…Cai and Miao [1] proposed an explicit multisymplectic Fourier pseudospectral scheme for the CMKDV equation. In a recent study, Yan and Zheng [20] derived an energy-preserving scheme and a momentum-preserving scheme by combining a finite volume element method (FVEM) and a discrete variational derivative method. But, their energy-preserving method is a nonlinear scheme, which needs to solve a nonlinear system at each time step.…”

Numerical solutions and stability analysis for solitary waves of complex modified Korteweg–de Vries equation using Chebyshev pseudospectral methods

“…Some methods are available in the literature for the analytical solutions of CMKdV equations with special cases, such as inverse scattering method [11] and tanh and sine-cosine methods [12]. However, several numerical methods are essential to introduce for understanding the properties of the interaction of solitary waves for CMKdV equations such as quintic B-spline collocation method [13], differential quadrature method [14], classical lie method [15][16][17], finite difference method [18,19], reductive perturbation method [20], multisymplectic box schemes [21,22], finite volume element method [23], collocation method [24], mesh-free method [25], projection differential method [26], degenerate Darboux transformation method [27], Bernoulli wavelet method [28], and operational matrix method [29,30]. Samadyar and Mirzaee [31] have studied implicit meshless method based on radial basis functions for the numerical solution of time-fractional stochastic Korteweg-de Vries equation.…”