A Time Two-Mesh Finite Difference Numerical Scheme for the Symmetric Regularized Long Wave Equation

Abstract

The symmetric regularized long wave (SRLW) equation is a mathematical model used in many areas of physics; the solution of the SRLW equation can accurately describe the behavior of long waves in shallow water. To approximate the solutions of the equation, a time two-mesh (TT-M) decoupled finite difference numerical scheme is proposed in this paper to improve the efficiency of solving the SRLW equation. Based on the time two-mesh technique and two time-level finite difference method, the proposed scheme can calculate the velocity u(x,t) and density ρ(x,t) in the SRLW equation simultaneously. The linearization process involves a modification similar to the Gauss-Seidel method used for linear systems to improve the accuracy of the calculation to obtain solutions. By using the discrete energy and mathematical induction methods, the convergence results with O(τC2+τF+h2) in the discrete L∞-norm for u(x,t) and in the discrete L2-norm for ρ(x,t) are proved, respectively. The stability of the scheme was also analyzed. Finally, some numerical examples, including error estimate, computational time and preservation of conservation laws, are given to verify the efficiency of the scheme. The numerical results show that the new method preserves conservation laws of four quantities successfully. Furthermore, by comparing with the original two-level nonlinear finite difference scheme, the proposed scheme can save the CPU time.

New Two-Level Time-Mesh Difference Scheme for the Symmetric Regularized Long Wave Equation

“…In this section, we conducted several numerical simulations of the proposed scheme for solving the SRLW equation. On the one hand, we present the computational efficiency and numerical accuracy of the proposed scheme and compare the obtained results with the nonlinear scheme in [16] and the TT-M difference scheme in [27], respectively. On the other hand, we focus on the conservation laws and the long-time behavior simulation of the proposed scheme.…”

“…The proposed scheme has several advantages: (i) Combined with the two level time-mesh technique, the scheme utilizes the nonlinear scheme on a coarse time-mesh and then constructs a linear difference system on a fine time-mesh, which more efficiently solves the SRLW equation than the nonlinear scheme in [16]; (ii) The new scheme obtains a high accuracy in solving the SRLW equation. The proposed scheme has a second-order convergence rate in time and a fourth-order convergence rate in space, which is higher than that of the scheme in [27]; (iii) The convergence and stability of the scheme have been verified through detailed proofs. Theoretical analysis of the scheme is more intricate compared to existing TT-M methods since a function with three variables is used in the process of the linear system construction.…”

“…In conclusion, in contrast to the nonlinear method presented in [16], the proposed scheme not only preserves nearly the same accuracy and convergence rate as the nonlinear scheme but also substantially decreases the CPU time needed to obtain numerical solutions. Next, we compare the accuracy of two schemes for the SRLW equation: the previous TT-M scheme in [27] and the proposed scheme. The former scheme exhibits first-order convergence in time and second-order convergence in space.…”

“…The combination of the time two-mesh (TT-M) technique [21][22][23][24][25][26][27] with other numerical methods also can improve the efficiency of solving nonlinear partial differential equations. Liu et al [21] investigated a finite element method with the TT-M technique, which was successfully applied to solve the fractional water wave model and other fractional models.…”

“…Afterward, other authors [22][23][24][25][26] used the TT-M method to study the numerical solutions for the partial differential equations such as the Allen-Cahn model, Sobolev model and the nonlinear Schrödinger equation. Gao et al [27] introduced a TT-M finite difference scheme for the SRLW equation, achieving first-order accuracy in time and second-order accuracy in space. However, the error in the numerical solutions of the scheme increases rapidly over a long time period, making it hard to simulate the long-time behavior of Equation (1).…”