A Conservative and Implicit Second-Order Nonlinear Numerical Scheme for the Rosenau-KdV Equation

In this paper, for solving the nonlinear Rosenau-KdV equation, a conservative implicit two-level nonlinear scheme is proposed by a new numerical method named the multiple integral finite volume method. According to the order of the original differential equation's highest derivative, we can confirm the number of integration steps, which is just called multiple integration. By multiple integration, a partial differential equation can be converted into a pure integral equation. This is very important because we can effectively avoid the large errors caused by directly approximating the derivative of the original differential equation using the finite difference method. We use the multiple integral finite volume method in the spatial direction and use finite difference in the time direction to construct the numerical scheme. The precision of this scheme is O(tau(2)+h(3)). In addition, we verify that the scheme possesses the conservative property on the original equation. The solvability, uniqueness, convergence, and unconditional stability of this scheme are also demonstrated. The numerical results show that this method can obtain highly accurate solutions. Further, the tendency of the numerical results is consistent with the tendency of the analytical results. This shows that the discrete scheme is effective.

Automated summary

In this paper, a new numerical method using multiple integration to convert a partial differential equation into a pure integral equation is proposed for solving the nonlinear Rosenau-KdV equation. By avoiding the large errors caused by finite difference methods, the method shows high accuracy and conservativeness. The numerical results are consistent with analytical results, demonstrating the effectiveness of the discrete scheme.