A high-accuracy conservative difference approximation for Rosenau-KdV equation

Abstract

In this paper, we study the initial-boundary value problem of Rosenau-KdV equation. A conservative two level nonlinear Crank-Nicolson difference scheme, which has the theoretical accuracy O(τ 2 + h 4 ), is proposed. The scheme simulates two conservative properties of the initial boundary value problem. Existence, uniqueness, and priori estimates of difference solution are obtained. Furthermore, we analyze the convergence and unconditional stability of the scheme by the energy method. Numerical experiments demonstrate the theoretical results.

Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

“…It is clearly seen that both schemes preserve the invariants very well and remain almost constant during the run time, and also in very good agreement with those given in Ref. [14] and their exact values obtained from Equations (15) and (16). For example, the exact values of the invariants are Q(t) = 5.49817368082 and E(t) = 1.98978293960 whereas their corresponding computed discrete values are Q(t) = 5.49817368131 and E(t) = 1.98978294643 from the Lie-Trotter splitting scheme, and Q(t) = 5.49817368082 and E(t) = 1.98978293990 from the Strang splitting scheme for h = Δt = 0.1 at t = 20.…”

“…All computations have been done using MATLAB R2011a on Intel (R) Core(TM) i7 CPU Q 720@1.60Ghz machine with 4 GB of memory. The initial boundary value problem given by Equations (1)-(3) has the following invariants [6,14]…”

“…Table 4 displays a comparison of some values of discrete invariants Q(t) and E(t) computed by the Lie-Trotter splitting and the Strang splitting techniques with those given in Ref. [14] of Example 1 at various values of h = Δt at t = 0, 10, 20. It is clearly seen that both schemes preserve the invariants very well and remain almost constant during the run time, and also in very good agreement with those given in Ref.…”

“…Then, we can proceed by calculating the eigenvalues i of the matrix L. For the schemes to be stable, all eigenvalues must be consistent with the following stability criteria [30]. The eigenvalues i for various values of parameters in matrix (14) are computed using symbolic programming language Matlab. Figure 1 shows the RK-4 stability region.…”

“…Zhou et al [13] studied the generalized Rosenau-KdV equation with a conservative nonlinear implicit finite difference scheme and also proved the convergence and the stability of the scheme by a discrete energy method. Hu et al [14] proposed conservative two-level nonlinear Crank-Nicolson difference scheme for the initial-boundary value problem of the Rosenau-KdV equation. They also proved that the scheme has the theoretical accuracy O( 2 + h 4 ).…”