Error propagation in the numerical integration of solitary waves. The regularized long wave equation

Abstract

We study the error propagation of time integrators of solitary wave solutions for the regularized long wave equation, u t + u x + 1 2 (u 2 ) x − u xxt = 0, by using a geometric interpretation of these waves as relative equilibria. We show that the error growth is linear for schemes that preserve invariant quantities of the problem and quadratic for 'nonconservative' methods. Numerical experiments are presented.

A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation

“…The corresponding modified equation is given by (30) and hence (31), respectively. We substitute the single solitary wave solution (8) with (31) and plot the resulting function in Figure 3. From the results plotted in Figure 3 we observe that the second-order modified equation has the same order error as the first-order modified equations plotted in Figure 2.…”

“…Irk [30] extends the work of Bhardwaj and Shankar [25] by considering an Adams-Moulton time-integration scheme coupled with a quintic spline collocation method for the spatial variable. Araújo and Durán [31] have discussed the importance of using numerical schemes that conserve the invariants of (1). More specifically, they show that the error growth in schemes that are conservative is linear whilst nonconservative schemes have quadratic growth in the errors.…”

“…More specifically, they show that the error growth in schemes that are conservative is linear whilst nonconservative schemes have quadratic growth in the errors. Araújo and Durán [31] base their results on the fact that the RLW equation admits a Hamiltonian structure. Also, one of its invariants is generated by a symmetry group.…”

“…Also, one of its invariants is generated by a symmetry group. Solitary wave solutions admitted by the RLW equation then occur as critical points of the Hamiltonian function [31]. This geometric interpretation of the solitary wave solution is used to determine the relationship between numerical schemes that conserve quantities (5)-(7) and the linear growth in error.…”

Numerical detection and generation of solitary waves for a nonlinear wave equation

“…On the other hand, it is crucial a good behavior of the error for long time integration of solitary waves of nonlinear wave equations. For this, the geometric integrators are suitable [23][24][25].…”