Analysis of numerical approximation algorithms for nonlinear differential equations using a discrete multiple scales technique

[摘要] English: Perturbation techniques for the solution of differential equations form an essentialingredient of the tools of mathematics as applied to physics, engineering, financeand other areas of applied mathematics. A natural extension would be to seekperturbation-type solutions for discrete approximations of differential equations.The main objective of the research project is to develop a perturbation technique forapproximation. The spurious behavior, predicted theoretically, is shown to be presentexperimentally, independent of temporal discretization.We also detail some comparisons of central difference solutions of different ordersof approximation to the KdV equation. The results show a clear benefit of higherorder central differences relative to lower order methods. The benefit of the centraldifference methodology would also extend to more general regions over which wewould solve partial differential equations.We also show that the method of multiple scales can provide an adequate explanationfor spurious behavior in a difference scheme for the Van der Pal equation.the solution of discrete equations.We discuss the well-known method of multiple scales and show its use for the solutionof the Korteweg-de Vries (KdV), Regularized Long Wave (RLW) and Van der Polequations. In particular, for the KdV and RLW equations the analysis shows thatthe envelopes of modulated waves are governed by the nonlinear Schrödinger equation.We present a variation of the multiple scales technique which presents an idealframework from which we devise a discrete multiple scales analysis methodology.We discuss a discrete multiple scales methodology derived by Schoombie [111], asapplied to the Zabusky-Kruskal approximation of the KdV equation. This discretemultiple scales analysis methodology is generalized and applied to the solution of ageneralized finite difference approximation of the KdV equation. We show the consistencyof the method with the continuous analysis as the discretization parameterstend to zero.The discrete multiple scales technique is a powerful tool for the examination of modulationalproperties of the KdV equation. In the case of certain modes of the carrierwave, the discrete multiple scales analysis breaks down, indicating that the numericalsolution deviates in behavior from that of the KdV equation. Several numericalexperiments are performed to examine the spurious behavior for different orders of approximation. The spurious behavior, predicted theoretically, is shown to be presentexperimentally, independent of temporal discretization.We also detail some comparisons of central difference solutions of different ordersof approximation to the KdV equation. The results show a clear benefit of higherorder central differences relative to lower order methods. The benefit of the centraldifference methodology would also extend to more general regions over which wewould solve partial differential equations.We also show that the method of multiple scales can provide an adequate explanationfor spurious behavior in a difference scheme for the Van der Pal equation.

[发布机构] University of the Free State