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

Abstract

English: Perturbation techniques for the solution of differential equations form an essential ingredient of the tools of mathematics as applied to physics, engineering, finance and other areas of applied mathematics. A natural extension would be to seek perturbation-type solutions for discrete approximations of differential equations. The main objective of the research project is to develop a perturbation technique for approximation. The spurious behavior, predicted theoretically, is shown to be present experimentally, independent of temporal discretization. We also detail some comparisons of central difference solutions of different orders of approximation to the KdV equation. The results show a clear benefit of higher order central differences relative to lower order methods. The benefit of the central difference methodology would also extend to more general regions over which we would solve partial differential equations. We also show that the method of multiple scales can provide an adequate explanation for 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 solution of the Korteweg-de Vries (KdV), Regularized Long Wave (RLW) and Van der Pol equations. In particular, for the KdV and RLW equations the analysis shows that the envelopes of modulated waves are governed by the nonlinear Schrödinger equation. We present a variation of the multiple scales technique which presents an ideal framework from which we devise a discrete multiple scales analysis methodology. We discuss a discrete multiple scales methodology derived by Schoombie [111], as applied to the Zabusky-Kruskal approximation of the KdV equation. This discrete multiple scales analysis methodology is generalized and applied to the solution of a generalized finite difference approximation of the KdV equation. We show the consistency of the method with the continuous analysis as the discretization parameters tend to zero. The discrete multiple scales technique is a powerful tool for the examination of modulational properties of the KdV equation. In the case of certain modes of the carrier wave, the discrete multiple scales analysis breaks down, indicating that the numerical solution deviates in behavior from that of the KdV equation. Several numerical experiments are performed to examine the spurious behavior for different orders of approximation. The spurious behavior, predicted theoretically, is shown to be present experimentally, independent of temporal discretization. We also detail some comparisons of central difference solutions of different orders of approximation to the KdV equation. The results show a clear benefit of higher order central differences relative to lower order methods. The benefit of the central difference methodology would also extend to more general regions over which we would solve partial differential equations. We also show that the method of multiple scales can provide an adequate explanation for spurious behavior in a difference scheme for the Van der Pal equation.

Afrikaans: Perturbasie tegnieke vorm 'n integrale deel van die gereedskap van wiskundige tegnieke om differensiaal vergelykings op te los in fisika, ingenieurswese, finansiële en verwante areas in toegepaste wiskunde. Dit is gevolglik 'n natuurlike uitbreiding om perturbasie oplossings te soek vir die numeriese benaderings van differensiaal vergelykings. Die hoofdoel van dié navorsingsprojek is om perturbasie tegnieke te vind vir die oplossing van diskrete vergelykings. Ons bespreek die bekende veelvuldige skale tegniek en toon die gebruik daarvan aan vir die oplossing van die Korteweg-de Vries (KdV), RLW en Van der Pol vergelykings. Vir die KdV en RLW vergelykings is 'n gevolg van die analise dat die omhulsel van gemoduleerde golwe beheer word deur die nie-lineêre Schródinger vergelyking. Ons bespreek 'n spesifieke veelvuldige skale tegniek wat 'n ideale raamwerk bied om 'n diskrete tegniek te skep. Ons ondersoek 'n diskrete veelvuldige skale tegniek soos deur Schoombie [Ll l] ontwikkel en toegepas op die Zabusky-Kruskal benadering van die KdV vergelyking. Die tegniek word veralgemeen en toegepas op 'n algemene sentraal verskil benadering van die KdV vergelyking. Ons toon aan dat die diskrete metode konsistent is met die kontinue geval as die diskretiserings parameters na nul neig. Die diskrete skale tegniek is 'n geskikte tegniek vir die ondersoek van modulasie eienskappe van die KdV vergelyking. Vir spesifieke modes van die draer golf word oplossings van die diskrete skale tegniek onwenslik wat aandui dat die numeriese oplossing wat ons ondersoek verskil van die oplossing van die KdV vergelyking. Verskeie numeriese eksperimente word uitgevoer om die vals oplossings te ondersoek. Die vals oplossings, soos teoreties voorspel, word eksperimenteel aangetoon, onafhanklik van die diskretisasie tegniek in tyd. Ons benadruk ook oplossings van sentraal verskil benaderings met verskillende ordes van akkuraatheid vir die KdV vergelyking. Die resultate toon 'n duidelike voordeel aan van hoër orde metodes teenoor laer orde metodes. Die voordeel van die sentraal verskil vergelykings is dat ons dit op 'n veralgemeende gebied kan gebruik vir die oplossing van parsiële differensiaal vergelykings. Ons beskou ook 'n eindige verskil benadering van die Van der Pol vergelyking en toon aan dat die diskrete skale tegniek 'n bevredigende verduideliking bied vir vals oplossings veroorsaak deur die spesifieke metode.