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.
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