# Adaptive method of lines by Alain Vande Wouwer, A Vande Wouwer, Ph. Saucez, W.E.

By Alain Vande Wouwer, A Vande Wouwer, Ph. Saucez, W.E. Schiesser

The final approach to traces (MOL) process presents a versatile structure for the answer of all of the significant sessions of partial differential equations (PDEs) and is especially well matched to evolutionary, nonlinear wave PDEs. regardless of its application, besides the fact that, there are really few texts that discover it at a extra complicated point and replicate the method's present kingdom of development.Written by means of extraordinary researchers within the box, Adaptive approach to strains displays the variety of thoughts and functions relating to the MOL. such a lot of its chapters specialise in a specific program but in addition offer a dialogue of underlying philosophy and procedure. specific awareness is paid to the idea that of either temporal and spatial adaptivity in fixing time-dependent PDEs. Many vital rules and techniques are brought, together with relocating grids and grid refinement, static and dynamic gridding, the equidistribution precept and the concept that of a computer screen functionality, the minimization of a useful, and the relocating finite aspect procedure. functions addressed comprise shallow water move, combustion and flame propagation, delivery in porous media, fuel dynamics, chemical engineering techniques, solitary waves, and magnetohydrodynamics.As the 1st complex textual content to symbolize the fashionable period of the tactic of traces, this monograph deals an excellent chance to find new strategies, research new innovations, and discover a variety of purposes.

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Extra resources for Adaptive method of lines

Example text

Interpolation of the solution to produce new initial conditions In Step 1, the spatial derivatives are approximated using finite difference approximations up to any level of accuracy on a nonuniform grid as implemented in the standard Fortran subroutine WEIGHTS by Fornberg [3]. This algorithm is used for generating “direct” as well as “stagewise” schemes. In the latter case, higherorder derivatives are obtained by successive numerical differentiations of lower-order derivatives. An example of the use of these particular schemes will be given in the section on the Korteweg-de Vries equation.

Solitons exist in some particular equations such as the nonlinear Schrödinger (NLS) equation and the Korteweg-de Vries (KdV) equation, which will be studied numerically in the continuation of this chapter. The importance of solitons in today’s literature is explained by the fact that very general nonlinear wave equations have particular regimes in which their long time behavior is modeled by an equation that has solitons. This modeling procedure, called the reductive perturbation method, may be compared to a linearization in which solitons play the role of exponential solutions [13].

1 Introduction In recent years, much interest has developed in the numerical treatment of PDEs giving rise to nonlinear wave phenomena, and particularly, solitary waves. , the cubic Schrödinger equation (CSE) and the derivative nonlinear Schrödinger equation (DNLS), as well as several Korteweg-de Vries (KdV)-like equations in one space dimension. These equations have been used extensively to model nonlinear dispersive waves in a wide range of application areas, such as water wave models, laser optics, and plasma physics.