Article

Symmetry Based Numerical Methods for Partial Differential Equations

12/1997;
Source: CiteSeer

ABSTRACT We look at numerical methods for differential equations which are invariant under the action of a symmetry group. We show that numerical methods which preserve this symmetry give excellent results when used to compute problems with singularities and with free boundaries 1 Introduction It has long been recognised that many important partial differential equations arising in mathematical physics and applied mathematics are invariant with respect to a Lie group of transformations and that exact solutions can be determined in many case by exploiting this invariance. Remarkably such solutions are often also global attractors for more general solutions of the equations, given a variety of initial data, and give an accurate indication of the intermediate asymptotic behaviour of the solution after initial effects have died down and before boundary effects become important [2]. Symmetry also plays an important role in studies of ordinary differential equations describing the evolution of syst...

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Keywords

accurate indication
 
boundary effects
 
excellent results
 
initial data
 
initial effects
 
intermediate asymptotic behaviour
 
Lie group
 
mathematical physics
 
numerical methods
 
partial differential equations
 
singularities
 
symmetry group
 
syst
 
transformations