Numerical computation of relative equilibria with Femlab

Relative equilibria are special solutions of partial differential equations (PDEs), which are stationary in an appropriate comoving frame of reference. Such solutions occur frequently in biological and chemical models, e.g. when describing pattern formation of reaction-diffusion equations. Examples are traveling waves in 1d, planar and spiral waves in 2d and scroll waves in 3d. If the equation has a special symmetry property equivariance, then one can transform the equation into the comoving frame during the computation of the solution. We will show that this can be very convenient for numerical computations. We have implemented the resulting “frozen equation” in Femlab and conducted several numerical experiments for different examples in 1d and 2d such as traveling waves and spirals in the FitzHugh-Nagumo and the complex Ginzburg-Landau system. Moreover, we describe some of the problems which arise because of the additional convective terms which have been introduced by the freezing transformation.