Numerical Analysis of equivariant evolution equations
08-036 Veerle Ledoux, Simon Malham, Jitse Niesen, Vera Thümmler.
Computing Stability of Multidimensional
Travelling Waves
We present a numerical method for
computing the pure-point spectrum associated with the linear stability
of multi-dimensional travelling fronts to parabolic nonlinear
systems. Our method is based on the Evans function shooting
approach. Transverse to the direction of propagation we project the
spectral equations onto a finite Fourier basis. This generates a
large, linear, one-dimensional system of equations for the
longitudinal Fourier coefficients. We construct the stable and
unstable solution subspaces associated with the longitudinal far-field
zero boundary conditions, retaining only the information required for
matching, by integrating the Riccati equations associated with the
underlying Grassmannian manifolds. The Evans function is then the
matching condition measuring the linear dependence of the stable and
unstable subspaces and thus determines eigenvalues. As a model
application, we study the stability of two-dimensional wrinkled front
solutions to a cubic autocatalysis model system. We compare our
shooting approach with the continuous orthogonalization method of
Humpherys and Zumbrun. We then also compare these with standard
projection methods that directly project the spectral problem onto a
finite multi-dimensional basis satisfying the boundary conditions.
