The chebop system for automatic solution of differential equations

In MATLAB, it would be good to be able to solve a linear differential equation by typing u = L\f, where f, u, and L are representations of the right-hand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to be able to exponentiate an operator with expm(L) or determine eigenvalues and eigenfunctions with eigs(L). A system is described in which such calculations are indeed possible, based on the previously developed chebfun system in object-oriented MATLAB. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution

ADIPLS -- the Aarhus adiabatic oscillation package

Development of the Aarhus adiabatic pulsation code started around 1978. Although the main features have been stable for more than a decade, development of the code is continuing, concerning numerical properties and output. The code has been provided as a generally available package and has seen substantial use at a number of installations. Further development of the package, including bringing the documentation closer to being up to date, is planned as part of the HELAS Coordination Action.Comment: Astrophys. Space Sci., in the pres

A multigrid continuation method for elliptic problems with folds

We introduce a new multigrid continuation method for computing solutions of nonlinear elliptic eigenvalue problems which contain limit points (also called turning points or folds). Our method combines the frozen tau technique of Brandt with pseudo-arc length continuation and correction of the parameter on the coarsest grid. This produces considerable storage savings over direct continuation methods,as well as better initial coarse grid approximations, and avoids complicated algorithms for determining the parameter on finer grids. We provide numerical results for second, fourth and sixth order approximations to the two-parameter, two-dimensional stationary reaction-diffusion problem: Δu+λ exp(u/(1+au)) = 0. For the higher order interpolations we use bicubic and biquintic splines. The convergence rate is observed to be independent of the occurrence of limit points