Fully pseudospectral time evolution and its application to 1+1   dimensional physical problems
  It was recently demonstrated that time-dependent PDE problems can numerically be solved with a fully pseudospectral scheme, i.e. using spectral expansions with respect to both spatial and time directions (Hennig and Ansorg, 2009 [15]). This was done with the example of simple scalar wave equations in Minkowski spacetime. Here we show that the method can be used to study interesting physical problems that are described by systems of nonlinear PDEs. To this end we consider two 1+1 dimensional problems: radial oscillations of spherically symmetric Newtonian stars and time evolution of Gowdy spacetimes as particular cosmological models in general relativity. 
J\"org Hennig
gr-qc physics.comp-ph