Collocation in space and time: experience with the Korteweg-de Vries equation Original Research Article

We consider, from a computational point of view, the use of collocation in both space and time for the solution of certain evolutionary partial differential equations (PDEs). Our approximating function is drawn from a tensor product of polynomial spline spaces. Analyses of such methods have appeared in the literature, albeit infrequently, but few, if any, numerical results have been reported. Here we take a complementary approach, forgoing analysis and focussing on our numerical experience. We are interested primarily in equations having smooth travelling wave solutions, and so adopt solitons of the Korteweg-de Vries (KdV) equation as our model problem. Collocation at the Gauss points in both spatial and temporal subintervals leads to efficient, accurate solutions. However, collocating at the Radau II points in time proves to be nonconservative, leading to significant amplitude and phase errors. To help provide perspective for our numerical results, we include comparisons with a finite difference code and a pseudospectral collocation code with leapfrog timestepping. Additionally, we discuss some interesting implementation issues.