Coordinate Transformation for the Porous Media Equation that Renders the Free-Boundary Stationary.

摘要

We consider the one-dimensional porous media problem ut = u square xx' u(0, x) = u0(x) for initial data sub uo which vanishes outside a finite interval. Solutions of this problem contain a free boundary separating points (t,x) with u(t,x) > 0 from those with u(t,x) = 0. We deduce a coordinate transformation x = X(t,p) which renders this free boundary stationary in time. We describe a simple difference scheme based on this transformation and show that, even with a fairly crude mesh, the free boundary is tracked quite accurately.