An interface tracking algorithm for the porous medium equation

We study the convergence of a finite difference scheme for the Cauchy problem for the porous medium equation u t = ( u m ) x x , m > 1 {u_t} = {({u^m})_{x\,x}},m > 1 . The scheme exhibits the following two features. The first is that it employs a discretization of the known interface condition for the propagation of the support of the solution. We thus generate approximate interfaces as well as an approximate solution. The second feature is that it contains a vanishing viscosity term. This term permits an estimate of the form ∥ ( u m − 1 ) x x ∥ 1 , R ⩽ c / t \parallel {({u^{m - 1}})_{x\,x}}\;\parallel _{1,{\mathbf {R}}} \leqslant c/t . We prove that both the approximate solution and the approximate interfaces converge to the correct ones. Finally error bounds for both solution and free boundaries are proved in terms of the mesh parameters.