An invariant moving mesh scheme for the nonlinear diffusion equation

Abstract

We consider the Cauchy problem for the nonlinear diffusion equation, u t = ( u mu x ) x which is posed on an infinite domain. The PDE and a conservation law are invariant to a Lie group of stretchings which is used to construct the invariant quantities, xt −1/(2+ m) and ut 1/(2+ m) . Using these invariants as similarity variables the problem is reduced to a second order ODE and then integrated to give the well known similarity solutions. The problem is semi-discretized using the method of lines. The mesh movement is governed by the conservation of mass law, so that the computational domain expands as the solution diffuses. The resulting semi-discretization is a system of ODEs which is invariant to the same Lie group as the PDE and so the mesh X i ( t) behaves like the discretized invariant Y it 1/(2+ m) and the solution U i ( t) like V i ( t) −1/(2+ m) . Furthermore, it is shown that, using these invariants, the same reduction and integration is possible in the semi-discrete case as in the continuous case.