An invariant moving mesh scheme for the nonlinear diffusion equation Original Research Article

We consider the Cauchy problem for the nonlinear diffusion equation, ut = (umux)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 ut1/(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 Xi(t) behaves like the discretized invariant Yit1/(2+m) and the solution Ui(t) like Vi(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.