An invariant moving mesh scheme for the nonlinear diffusion equation

✍ Scribed by C.J. Budd; G.J. Collins

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 698 KB
• Volume: 26
• Category: Article
• ISSN: 0168-9274
• DOI: 10.1016/s0168-9274(97)00079-2

✦ Synopsis

We consider the Cauchy problem for the nonlinear diffusion equation, '~zt = (~"ux)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-I/(2+'~') and ut j/(2+~0. 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~(t) behaves like the discretized invariant ~t ~/(2+'~) and the solution Ui(t) like ~ t/(2+,~). 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.