THE NUMERICAL SOLUTION OF QUASILINEAR ELLIPTIC EQUATIONS

Publisher Summary This chapter discusses the numerical solution of quasilinear elliptic equations. It describes the numerical solution of nonlinear diffusion equations by implicit finite difference methods. There are certain nonlinear problems that impose severe stability restrictions on explicit methods but not on implicit methods. Notable among these are conductive heat transfer problems with change of phase, the so-called Stefan problems. An implicit method for the diffusion equation means the approximation of the time derivative by a suitable backward difference quotient. It is generally observed that the solution of a parabolic equation by a sequence of elliptic equations is not subject to stability restrictions on the admissible time step. This is an essential feature as in many applications the behavior of a diffusion system has to be modeled over long time periods with locally fine spacial resolution.