Discretization based indices for semilinear partial differential algebraic equations

Semilinear partial differential algebraic equations (PDAEs) of the form Au-t(t, x) + Bu-xx(t, x) + F(u) = f(t, x) (F(u) is an element of R-n is a nonlinear function of u is an element of R-n), where at least one of the matrices A, B is an element of R-nxn is singular, are studied. For these systems we propose definitions of a differential time index and of a differential space index which are based on vertical and horizontal semidiscretization schemes, respectively. In these definitions an essential assumption is that the corresponding semidiscretizations converge. For two often used methods we show the convergence for index-one systems. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.