Die Linienmethode bei nichtlinearen parabolischen Differentialgleichungen

The method of lines for parabolic differential equations consists in the discretization of the spatial variable only. In this way the first boundary value problem for a parabolic differential equation is transformed into an initial value problem for a system of ordinary differential equations. In this paper it is proved that for the general nonlinear parabolic equation the solution of the discrete problem converges to the solution of the original problem, when the mesh size tends to zero. The principal tool in this investigation is the theory of ordinary differential inequalities and especially the concept of quasimonotonicity.