Nichtnegativitäts-und substanzerhaltende Differenzenschemata für lineare Diffusionsgleichungen

Explicit difference schemes preserving non-negativity and mass as does the described diffusion process are given for the Fokker-Planck partial differential equation (Kolmogorov's forward differential equation) in ℝⁿ. These schemes can be interpreted as descriptions of discrete non-stationary inhomogeneous random walks. They are stable in the maximum norm, and if an initial condition is given, they allow approximate solution of the differential equation as well as approximate Monte-Carlo simulation of the basic diffusion process. Essential conditions are the boundedness of the coefficients of the differential equation and of some of their derivatives and a strong diagonal dominance of the diffusion matrix. The results can be generalized to general linear parabolic differential equations (conservation of mass being lost, of course). By means of the developed method it is also possible to construct difference approximations of non-negative type to linear elliptic operators in ℝⁿ if these operators satisfy analogous conditions.