engleski

Explicit stable methods for second order parabolic systems

We show that it is possible to construct stable, explicit finite difference approximations for the classical solution of the initial value problem for the parabolic systems of the form $\partial_tu=A(t,{\bf x})u+f$ on $\R^d$, where $A(t,{\bf x}) \ = \ \sum_{ij} a_{ij}(t,{\bf x}) \partial_i\partial_j \ + \ \sum_i b_i(t,{\bf x}) \partial_i \ + \ c(t,{\bf x})$. The numerical scheme relies on an approximation of the elliptic operator $A(t,{\bf x})$ on an equidistant mesh by matrices that possess structure of a generator of Markov jump process. In the case of ${\R}^2$ scaling of second difference operators can be applied to get the necessary structure of approximations, while in the case of $\R^d, \: d > 2$, rotations at grid-knots are performed in order to get the mentioned structure. Numerical experiments illustrate the theory.

Parabolic systems; finite difference schemes; Markov chains

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