Manchester Geometry Seminar 2005/2006
Numerical Analysis Seminar
(Joint session)

The Difference Potential Method (DPM) is a numerical method possessing many remarkable properties. In particular, it allows reducing the solution of a boundary-value problem in an arbitrary domain to the solution of a boundary equation without the knowledge of Green's function. The main ingredient of the method, the difference potential, plays for general difference schemes the role similar to that of the Cauchy-type integral in the theory of analytic functions.

The lecture is devoted to ideas, constructions, and new general opportunities that the DPM provides in discrete modelling and for solving some problems of mathematical physics.

In particular, we consider the boundary-value problems in the domains in Euclidean space or manifolds such as sphere, torus, and the Möbius one-side surface. The new possibilities of DPM are illustrated by the applications to numerical solution of the Stokes problem for viscous fluid flow, and to construction of non-local artificial boundary conditions for calculating of gas flow around an airfoil and non-reflecting artificial boundary conditions for the long-time calculating of wave propagation. We shall also present the difference model of the active shielding of a space domain against outside time-harmonics noise. The simplicity of this model makes it the foundation for creating a technical shielding system.

REFERENCE:

V.S. Ryaben'kii, Method of Difference Potentials and its Applications. Springer Series in Comput. Mathem., V.30, 2002