Numerical treatment of boundary conditions

Abstract

In chapter 5, you have seen the number and type of boundary conditions needed to get a well-posed problem. In the case of discretized equations (finite-difference, finite-element, spectral, etc.) the same boundary conditions are applied, but you need something additional because the number of boundary conditions is usually less than the number of unknowns at the boundaries. You could try to supplement this by using some of the discrete equations, but these usually cannot be applied right at the boundaries. This is most clearly seen for finite-difference equations, which involve some stencil of grid points around the central point. If applied at a boundary, you would need values from grid points located outside the region, which is impossible. Various approaches are possible to get additional equations on the boundary:

• (i) apply some kind of extrapolation from the inner region for those variables that are not specified by boundary conditions;

• (ii) introduce additional grid points outside the region such that the finite-difference equations can be used up to and including the boundary; however, this requires some way to define values at the added grid points (e.g. by symmetry arguments);

• (iii) apply adapted finite-difference equations at the boundary, which use only available grid points, and consequently have an asymmetric grid stencil.