Estimates are established for the width of the " a-shadow" -the region outside which the solution of a given difference boundary-value problem differs from the solution of the corresponding left-hand difference boundary-value problem by a quantity O(h q+') where q is the order of approximation of the left-hand difference boundary-value problem.
Consider a system of differential equations of hyperbolic type which has characteristics of one sign.
To solve a boundary-value problem with two boundaries for a system of this type, it is suffficient to prescribe boundary conditions on one boundary only /i/ -the source from which the characteristics emanate.
To fix our ideas, let us assume that all the characteristics emanate from the left-hand boundary. Then any problem with two boundaries is equivalent to a left-hand boundary-value problem.
When such a problem is tackled using finite differences, however, it may become necessary to prescribe conditions on both boundaries, formulating additional conditions for the right-hand boundary. When that is done one has to estimate the influence of the additional boundary conditions.
Let the order of approximation of the left-hand difference boundary-value problem (d.b.p.) be O(hq), where h is the grid size in Z. We define the " ~-shade" of the additional boundary conditions to be the region outside which the solution of the d.b.p, wit h two boundaries differs from that of the left-hand d.b.p, by O(hq+'). It is assumed that the left-and right-hand d.b.p, are stable in C. We shall prove that if n~exp (bN), N=h-', the 8 -shade is a boundary layer