Unformatted text preview: circumstances than stated here. It is, furthermore, usual to state it as a stability condition
rather than one on the convergence of a di erence scheme. We'll consider other forms of
the theorem in Chapter 6.
a a x= t a x= t a Proof. We use a straight forward contradiction argument. Since the di erence scheme is
required to converge for all initial conditions, consider space-time points where the two
domains of dependence are disjoint and choose initial data that is nonzero on the domain
of dependence of the partial di erential equation and zero on the domain of dependence
of the di erence scheme. The solution of the di erence scheme must be trivial at these
points, whereas the solution of the partial di erential equation at the same points will be
nonzero. Re ning the mesh so that the two domains of dependence remain disjoint does
not alter this conclusion hence, the solution of the di erence equation cannot possibly 16 Finite Di erence Methods converge to that of the partial di erential equation. Therefore, the domain of dependence
of the di erence scheme must contain that of the partial di erential equation. Problems
1. Suppose is a positive con...
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