Stability of a difference scheme

We will not attempt to encompass a wide variety of situations, but instead look in more detail at several exhaustive examples before formulating the definition of stability of a difference scheme with respect to the input data, the concept of which we have intuitively developed earlier. 

It is straightforward to verify that the function u(x) = Ug exp —ax gives the exact solution of problem (1). This solution does not increase with increasing x u x) Mq for a 0, so that M(a ) continuously depends on Ug. An excellent start in this direction is to approximate problem (1) on the equidistant grid = [x = ih, z = 0,1. by the difference problem 

Marcel Dekker, Inc. 270 Madison Avenue. New York, New Yoik 10016 

We regard a point x to be fixed and take a sequence of steps h so that X would always belong the set of grid nodes x — i h. The attached number may be made arbitrarily large once we will refine the grid in any convenient way, that is, letting ft — 0. The value of y at this point becomes 

The last inequality implies that the solution of the difference problem (2) continuously depends on the input data. In such cases we say that a difference scheme is stable with respect to the input data. 

Example 2. An unstable scheme. For problem (1) we rely on the scheme 

---

A rigorous definition of stability of a difference scheme will be formulated in the next section. The improvement of the approximation order for a difference scheme on a solution of a differential equation will be of great importance since the scientists wish the order to be as high as possible. 

Stability of a difference scheme. Let two normed vector spaces and be given with parameter h being a vector of some normed space with the norm /i > 0. In dealing with a linear operator with the domain V Ah) — and range TZ Af ) C B we consider the equation... 

Methods for the convergence rates of additive schemes. So far we have established many times that approximation and stability of a difference scheme provide its convergence. For additive schemes we shall need stability with respect to the right-hand side so that it follows from the condition of summarized approximation... 

---