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##

4.3.2 Relaxation Method

Discretize to transform a given DE
into a set of algebraic equations. For example, applying DDST to

we find

which leads to the set of equations

Since we have a BVP, and will be given.

- Let
be an inaccurate (estimated?)
solution. The error components

together with then define an error vector
.
- How to modify
to make
disappear?
Expand linearly:

This modified error vector is called
. We want it to
vanish,
:

Thus our system of equations is *tridiagonal*:
*Recursion technique!*

__EXAMPLE:__

DDST leads to
. Expand:

Start the downwards recursion:
and
.

brings us down to . Putting

we take the upwards recursion

Improve
and iterate.

** Next:** 5. Partial Differential Equations
**Up:** 4.3 Boundary Value Problems
** Previous:** 4.3.1 Shooting Method
* Franz J. Vesely Oct 2005*

See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001