Scientific Computing

This module illustrates the finite difference method for numerically
solving boundary value problems for ordinary differential equations.
A general boundary value problem (BVP) consists of an ordinary
differential *u*″ =
*f*(*t*, *u*, *u*′)*a*, *b*]*u*(*a*) = *α**u*(*b*) = *β**finite
difference method* approximates the solution to the BVP by
approximating the derivatives of the solution *u* at a set of mesh
points within *a*, *b*]*t*_{i}, *i* =
0,…,*n*+1*t*_{0} =
*a**t*_{n+1} =
*b**y*_{i} denote the
approximation to *u*(*t*_{i} )*y*_{i}^{(1)}(
** y**)

The user begins by selecting from the menu provided an ODE and a
specific solution to be sought (if there is more than one). Boundary
values *u*(*a*) = *α**u*(*b*) = *β**Initialize* to begin the process of
solving the BVP. A predetermined starting guess for the solution is
drawn, represented as a piecewise linear function connecting the
approximate solution values at the mesh points. The user clicks
*Iterate* repeatedly to perform successive iterations of Broyden's
Method to solve the system of algebraic equations. At each iteration,
the piecewise linear representation of the next approximate solution is
added to the graph, and the Euclidean norm of the residual is printed.
The approximate solution curves are color coded according to their
residuals, so as the solutions converge their colors change from blue
to red.

To compare approximate solutions obtained using different method orders or different numbers of mesh points, see the alternative Finite Difference Method module.

**Reference:** Michael T. Heath, *Scientific Computing,
An Introductory Survey*, 2nd edition, McGraw-Hill, New York,
2002. See Section 10.4.

**Developers:** Evan VanderZee and Michael Heath