Finite Difference Method for Boundary Value Problems
This module enables the user to compare different approximate
solutions computed using the finite difference method for boundary
value problems for ordinary differential equations. A general boundary
value problem (BVP) consists of an ordinary differential equation
(ODE) with side conditions specified at more than one point.
This module illustrates the solution of second-order scalar ODEs of the
form u″ = f(t, u,
u′) on an interval [a,
b] with boundary conditions u(a) =
α and u(b) =
β. The 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] using finite difference quotients, which transforms
the BVP into a system of algebraic equations. Denote the fixed mesh
points by ti, i =
0,…,n+1, where t0 =
a, and tn+1 =
b. Let yi denote the
approximation to u(ti ),
and let yi(1)(
y) and
yi(2)(
y) denote finite difference approximations to
u′(ti ) and
u″(ti ),
respectively, where y denotes the vector of approximate
solution values at the mesh points, on which the finite difference
approximations depend. The finite difference method seeks to find a
y such that y0 =
α, yn+1 =
β, and
yi(2)(y) =
f(ti, yi,
yi(1)(y)) for
i = 1,…,n. The latter is a system of
algebraic equations to be solved for 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 and
u(b) = β are indicated by black
dots on the graph. Next the user specifies the order of the finite
difference approximations and the number of mesh points (including
boundary points). The mesh points are equally spaced, and the finite
difference formulas used to approximate derivatives are centered
(except near boundaries). When the user clicks Solve, the
resulting approximate solution to the BVP is drawn on the graph. To
compare this solution with other approximate solutions, the user can
make additional choices of difference order and number of mesh points,
then click Solve to add each new approximate solution to the
graph. The solutions for different parameters are color coded so that
the color changes from blue to red as the number of mesh points
increases and from light to dark as the difference order increases.
To clear all of the solutions from the graph, click Reset.
To illustrate the details of the individual steps of the finite
difference method for a particular choice of parameters, 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
Department of Computer Science