Collocation Method for Boundary Value Problems
This module illustrates the collocation method for numerically
solving 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 collocation
method approximates the solution to the BVP by a linear combination
of basis functions determined by requiring that the ODE be satisfied at
each of a discrete set of mesh points within [a,
b], and that the boundary conditions be satisfied.
Denote the fixed mesh points by ti,
i = 1,…,n, where
t1 = a, and
tn = b. Denote the basis
functions by φi, i =
1,…,n. Let v(t,
x) denote a linear combination of basis functions
φi with coefficients
xi. The collocation method seeks a solution
x to the system of equations
v(t1, x) =
α, v(tn,
x) = β,
v″(ti, x) =
f(ti,
v(ti, x),
v′(ti, x))
for i = 2,…,n − 1. Note that
the basis functions must be twice differentiable.
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 chooses either polynomials (Chebyshev
polynomials in this module) or cubic B-splines as the set of basis
functions. The user also chooses the number and distribution of mesh
points. If the cubic spline basis is used, the selected distribution
determines the set of knots defining the B-splines as well as the set
of collocation points.
After the parameters for collocation have been specified, the user next
clicks Initialize to begin the process of solving the BVP. A
predetermined starting guess x0 is used to
initialize Broyden's method for solving the system of equations, and
the corresponding approximate solution to the BVP,
v(t, x0), is drawn
in blue. The user then clicks Iterate repeatedly to execute
successive iterations of Broyden's method to solve the system of
equations. For each iterate xk, the
corresponding function v(t,
xk) 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 basis
functions or different numbers or distribution of collocation points,
see the alternative Collocation Method
module.
Reference: Michael T. Heath, Scientific Computing,
An Introductory Survey, 2nd edition, McGraw-Hill, New York,
2002. See Section 10.5.
Developers: Evan VanderZee and Michael Heath
Department of Computer Science