Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
1
Ordinary Dierential Equations
1.0
Mathematical Background
1.0.1
Smoothness
Denition 1.1 A function f dened on [a, b] is continuous at x0 [a, b] if lim f (x) =
xx0
f (x0 ).
Remark Note that this implies existence of the quantities on both sides of the eq
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
12
Galerkin and Ritz Methods for Elliptic PDEs
12.1
Galerkin Method
We begin by introducing a generalization of the collocation method we saw earlier for
twopoint boundary value problems. Consider the elliptic PDE
Lu(x) = f (x),
(110)
where L is a linear
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
11
Pseudospectral Methods for TwoPoint BVPs
Another class of very accurate numerical methods for BVPs (as well as many timedependent PDEs) are the socalled spectral or pseudospectral methods. The basic idea
is similar to the collocation method described
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
10
Gaussian Quadrature
So far we have encountered the NewtonCotes formulas
n
b
f (x)dx
a
b
Ai f (xi ),
i (x)dx,
Ai =
a
i=0
which are exact if f is a polynomial of degree at most n.
It is important to note that in the derivation of the NewtonCotes formu
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
9
Boundary Value Problems: Collocation
We now present a dierent type of numerical method that will yield the approximate
solution of a boundary value problem in the form of a function, as opposed to the
set of discrete points resulting from the methods st
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
8
Boundary Value Problems for PDEs
Before we specialize to boundary value problems for PDEs which only make sense
for elliptic equations we need to explain the terminology elliptic.
8.1
Classication of Partial Dierential Equations
We therefore consider ge
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
7
Boundary Value Problems for ODEs
Boundary value problems for ODEs are not covered in the textbook. We discuss this
important subject in the scalar case (single equation) only.
7.1
Boundary Value Problems: Theory
We now consider secondorder boundary val
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
6
Nonlinear Algebraic Systems
We saw earlier that the implementation of implicit IVP solvers often requires the solution of nonlinear systems of algebraic equations. Nonlinear systems also come up in
the solution of boundary values problems of ODEs and PD
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
5
5.1
Error Control
The Milne Device and PredictorCorrector Methods
We already discussed the basic idea of the predictorcorrector approach in Section 2.
In particular, there we gave the following algorithm that made use of the 2ndorder
AB and AM (trape
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
4
Stiness and Stability
In addition to having a stable problem, i.e., a problem for which small changes in the
initial conditions elicit only small changes in the solution, there are two basic notions
of numerical stability. The rst notion of stability is
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
3
RungeKutta Methods
In contrast to the multistep methods of the previous section, RungeKutta methods
are singlestep methods however, with multiple stages per step. They are motivated
by the dependence of the Taylor methods on the specic IVP. These new
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
2
Multistep Methods
Up to now, all methods we studied were single step methods, i.e., the value y n+1 was
found using information only from the previous time level tn . Now we will consider
socalled multistep methods, i.e., more of the history of the sol
Numerical Methods for Differential Equations/Computational Mathematics II
MATH 478

Spring 2007
13
Fast Fourier Transform (FFT)
The fast Fourier transform (FFT) is an algorithm for the ecient implementation of
the discrete Fourier transform. We begin our discussion once more with the continuous
Fourier transform.
13.1
Continuous and Discrete Fourier