Finite Dierence Methods
Summary and References to the textbook Borse,
G.J.: Numerical Methods with MATLAB
Anders Sjoberg
May 28, 1998
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1. Examples of mathematical models
The equation au = @=@x (k @u=@x) + @=@y (k @u=@y) + f where a and
k are functions of x and t, and u describes the heat ow in a plate. The
equation can be simplied in the following ways:
(a) to a linear problem if a k and f do not depend on u,
(b) to a problem with constant coecients (a k and f are constants),
(c) to a one-dimensional problem, and
(d) to a stationary problem, by setting u = 0:
When a complicated mathematical model is of interest it may be relevant
to study a simplied model.
2. Classication
The classical model equations are
u =u
The heat equation (parabolic)
u =u
The wave equation (hyperbolic)
u + u = 0
Kreiss' equation (hyperbolic)
u +u =0
Laplace's equation (elliptic)
u + u = f (x y ) Poisson's equation (elliptic)
See Borse, page 476-478. Note that the classication is the same also if
the coordinate system is changed. The three types of PDE:s have radically
dierent properties and describe dierent physical phenomena.
For systems of PDE:s, the classication is determined by the eigenvalues of
the principal coecient matrix.
3. Well-posed PDE-problems
An important concept is well-posed problems. The type of the PDE and
the boundary and initial conditions have to match each other. If a problem is well-posed or not can be shown by the \energy method" (parabolic
and hyperbolic equations), the maximum principle (parabolic and elliptic),
separation of variables (parabolic and hyperbolic), and the method of characteristics (hyperbolic).
4. Characteristic curves
See Borse, page 490-492. The characteristics show in which direction and
how fast the information propagates.
5. Dierence approximations for time dependent problems
The important concepts are consistency, truncation error, convergence and
stability. The Lax-Richtmyer equivalence theorem is fundamental.
Three kinds of stability tests are mentioned: perturbation scheme, the
Fourier method, and the energy method.
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Implicit dierence methods give rise to a system of equations in every time
step. The advantage of implicit methods is that the stability properties
are better which means that a larger time step can be used compared to
explicit methods.
If there are three time levels or more in the dierence scheme, a specic
starting method is needed. If the boundary conditions contain derivatives
new, computational, problems arise. The stability theory becomes more
dicult as well.
6. Common dierence methods for time dependent problems
See Borse, pages 481-487 and 482-494. Methods of higher order, methods
for systems of PDE:s and methods for problems with two spatial dimensions
can also be dened.
7. Generalizations
Time dependent PDE:s may contain many diculties: complicated geometry, unlimited computational area, complicated boundary conditions and
initial conditions, several dimensions, non-linearities, dierent time scales
etc. However, the basic ideas and concepts of nite dierence methods can
be used also in these cases.
8. Elliptic problems
Dirichlet's problem for the Laplace equation
(
u + u = 0 in an area (1)
u(x y ) = g (x y ) on the boundary
is our simplest model problem and it is well-posed. The solution has its
maximum and the minimum on the boundary.
In the Neumann problem a linear combination of u and the normal derivative of u is specied on the boundary.
Elliptic PDE-problems can be solved by nite dierence methods or nite
element methods (FEM).
9. Dierence methods for elliptic problems
When derivatives are replaced by dierence quotients a system of equations
with the unknown values of u, is created. Since the dierence operators
only connect neighboring grid points, the coecient matrix will be sparse.
A simple example is when the standard ve-point-formula is applied to the
problem (1) above. This yields a block-tridiagonal system where the blocks
on the diagonal are tridiagonal matrices. On each row a maximum of ve
nonzero elements can be found.
The large sparse linear systems can be solved by a direct method or an
iterative method. With a direct method zeros may change into non-zeros,
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i.e. ll-in occurs. A smart numbering of the unknowns may reduce the
ll-in, but normally iterative methods are considered to be more ecient
for large sparse systems.
If the boundaries are curved, the dierence approximations have to be modied, in order to preserve the order of approximation.
The same ideas and methods used for the model equations may be used
for more general elliptic PDE-problems. Note that there is no problem
of stability when solving elliptic PDE:s, as when solving parabolic and
hyperbolic PDE:s.
Reference literature
Golub, G.H. & Ortega, J.M.: Scientic Computing and Dierential Equations, Academic Press, 1992 (ISBN 0-12-289255-0)
Morton, K.W. & Mayers. D.F.: Numerical Solution of Partial Dierential
Equations, Cambridge University Press, 1994 (ISBN 0 521 42922 6)
Strikwerda, J.C.: Finite Dierence Schemes and Partial Dierential Equations, Wadsworth & Brooks/Cole, 1989.
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