Unformatted text preview: ectly address the solution of implicit systems of the form
f (t y(t) y(t)) = 0 y(0) = y0: (9.2.7) One of the best of these is the code DASSL written by Petzold 3]. DASSL uses variablestep, variable-order backward di erence methods to solve problems without needing M;1
Let us illustrate these concepts by applying some simple one-step schemes to problems
having the forms (9.2.1) or (9.2.4). However, implementation of these simple methods
is only justi ed in certain special circumstances. In most cases, it is far better to use
existing ODE software in a method of lines framework.
For simplicity, we'll assume that all boundary data is homogeneous so that the boundN
ary inner product in (9.2.2a) vanishes. Selecting a nite-dimensional space S0 H01, we
then determine U as the solution of
(V Ut ) + A(V U ) = (V f ) 8v 2 S0N : (9.2.8) Evaluation leads to ODEs having the form of (9.2.4a) regardless of whether or not the
system is one-dimensional or the coe cients are constant. The actual matrices M and K
and load vector l will, of course, di er from those of Example 9.2.1 in these cases. The
systems (9.2.4a) or (9.2.8) are called semi-discrete Galerkin equations because time has
not yet been discretized.
We discretize time into a sequence of time slices (tn tn+1] of duration t with tn =
n t, n = 0 1 : : : . For this discussion, no generality is lost by considering uniform time
steps. Let: u(x tn) be the exact solution of the Galerkin problem (9.2.2a) at t = tn.
U (x tn ) be the exact solution of the semi-discrete Galerkin problem (9.2.8) at t = tn.
U n (x) be the approximation of U (x tn) obtained by ODE software. 10 Parabolic Problems cj (tn ) be the Galerkin coe cient at t = tn thus, for a one-dimensional problem
U (x tn ) = X N ;1
j =1 cj (tn) j (x): For a Lagrangian basis, cj (tn) = U (xj tn). cn be the approximation of cj (tn) obtained by ODE software. For a one-dimensional
j problem U n (x) = X N ;1
j =1 cn j (x):
j We suppose that all solutions are known at time tn and that we seek to determine
them at time tn+1. The simplest numerical scheme for doing this is the forward Euler
method where (9.2.8) is evaluated at time tn and
Ut (x tn) U (x) ; U (x) :
t (9.2.9) A simple Taylor's series argument reveals that the local discretization error of such an
approximation is O( t). Substituting (9.2.9) into (9.2.8) yields
(V U ; U ) + A(V U n ) = (V f n)
t 8v 2 S0N : (9.2.10a) Evaluation of the inner products leads to Mc n+1 ; cn + Kncn = ln: (9.2.10b) t We have allowed the sti ness matrix and load vector to be functions of time. The mass
matrix would always be independent of time for di erential equations having the explicit
form of (9.2.1a) as long as the spatial nite element mesh does not vary with time.
The ODEs (9.2.10a,b) are implicit unless M is lumped. If lumping were used and, e.g.,
M hI then cn+1 would be determined as cn+1 = cn + ht ln ; Kncn]: Assuming that cn is known, we can determine cn+1 by inverting M.
Using the backward Euler meth...
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- Spring '14
- The Land, Tn, Boundary value problem, Numerical ordinary differential equations, nite element, parabolic problems