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The latter case can occur when eg a heat conduction

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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 to exist. 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 n+1 n 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 n+1 n (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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