24f if the initial data is consistent with the

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Unformatted text preview: 1 2 3 : : : N ; 1: Had we solved the wave equation (9.1.2) instead of the heat equation (9.1.1) using a piecewise-linear nite element basis, we would have found the discrete system Mc + Kc = 0 (9.2.5) with p in (9.2.4c) replaced by c2. The resulting initial value problems (IVPs) for the ordinary di erential equations (ODEs) (9.2.4a) or (9.2.5) typically have to be integrated numerically. There are several excellent software packages for solving IVPs for ODEs. When such ODE software is used with a nite element or nite di erence spatial discretization, the resulting procedure is called the method of lines. Thus, the nodes of the nite elements appear to be \lines" in the time direction and, as shown in Figure 9.2.2 for a one-dimensional problem, the temporal integration proceeds along these lines. 8 Parabolic Problems t x 0 = x0 x1 xj xN-1 xN = 1 Figure 9.2.2: \Lines" for a method of lines integration of a one-dimensional problem. Using the ODE software, solutions are calculated in a series of time steps (0 t1], (t1 t2], : : : . Methods fall into two types. Those that only require knowledge of the solution at time tn in order to obtain a solution at time tn+1 are called one-step methods. Correspondingly, methods that require information about the solution at tn and several times prior to tn are called multistep methods. Excellent texts on the subject are available 2, 6, 7, 8]. One-step methods are Runge-Kutta methods while the common multistep methods are Adams or backward di erence methods. Software based on these methods automatically adjusts the time steps and may also automatically vary the order of accuracy of a class of methods in order to satisfy a prescribed local error tolerance, minimize computational cost, and maintain numerical e ciency. The choice of a one-step or multistep method will depend on several factors. Generally, Runge-Kutta methods are preferred when time integration is simple relative to the spatial solution. Multistep methods become more e cient for complex nonlinear problems. Implicit Runge-Kutta methods may be e cient for problems with high-frequency oscillations. The ODEs that arise from the nite element discretization of parabolic problems are \sti " 2, 8] so backward di erence methods are the preferred multistep methods. Most ODE software 2, 7, 8] addresses rst-order IVPs of the explicit form _ y(t) = f (t y(t)) y(0) = y0: (9.2.6) Second-order systems such as (9.2.5) would have to be written as a rst-order system by, e.g., letting _ d=c 9.2. Semi-Discrete Galerkin Problems and, hence, obtaining 9 _ c=d: _ ;Kc Md Unfortunately, systems having the form of (9.2.4a) or the one above are implicit and would require inverting or lumping M in order to put them into the standard explicit form (9.2.6). Inverting M is not terribly di cult when M is constant or independent of t however, it would be ine cient for nonlinear problems and impossible when M is singular. The latter case can occur when, e.g., a heat conduction and a potential problem are solved simultaneously. Codes for di erential-algebraic equations (DAEs) dir...
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