1 start the code with a rst order method for the

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Unformatted text preview: of order k before considering an order change. 56 4. Starting values when changing step size or order may be computed by a Taylor's series when using the Nordsieck representation (5.7.10). There are also variable step size LMMs ( 12], Section III.5). The coe cients of these formulas are functions of h. The variable step formulas are generally more stable than the uniform step formulas but are less e cient. Some available LMM codes are 1. DEABM is a modi cation of a code developed by Shampine and Gordon 16]. This code implements a variable step size divided di erence representation of the Adams formulas. It uses a PECE strategy and includes order variation. Let's go over the order variation scheme to illustrate the technique. After performing a step with an order k method, compute estimates dk;2, dk;1, and dk of the local error of n n n solutions with methods of order k ; 2, k ; 1, and k, respectively. Reduce the order to k ; 1 if max kdk;2k kdk;1k] kdk jj: n n n (5.7.14) Increase the order when a step is successful, (5.7.14) is violated, and a constant step size is used. (Remember, these are variable step-size methods.) Estimates of local discretization errors are obtained using approaches similar to (5.7.13). Norms are used for vector systems. 2. EPISODE, developed by Byrne and Hindmarsh 7], is a variable step, variable order implementation of the constant-step Adams and BDF methods using the Nordsieck representation. For nonsti problems, functional iteration uses a P (EC ) strategy. Newton's method is used for sti problems. 3. LSODE is another implementation of the constant-step Adams and BDF methods. It is similar to EPISODE. 4. VODE is a variable step, variable order code based on the variable-step Adams and BDF formulas. It was developed by Brown et al. 5] and is an extension of EPISODE. 57 Code Symbol Storage/eqn. DEABM ---22 EPISODE -..-..18 LSODE -.-.-.17 D02CAF : : : : : : . 19 DOPRI8 ; ; ; ; ; 9 Table 5.7.1: Legends for Figures 5.7.1 and 5.7.2 and code storage 12]. 5. DASSL, devel...
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