# 1 start the code with a rst order method for the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online