A-Stable Method for the Solution of the Cauchy Problem for Stiff Systems of Ordinary Differential Equations

Artem'ev, S. S.; Demidov, G. V.

doi:10.1007/978-3-662-38527-2_36

Cited by 4 publications

(3 citation statements)

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“…In order to obtain the reference formula of third order, they used an extra two function evaluations, which are saved if the step size remains unchanged. If the step size changes, the process becomes (2,3,6,3), however the algorithms for Type 1 methods usually try to keep the step size constant for reasonable periods, so that the process is essentially (2, 2, 4, 2) for a good part of the integration. This (2, 2, 4, 2) method thus requires slightly more work than the (2, 2, 5, 0) method described in this paper and does not share the same level of stability.…”

Section: =1mentioning

confidence: 99%

“…For example, if we use a rational polynomial function such as (1. 3) ' + 1 «oWn« r+l I + 2 Kbkj« q=\ where a , q = 1,... ,r, and bq, q = 1, ...,r+ I, and a{j are real scalars, and / > 1 for stability reasons (see Theorem 1), then at every step one or more matrices must be factorized and one or more systems of linear algebraic equations must be solved with this factorized matrix. In the interests of efficiency, we will use a matrix of the form…”

mentioning

confidence: 99%

“…Type 2. To obtain consistency for Type 2 processes, we expand the Jacobian J"_k = df(y(x))/dy \x=Xn_k in the Taylor series 00 (3)(4)(5)(6)(7)(8)(9) Jn-k = Jn+ 2 \/j\ (-pkKYj\J\ 7=1 where xn = xn_k + pkh", xn+, = x" + «", with k (3.10) P* = 2 *■-//*«• *>l,/»o = 0.…”

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confidence: 99%

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Two classes of internally 𝑆-stable generalized Runge-Kutta processes which remain consistent with an inaccurate Jacobian

Day¹,

Murthy²

1982

Math. Comp.

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Abstract. Generalized Runge-Kutta Processes for stiff systems of ordinary differential equations usually require an accurate evaluation of a Jacobian at every step. However, it is possible to derive processes which are Internally S-stable when an accurate Jacobian is used but still remain consistent and highly stable if an approximate Jacobian is used. It is shown that these processes require at least as many function evaluations as an explicit Runge-Kutta process of the same order, and second and third order processes are developed. A second class of Generalized Runge-Kutta is introduced which requires that the Jacobian be evaluated accurately less than once every step. A third order process of this class is developed, and all three methods contain an error estimator similar to those of Fehlberg or England.

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Section: =1mentioning

confidence: 99%

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confidence: 99%