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Butcher, J. C.

High order A-stable numerical methods for stiff problems. (English) Zbl 1203.65106

J. Sci. Comput. 25, No. 1-2, 51-66 (2005).
Summary: Stiff problems pose special computational difficulties because explicit methods cannot solve these problems without severe limitations on the stepsize. This idea is illustrated using a contrived linear test problem and a discretized diffusion problem. Even though the Euler method can solve these problems if the stepsize is small enough, there is no such limitation for the implicit Euler method. To obtain high order A-stable methods, it is traditional to turn to Runge-Kutta methods or to linear multistep methods. Each of these has limitations of one sort or another and we consider, as a middle ground, the use of general linear (or multivalue multistage) methods. Methods possessing the property of inherent Runge-Kutta stability are identified as promising methods within this large class, and an example of one of these methods is discussed. The method in question, even though it has four stages, out-performs the implicit Euler method if sufficient accuracy is required, because of its higher order.

Cited in 8 Documents

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations

Keywords:

stiff differential equations; A-stable methods; general linear methods; inherent RK stability

Software:

RODAS
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\textit{J. C. Butcher}, J. Sci. Comput. 25, No. 1--2, 51--66 (2005; Zbl 1203.65106)
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References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.