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The Study of Some Linear Multi-step Methods with the More Bigger Absolutely Stable Region

Author: YinFangZuo
Tutor: LiuBo
School: Jilin University
Course: Computational Mathematics
Keywords: Stiff differential equations Linear multi-step Absolutely stable area
CLC: O241.8
Type: Master's thesis
Year: 2011
Downloads: 25
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Abstract


Often encountered stiff differential equations in the process of control systems engineering, electronic networks, biological, physical and chemical kinetics. Stiff differential equations suitable for solving the implicit method, which is significantly different from the ordinary differential equation of a so Stiff differential equations with implicit method with a larger stable region in recent years, many of the theory of rigid algorithm generated a strong interest and in-depth inquiry, the construction of efficient numerical algorithm typical algorithm is one of the important problem of linear multi-step Runge-Kutta method for solving initial value problems of ordinary differential equations, linear multi-step form is simple, easy calculation, is one of the widely used method of Runge- Kutta methods are a class of important classical algorithm for solving stiff differential equations. general form of linear multistep methods, the research is absolutely stable region greater linear multi-step a good calculation method should be a small amount of calculation, and obtain meet the specified accuracy requirements of the calculations, so the error is an important indicator of the discrimination of pros and cons. consider here only the theoretical error that truncation error is not enough, you must also consider the calculation error. compatibility determines the size of the local discretization error, cause poor accuracy, which is not enough to consider only the zero stability in order to achieve a given accuracy, find the appropriate length calculation step, the introduction of the concept of absolute stability, this stability is also guiding significance for the actual calculation. can see the size and shape of the absolute stability region is important. Valuation A numerical method and the method by comparison between the absolute stability of the larger the area, the specific characteristic values, step limit smaller, especially stiff differential equations, eigenvalues, when the absolute stability region is far less than the characteristic value, the step size will limit small, resulting in the above-mentioned problems, so we hope that the absolute stability of the region as large as possible to minimize the step limit this paper first describes the characteristics of rigidity problem, recalled some important algorithms for solving stiff problems, including the famous Gear construct a class of k-step k-order linear multi-step formula and the Runge-Kutta method, as well as a brief introduction in recent years about solving stiff problems constructor. On this basis, the new algorithm are linear multistep method in the range of one-dimensional, through the analysis of absolute stability interval, that is absolutely stable region real negative Axle intersect for analysis. raised a family of two-step with a free parameter third-order linear multi-step method, three, four, containing two parameters-order linear multistep method, and contains a parameter of the three-step Third-order. Adams-Moulton method in the case of methods to meet the zero stability condition, the selection of free parameters, to seek the greatest possible absolute stability range. Draw the boundary locus method corresponds to an absolute stable region, and absolute stable region with some of the traditional method of comparison results show that the obtained having greater absolute stable region, more than Adams-Moulton methods suitable rigid Differential Gear method, although not its absolute stability region, but than its high accuracy for solving some stiff differential equations or meaningful. Finally, numerical experiments were one-dimensional and two-dimensional stiff differential equations (group), the data is consistent with the theoretical results.

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Numerical Analysis > The numerical solution of differential equations, integral equations
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