# zbMATH — the first resource for mathematics

Stiffness in numerical initial-value problems: $$A$$ and $$L$$-stability of numerical methods. (English) Zbl 1007.65053
Summary: The main aim of this note is to help students to gain an insight into the asymptotic stability concept by means of visual representation of the stability regions of different numerical methods. This facilitates understanding of the meaning of stability for constant step sizes and the related concept of stiffness in numerical initial-value problems. Moreover, the distinction between $$A$$ and $$L$$-stability of numerical methods can be more easily understood.
##### MSC:
 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems
Full Text:
##### References:
 [1] DOI: 10.1073/pnas.38.3.235 · Zbl 0046.13602 [2] HAIRER E., Solving Differential Equations II: Stiff and Differential-Algebraic Problems (1991) · Zbl 0729.65051 [3] LAMBERT J. D., Numerical Methods for Ordinary Differential Systems (1991) · Zbl 0745.65049 [4] BRUGNANO L., Discrete Impulsive Syst. 2 pp 317– (1999) [5] HENRICI P., Discrete Variable Methods in Ordinary Differential Equations (1962) · Zbl 0112.34901 [6] ASCHER U. M., Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations (1998) · Zbl 0908.65055 [7] BRUGNANO L., Solving Differential Problems by Multistep Initial and Boundary Methods (1998) · Zbl 0934.65074 [8] BUTCHER J. C., The Numerical Analysis of Ordinary Differential Equations Runge-Kutta and General Linear Methods (1987) · Zbl 0616.65072 [9] GAUTSCHI W., Numerical Analysis. An Introduction (1997) · Zbl 0877.65001 [10] ISERLES A., A First Course in the Numerical Analysis of Differential Equations (1996) · Zbl 0841.65001 [11] SHAMPINE L. F., Numerical Solution of Ordinary Differential Equations (1994) · Zbl 0832.65063 [12] DOI: 10.1080/0020739940250402 · Zbl 0822.65002 [13] MATHEWS J. H., Comput. Educ. J. pp 58– (1994) [14] MATHEWS J. H., Numerical Methods Using MATLAB,, 3. ed. (1999) [15] QUARTERONI A., Numerical Mathematics (2000) · Zbl 0957.65001 [16] VAN LOAN C. F., Introduction to Scientific Computing, a Matrix-Vector Approach Using MATLAB (1997) [17] GARCIA A. L., Numerical Methods for Physics, 2. ed. (2000) [18] GEAR C. W., Numerical Initial Value Problems in Ordinary Differential Equations (1971) · Zbl 1145.65316 [19] DOI: 10.1137/S1064827594276424 · Zbl 0868.65040
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.