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The Stability of Numerical Methods for Second Order Ordinary Differential Equations
C. William Gear
SIAM Journal on Numerical Analysis
Vol. 15, No. 1 (Feb., 1978), pp. 188-197
Published by: Society for Industrial and Applied Mathematics
Stable URL: http://www.jstor.org/stable/2156571
Page Count: 10
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An important characterization of a numerical method for first order ODE's is the region of absolute stability. If all eigenvalues of the linear problem y' = Ay are inside this region, the numerical method is stable. If the second order system y" = 2Ay' - By is solved as a first order system, the same result applies to the eigenvalues of the generalized eigenvalue problem λ2I - 2λ A + B. No such region exists for general methods for second order equations, but in some cases a region of absolute stability can be defined for methods for the single second order equation y" = 2ay' - by. The absence of a region of absolute stability can occur when different members of a system of first order equations are solved by different methods.
SIAM Journal on Numerical Analysis © 1978 Society for Industrial and Applied Mathematics