ISSN:
1572-9125
Keywords:
Linear stability
;
iteration schemes
;
implicit Runge-Kutta methods
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract This article examines stability properties of some linear iterative schemes that have been proposed for the solution of the nonlinear algebraic equations arising in the use of implicit Runge-Kutta methods to solve a differential systemx′ =f(x). Each iteration step requires the solution of a set of linear equations, with constant matrixI −hλJ, whereJ is the Jacobian off evaluated at some fixed point. It is shown that the stability properties of a Runge-Kutta method can be preserved only if λ is an eigenvalue of the coefficient matrixA. SupposeA has minimal polynomial (x − λ) m p(x),p(λ) ≠ 0. Then stability can be preserved only if the order of the method is at mostm + 2 (at mostm + 1 except for one case).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01740545
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