Linear and non-linear stability for general linear methods
Copyright policy )Linear and non-linear stability for general linear methods
Stability in a numerical method prevents the growth of the approximate solution of a differential equation for which the solution is bounded. A variety of concepts of stability have been developed. For one-leg methods, two extremes, namely A-stability and algebraic stability together with a spectrum between these extremes are equivalent. For Runge-Kutta methods, A-stability is distinguished from the others. In this paper it is shown that general linear methods distinguish more of these concepts. In particular, A-stability is weaker than weak AN- stability which is weaker than strong AN-stability, while Euclidean AN- stability is equivalent to algebraic stability for such methods. Sufficient conditions for the equivalence of strong AN-stability to algebraic stability are promised in a later paper.
- University of Auckland New Zealand
Runge-Kutta methods, one-leg methods, linear methods, strong AN- stability, algebraic stability, Nonlinear ordinary differential equations and systems, A-stability, Numerical methods for initial value problems involving ordinary differential equations, Stability and convergence of numerical methods for ordinary differential equations, AN-stability, comparison of stability concepts
Runge-Kutta methods, one-leg methods, linear methods, strong AN- stability, algebraic stability, Nonlinear ordinary differential equations and systems, A-stability, Numerical methods for initial value problems involving ordinary differential equations, Stability and convergence of numerical methods for ordinary differential equations, AN-stability, comparison of stability concepts
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