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Stability of Two-Step Methods for Variable Integration Steps

1983; Society for Industrial and Applied Mathematics; Volume: 20; Issue: 5 Linguagem: Inglês

10.1137/0720076

1095-7170

Germund Dahlquist, Werner Liniger, Olavi Nevanlinna,

Computational Fluid Dynamics and Aerodynamics

Two of the most commonly used methods, the trapezoidal rule and the two-step backward differentiation method, both have drawbacks when applied to difficult stiff problems. The trapezoidal rule does not sufficiently damp the stiff components and the backward differentiation method is unstable for certain stable variable-coefficient problems with variable-steps. In this paper we show that there exists a one-parameter family of two-step, second-order one-leg methods which are stable for any dissipative nonlinear system and for any test problem of the form $\dot x = \lambda (t)x$, $\operatorname{Re} \lambda (t) \leq 0$, using arbitrary step sequences.