Abstract
The aim of this paper is the analysis of exponential mean-square stability properties of nonlinear stochastic linear multistep methods. In particular it is known that, under certain hypothesis on the drift and diffusion terms of the equation, exponential mean-square contractivity is visible: the qualitative feature of the exact problem is here analysed under the numerical perspective, to understand whether a stochastic linear multistep method can provide an analogous behaviour and which restrictions on the employed stepsize should be imposed in order to reproduce the contractive behaviour. Numerical experiments confirming the theoretical analysis are also given.
Highlights
Adv Comput Math (2021) 47: 55 the property of the continuous problem is detected and identified through sufficient conditions, we aim to provide their numerical counterparts, equivalent to conditional stability properties for the numerical method
For a well-posed deterministic initial value problem based on the ordinary differential equation y (t) = f (y(t))
Where ξ ∈ C and Re(ξ ) ≤ 0, considered for the first time by Dahlquist [6]. The solution of this simple problem remains bounded when t goes to infinity and one needs to require that the numerical solution possesses an analogous stability property to that displayed by the exact solution: this fact is at the basis of the linear stability analysis of numerical methods
Summary
Adv Comput Math (2021) 47: 55 the property of the continuous problem is detected and identified through sufficient conditions, we aim to provide their numerical counterparts, equivalent to conditional stability properties for the numerical method. 1.1 Linear and nonlinear stability issues for deterministic differential equations. Where ξ ∈ C and Re(ξ ) ≤ 0, considered for the first time by Dahlquist [6] The solution of this simple problem remains bounded when t goes to infinity and one needs to require that the numerical solution possesses an analogous stability property to that displayed by the exact solution: this fact is at the basis of the linear stability analysis of numerical methods (see [4] and references therein). Extensive research has been carried out in order to provide numerical methods generating contractive numerical solutions for dissipative problems, giving rise to the notions of AN-stability, G-stability, algebraic stability and so on (see [4, 15] and references therein)
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