SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Weak second order mean-square stable integrators for stiff stochastic differential equations

G. Vilmart, A. Abdulle and K.C. Zygalakis

Abstract
We present two families of integrators for stiff Itô stochastic differential equations which exhibit simultaneously favourable mean-square stability properties and weak second order of accuracy. The first integrators are implicit with respect to the drift function and are shown to be mean-square A-stable for the usual complex scalar linear test problem with multiplicative noise. The second integrators are fully explicit and still have extended mean-square stability domains. These constructions inspired the design of a ``swiss-knife'' integrator for stiff diffusion-advection-reaction problems with noise.

Bibliography
[1] A. Abdulle and G. Vilmart, PIROCK: a swiss-knife partitioned implicit-explicit orthogonal Runge-Kutta Chebyshev integrator for stiff diffusion-advection-reaction problems with or without noise, J. Comp. Phys. 242 (2013), 869-888.
[2] A. Abdulle, G. Vilmart and K.C. Zygalakis, Weak second order explicit stabilized methods for stiff stochastic differential equations, to appear in SIAM J. Sci. Comput.
[3] A. Abdulle, G. Vilmart and K.C. Zygalakis, Mean-square A-stable diagonally drift-implicit integrators of weak second order for stiff Itô stochastic differential equations, to appear in BIT.

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