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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.