Computer simulation of liquids
Computer simulation of liquids
A first course in the numerical analysis of differential equations
A first course in the numerical analysis of differential equations
Numerical Methods for Second-Order Stochastic Differential Equations
SIAM Journal on Scientific Computing
Calculating effective diffusivities in the limit of vanishing molecular diffusion
Journal of Computational Physics
Modified Energy for Split-Step Methods Applied to the Linear Schrödinger Equation
SIAM Journal on Numerical Analysis
Long-Run Accuracy of Variational Integrators in the Stochastic Context
SIAM Journal on Numerical Analysis
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In this paper we describe a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence. Modified equations are derived for a variety of numerical methods, such as the Euler or the Milstein method. Existence of higher order modified equations is also discussed. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we derive an SDE which the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations is also discussed.