# A General Implicit Splitting for Stabilizing Numerical Simulations of Langevin Equations

## TL;DR

This paper introduces a versatile second-order weakly accurate splitting method to stabilize Monte Carlo simulations of Langevin equations, encompassing explicit, semi-implicit, and trapezoidal schemes, with proven generalizations.

## Contribution

It presents a unified, general splitting approach for Langevin equations that enhances stability and accuracy in stochastic simulation methods.

## Key findings

- The method achieves second-order weak accuracy.
- It includes explicit, semi-implicit, and trapezoidal schemes.
- The semi-implicit method is proven and generalizable.

## Abstract

In this paper is described a general 2-nd order accurate (weak sense) procedure for stablizing Monte-Carlo simulations of Ito stochastic differential equations. The splitting procedure includes explicit Runge-Kutta methods, semi-implicit methods, and trapezoidal Rule. We prove the semi-implicit method of Oettinger and note that it may be generalized for arbitrary splittings.

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Source: https://tomesphere.com/paper/hep-lat/9602008