Numerical time integration of stochastic differential equations

Time integration methods for stochastic differential equations are considered and compared for two models, namely the double pendulum and the Lagrangian drifter. The goal is to investigate how a choice in scheme has impact on the accuracy and correctness of the results. This is done by analysing the disturbance in the Poincaré sections for the double pendulum model, and using statistical tools for the model on the Lagrangian drifter. Two methods with a deterministic counter part of first order are considered, namely Euler-Maruyama and stochastic sympectic Euler, as well as two methods with deterministic counter part of a higher order, stochastic Störmer-Verlet and stochastic Runge-Kutta. The main result is that the result heavily relies on the type of model analysed. The stochastic Störmer-Verlet preserves the structure of the Poincaré section of the double pendulum better, and manages to preserve the total energy best. On the other hand, for the Lagrangian drifter the Euler-Maruyama method and the stochastic Runge-Kutta method yield generally identical dynamics.