page_title: Local linearization method
section_title: Order 2 SLL-RK schemes
hierarchical_section_title: Local linearization method / Strong LL methods for SDEs / Local Linearization schemes / Order 2 SLL-RK schemes
caption_attribution_description: English: img SSDEs
context_page_description: In mathematics, specifically in numerical analysis, the Local Linearization method is a general strategy for designing numerical integrators for differential equations based on a local linearization of the given equation on consecutive time intervals. The numerical integrators are then iteratively defined as the solution of the resulting piecewise linear equation at the end of each consecutive interval. The LL method has been developed for a variety of equations such as the ordinary, delayed, random and stochastic differential equations. The LL integrators are key component in the implementation of inference methods for the estimation of unknown parameters and unobserved variables of differential equations given time series of observations. The LL schemes are ideals to deal with complex models in a variety of fields as neuroscience, finance, forestry management, control engineering, mathematical statistics, etc.
context_section_description: For SDEs with a single Wiener noise (m=1)
  where
  with . Here,  for low dimensional SDEs, and  for large systems of SDEs, where , , ,  and  are defined as in the order-2 SLL-Taylor schemes, p+q>1 and .
caption: Fig. 4, Top: Evolution of domains in the phase plane of the harmonic oscillator (7.6), with ε=0 and ω=σ=1. Images of the initial unit circle (green) are obtained at three time moments T by the exact solution (black), and by the schemes SLL1 (blue) and Implicit Euler (red) with h=0.05. Bottom: Expected value of the energy (solid line) along the solution of the nonlinear oscillator (7.6), with ε=1 and ω=100, and its approximation (circles) computed via Monte Carlo with 10000 simulations of the SLL1 scheme with h=1/2 and p=q=6.
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