In numerical analysis, the local linearization (LL) method is a general strategy for designing numerical integrators for differential equations based on a local (piecewise) 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 (potentially noisy) 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.
Background Differential equations have become an important mathematical tool for describing the time evolution of several phenomenon, e.g., rotation of the planets around the sun, the dynamic of assets prices in the market, the fire of neurons, the propagation of epidemics, etc. However, since the exact solutions of these equations are usually unknown, numerical approximations to them obtained by numerical integrators are necessary. Currently, many applications in engineering and applied sciences focused in dynamical studies demand the developing of efficient numerical integrators that preserve, as much as possible, the dynamics of these equations. With this main motivation, the Local Linearization integrators have been developed.
High-order local linearization method High-order local linearization (HOLL) method is a generalization of the Local Linearization method oriented to obtain high-order integrators for differential equations that preserve the stability and dynamics of the linear equations. The integrators are obtained by splitting, on consecutive time intervals, the solution x of the original equation in two parts: the solution z of the locally linearized equation plus a high-order approximation of the residual r = x − z {\displaystyle \mathbf {r} =\mathbf {x} -\mathbf {z} } .
Local linearization scheme A Local Linearization (LL) scheme is the final recursive algorithm that allows the numerical implementation of a discretization derived from the LL or HOLL method for a class of differential equations.
LL methods for ODEs Consider the d-dimensional Ordinary Differential Equation (ODE)
with initial condition x ( t 0 ) = x 0 {\displaystyle \mathbf {x} (t_{0})=\mathbf {x} _{0}} , where f {\displaystyle \mathbf {f} } is a differentiable function. Let ( t ) h = { t n : n = 0 , . . , N } {\displaystyle \left(t\right)_{h}=\{t_{n}:n=0,..,N\}} be a time discretization of the time interval [ t 0 , T ] {\displaystyle [t_{0},T]} with maximum stepsize h such that t n < t n + 1 {\displaystyle t_{n}<t_{n+1}} and h n = t n + 1 − t n ≤ h {\displaystyle h_{n}=t_{n+1}-t_{n}\leq h} . After the local linearization of the equation (4.1) at the time step t n {\displaystyle t_{n}} the variation of constants formula yields
where
results from the linear approximation, and
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