Local linearization method: Difference between revisions

Revision as of 16:41, 8 December 2020

Revision as of 16:44, 8 December 2020

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=== Local Linearization schemes ===

[[File:FigureRDE1.png|thumb|391x391px377x377px|'''Fig. 3''' Phase portrait of trajectories of the ''Euler'' and ''LL'' schemes in the integration of the nonlinear RDE (6.2)-(6.3) with step size ''h=1/32'', and ''p=q=6''.]]

Depending of the approximations <math>\widetilde{\mathbf{\xi }}</math> to the process <math>\mathbf{\xi }</math> and of the algorithm to compute <math>\mathbf{\phi }</math>, different Local Linearizations schemes can be defined. Every numerical implementation <math>\mathbf{y}_{n}</math> of the Local Linear discretization <math>\mathbf{z}_{n}</math> is generically called ''Local Linearization scheme.''

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<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{L}(\mathbf{P}_{p,q}(2^{-k_{n}}

\mathbf{M}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r,} \quad </math> <ref name=″:24″ /> <ref name=":10" >Jimenez J.C.; Carbonell F. (2009). "Rate of convergence of local linearization schemes for random differential equations". BIT Numer. Math. 49 (2): 357–373. [https://doi.org/10.1007%2Fs10543-009-0225-0 doi:10.1007/s10543-009-0225-0]. S2CID 122003992.</ref> </div>

where the matrices <math>\mathbf{M}_{n}, \quad \mathbf{L} \quad and \quad \mathbf{r}</math> are defined as


<math>\mathbf{M}_{n}=\left[

\begin{array}{ccc}

\mathbf{f}_{\mathbf{x}}\left( \mathbf{y}_{n},\mathbf{\xi }(t_{n})\right) &

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0 & 0 & 0

\end{array}

\right]</math>

</math>

<math>\mathbf{L}=\left[