Local linearization method: Difference between revisions

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==== Order 2 SLL-RK schemes ====

[[File:Figure SSDE4.png|thumb|575x575px543x543px|'''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 method|Monte Carlo]] with ''10000'' simulations of the ''SLL1'' scheme with ''h=1/2'' and ''p=q=6''.]]

For SDEs with a single Wiener noise ''(m=1''''')''' <ref name=":4" />

<math>\mathbf{y}_{t_{n+1}}=\mathbf{y}_{n}+\widetilde{\mathbf{\phi }}(t_{n},\mathbf{

y}_{n};h_{n})+\frac{h_{n}}{2}\left( \mathbf{k}_{1}+\mathbf{k}_{2}\right) +</math>

<math>\quad \quad \quad +

\mathbf{g}\left( t_{n}\right) \Delta w_{n}+\frac{\left( \mathbf{g}\left(

t_{n+1}\right) -\mathbf{g}\left( t_{n}\right) \right) }{h_{n}}J_{\left(

0,1\right) } \qquadquad \qquad (7.5)</math>

where

<math>\mathbf{k}_{1} =\mathbf{f}(t_{n}+\frac{h_{n}}{2},\mathbf{y}_{n}+\widetilde{

\mathbf{\phi }}(t_{n},\mathbf{y}_{n};\frac{h_{n}}{2})+\gamma _{+})-\mathbf{f}

</math>

<math> \quad \quad -\mathbf{f}

_{\mathbf{x}}(t_{n},\mathbf{y}_{n})\widetilde{\mathbf{\phi }}(t_{n},\mathbf{y

}_{n};\frac{h_{n}}{2})-\mathbf{f}\left( t_{n},\mathbf{y}_{n}\right) -\mathbf{ </math>

f}_{t}\left( t_{n},\mathbf{y}_{n}\right) \frac{h_{n}}{2},

<math> \quad \quad -\mathbf{

</math><math>\mathbf{k}_{2} =\mathbf{f}(t_{n}+\frac{h_{n}}{2},\mathbf{y}_{n}+\widetilde{

\mathbff}_{t}\phileft( }}(t_{n},\mathbf{y}_{n};\right) \frac{h_{n}}{2})+\gamma _{-})-\mathbf{f},</math>

</math><math>\mathbf{k}_{2} =\mathbf{f}(t_{n}+\frac{h_{n}}{2},\mathbf{y}_{n}+\widetilde{

f}_\mathbf{t}\left(phi }}(t_{n},\mathbf{y}_{n}\right) ;\frac{h_{n}}{2}, )+\gamma _{-})</math>

<math> \quad \quad -\mathbf{f}

_{\mathbf{x}}(t_{n},\mathbf{y}_{n})\widetilde{\mathbf{\phi }}(t_{n},\mathbf{y

}_{n};\frac{h_{n}}{2})-\mathbf{f}\left( t_{n},\mathbf{y}_{n}\right) -\mathbf{</math>

<math>\quad \quad -\mathbf{

f}_{t}\left( t_{n},\mathbf{y}_{n}\right) \frac{h_{n}}{2},</math>

with <math>\gamma _{\pm }=\frac{1}{h_{n}}\mathbf{g}\left( t_{n}\right) \Bigl(

\widetilde{J}_{\left( 1,0\right) }\pm \sqrt{2\widetilde{J}_{\left(1,1,0\right) }h_{n}-

\widetilde{J}_{\left( 1,0\right) }^{2}} \Bigr) </math>.

\widetilde{J}_{\left( 1,0\right) }^{2}} \Bigr) </math>. Here, <math>\widetilde{\mathbf{\phi }}(t_{n},\mathbf{y}_{n};h_{n})=\mathbf{L}(\mathbf{P}_{p,q}(2^{-k_{n}}\mathbf{M}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r} </math> for low dimensional SDEs, and <math>\widetilde{\mathbf{\phi }}(t_{n},\mathbf{y}_{n};h_{n})=\mathbf{L\mathbf{k}}_{m_{n},k_{n}}^{p,q}(h_{n},\mathbf{M}_{n}, \mathbf{r}) </math> for large systems of SDEs, where <math>\mathbf{M}_{n} </math>, <math>\mathbf{L} </math>, <math>\mathbf{r} </math>, <math>\Delta \mathbf{w}_{n}^{i} </math> and <math>\widetilde{J}_{\alpha } </math> are defined as in the order-'''2''' SLL-Taylor schemes, ''p+q>1'' and <math>m_{n}>2 </math>.

\widetilde{J}_{\left( 1,0\right) }^{2}} \Bigr) </math>. Here, <math>\widetilde{\mathbf{\phi }}(t_{n},\mathbf{y}_{n};h_{n})=\mathbf{L}(\mathbf{P}_{p,q}(2^{-k_{n}}\mathbf{M}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r} </math> for low dimensional SDEs, and <math>\widetilde{\mathbf{\phi }}(t_{n},\mathbf{y}_{n};h_{n})=\mathbf{L\mathbf{k}}_{m_{n},k_{n}}^{p,q}(h_{n},\mathbf{M}_{n}, \mathbf{r}) </math> for large systems of SDEs, where <math>\mathbf{M}_{n} </math>, <math>\mathbf{L} </math>, <math>\mathbf{r} </math>, <math>\Delta \mathbf{w}_{n}^{i} </math> and <math>\widetilde{J}_{\alpha } </math> are defined as in the order-'''2''' SLL-Taylor schemes, ''p+q>1'' and <math>m_{n}>2 </math>.

==== Stability and dynamics <ref name=":4" /><ref name=":5">de la Cruz H.; Jimenez J.C.; Zubelli J.P. (2017). "Locally Linearized methods for the simulation of stochastic oscillators driven by random forces". BIT Numer. Math. 57: 123–151. [http://doi.org:10.1007%2Fs10543-016-0620-2 doi:10.1007/s10543-016-0620-2]. S2CID 124662762.</ref>====