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{{User sandbox|Lilian|Local Linearization Method}}
Consider the d-dimensional [[Ordinary differential equation|Ordinary Differential Equation]] (ODE).
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==== Computing integrals involving matrix exponential ====
Among a number of algorithms to compute the integrals <math>\phi _{j}</math>, those based on rational Padé and Krylov subspaces
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\mathbf{M}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r,} \qquad \qquad (6)</math>
where the matrices <math>\mathbf{M}_{n}</math>, '''L''' and '''r''' are deffined as
<math>\mathbf{M}_{n}=
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\mathbf{T}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r}_{1}\mathbf{,} \qquad \qquad (7)</math>
where for [[Autonomous system (mathematics)|autonomous]] ODEs the matrices <math>\mathbf{T}_{n}, \mathbf{L}_{1}</math> and <math>\mathbf{r}_{1}</math> are deffined as
<math>\mathbf{T}_{n}=\left[
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\right]</math>. Here, <math>\mathbf{f}_{\mathbf{xx}}</math> denotes the second derivative of '''f''' with respect to '''x''',
'''I''' the ''d''-dimensional identity matrix, and p + q > 2. For large systems of ODEs.
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{L\mathbf{k}}%
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with the same stepsize.
Consider the ''d''-dimensional [[Delay differential equation|Delay Differential Equation]] (DDE)
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with <math>p+q>1 \quad and \quad m_{n}>2.</math>
Consider the ''d-dimensional'' Random Differential Equation (RDE)
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The convergence rate of both schemes is <math>min\{2,2\gamma \}</math>, where is <math>\gamma</math> the exponent of the Holder condition of <math>\mathbf{\xi }</math>.
Consider the ''d''-dimensional [[Stochastic differential equation|Stochastic Differential Equation]] (SDE)
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By construction the strong LL and HOLL discretizations inherit the stability and [[Random dynamical system|dynamics]] of the linear ODEs, but it is not the case of the strong LL schemes in general. LL schemes '''(15)-(18)''' with <math>p\leq q\leq p+2 </math> are A-stable, which includes stiff and highly oscillatory linear equations. Moreover, for linear SDEs with [[Pullback attractor|random attractors]], these schemes also have a random attractor that [[Convergece of Random variables#Convergence in probability|converges in probability]] to the exact one as stepsizes decrease and preserve the [[ergodicity]] of these equations for any stepsize. These schemes also reproduce essential dynamical properties of simple and coupled harmonic oscillators such as the linear growth of energy along the paths, the oscillatory behavior around 0, the symplectic structure of Hamiltonian oscillators, and the mean of the paths. For nonlinear SDEs with small noise (e.g., '''(14)''' with <math>\mathbf{g}_{i}(t)\approx 0 </math>), the paths of these SLL schemes are basically the nonrandom paths of the LL scheme '''(6)''' for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the paths of the SLL scheme.
Consider the ''d''-dimensional stochastic differential equation
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By construction the weak LL and HOLL discretizations inherit the stability and [[Random dynamical system|dynamics]] of the linear ODEs, but it is not the case of the weak LL schemes in general. WLL schemes, with <math>p\leq q\leq p+2,</math> preserve the [[Moment (mathematics)|first two moments]] of the linear SDEs, and inherits the mean-square stability or instability that such solution may have. This includes, for instance, the equations of coupled harmonic oscillators driven by random force, and large systems of stiff linear SDEs that result from the method of lines for linear stochastic partial differential equations. Moreover these WLL schemes preserve the [[ergodicity]] of the linear equations, and are geometrically ergodic for some classes of nonlinear SDEs. For nonlinear SDEs with small noise (e.g., ('''19''') with <math>\mathbf{g}_{i}(t)\approx 0</math>), the solutions of these WLL schemes are basically the nonrandom paths of the LL scheme ('''6''') for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the mean of the WLL scheme.
Below is a time line of the main developments of the LL method.
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http://dx.doi.org/10.1007/s10543-010-0272-6
de la Cruz H., Biscay R.J., Carbonell F., Ozaki T., Jimenez J.C., A higher order Local Linearization method for solving ordinary differential equations, Appl. Math. Comput., 185 (2007) 197-212.
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