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== Local Linearization scheme ==
A ''Local Linearization (LL) scheme'' is the final [[Recursion (computer science)|recursive algorithm]] that allows the numerical implementation
of a [[discretization]] derived from the LL or HOLL method for a class of differential equations.
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<math>\frac{d\mathbf{x}\left( t\right) }{dt}=\mathbf{f}\left( t,\mathbf{x}\left(
t\right) \right) ,\qquad t\in \left[ t_{0},T\right], \qquad \qquad \qquad \qquad (1).</math>
with initial condition <math>\mathbf{x}(t_{0})=\mathbf{x}_{0}</math>, where <math>\mathbf{f}</math> is a differentiable function.
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<math>\mathbf{r}(t_{n},\mathbf{z}_{n};h)=\int\limits_{0}^{h}e^{\mathbf{f}_{\mathbf{%
x}}\left( t_{n},\mathbf{z}_{n}\right) (h-s)}\mathbf{g}_{n}(s,\mathbf{x}%
(t_{n}+s))ds, \qquad \qquad \qquad (2).</math>
is the residual of the linear approximation. Here, <math>\mathbf{f}_{\mathbf{x}}</math>and <math>\mathbf{f}_{t}</math> denote the partial derivatives of '''f''' with respect to the variables '''x''' and '''t''', respectively, and <math>\mathbf{g}
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}}(t_{n},\mathbf{z}_{n})\mathbf{z}_{n}</math>
For a time discretization <math>\left( t\right) _{h}</math>, the ''Local Linear discretization'' of the ODE (1) at each point <math>t_{n+1}\in \left(
t\right) _{h}</math> is deffined by the recursive expression
<math>\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }(t_{n},\mathbf{z}_{n};h_{n}),
\qquad with \quad \mathbf{z}_{0}=\mathbf{x}_{0}\text{.} \
The Local Linear discretization (3) [[Rate of convergence|converges]] with order 2 to the solution of nonlinear ODEs, but it match the
solution of the linear ODEs. The recursion (3) is also known as Exponential Euler discretization.
=== High Order Local Linear discretizations ===
For a time discretization <math>\left( t\right) _{h},</math> a ''High Order Local Linear (HOLL)'' discretization of the ODE (1) at each point <math>t_{n+1}\in \left( t\right) _{h}</math> is deffined by the recursive expression
<math>\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }(t_{n},\mathbf{z}_{n};h_{n})+
\widetilde{\mathbf{r}}(t_{n},\mathbf{z}_{n};h_{n}),\qquad with \quad
\mathbf{z}_{0}=\mathbf{x}_{0}, \qquad \qquad \qquad(4)</math>
where <math>\tilde{r}</math> is an approximation to the residual '''r''' of order <math>\alpha </math>higher than 2 <math>(i.e., \left\vert \mathbf{r}(t_{n},
\mathbf{z}_{n};h)-\widetilde{\mathbf{r}}(t_{n},\mathbf{z}_{n};h)\right\vert
\propto h^{\alpha }).</math> The HOLL discretization (4) [[Rate of convergence|converges]] with order � to the solution of nonlinear ODEs, but it match the solution of the linear ODEs.
HOLL discretizations can be derived in two ways: 1) by approximating the integral representation (2) of r; and 2) by using a numerical integrator for the di§erential representation of '''r''' deffined by
<math>\frac{d\mathbf{r}\left( t\right) }{dt}=\mathbf{q}(t_{n},\mathbf{z}_{n};t
\mathbf{,\mathbf{r}}\left( t\right) \mathbf{),}\qquad with \qquad \mathbf{r}
\left( t_{n}\right) =\mathbf{0,} \qquad \qquad \qquad (5)</math>
for all <math>t\in \lbrack t_{k},t_{k+1}]</math>, where
<small><math>\mathbf{q}(t_{n},\mathbf{z}_{n};s\mathbf{,\xi })=\mathbf{f}(s,\mathbf{z}_{n}+%
\mathbf{\phi }\left( t_{n},\mathbf{z}_{n};s-t_{n}\right) +\mathbf{\xi })-%
\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{z}_{n})\mathbf{\phi }\left( t_{n},%
\mathbf{z}_{n};s-t_{n}\right) -\mathbf{f}_{t}\left( t_{n},\mathbf{z}%
_{n}\right) (s-t_{n})-\mathbf{f}\left( t_{n},\mathbf{z}_{n}\right) .</math></small>
The resulting approximation is often called Locally Linearized discretization.
Known HOLL discretizations are the following.
Locally Linearized Runge Kutta discretization
<math>\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }(t_{n},\mathbf{z}
_{n};h_{n})+h_{n}\sum_{j=1}^{s}b_{j}\mathbf{k}_{j},\quad with \quad \mathbf{k}
_{i}=\mathbf{q}(t_{n},\mathbf{z}_{n};\text{ }t_{n}+c_{i}h_{n}\mathbf{,}\mathbf{
}h_{n}\sum_{j=1}^{i-1}a_{ij}\mathbf{k}_{j}), </math>
which is obtained by solving (5) via a s-stage RK scheme with coefficients <math>\mathbf{c}=\left[ c_{i}\right] , \mathbf{A}=\left[ a_{ij}\right] \quad and \quad \mathbf{b}=\left[ b_{j}\right]</math>
Local Linear Taylor discretization
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