Content deleted Content added
Lilianmm87 (talk | contribs) No edit summary |
Lilianmm87 (talk | contribs) |
||
Line 51:
Let <math>\left( t\right) _{h}=\{t_{n}:n=0,..,N\}</math> be a time discretization of the time interval <math>[t_{0},T]</math> with maximum stepsize '''h''' such that <math>t_{n}<t_{n+1} \quad and \quad h_{n}=t_{n+1}-t_{n}\leq h</math>. After the local linearization of the equation (1) at the time step <math>t_{n}</math> the [[Variation of parameters#First order equation|variation of constants formula]] yields
<div style="text-align: center;">
<math>\mathbf{x}(t_{n}+h)=\mathbf{x}(t_{n})+\mathbf{\phi }(t_{n},\mathbf{x}
(t_{n});h)+\mathbf{r}(t_{n},\mathbf{x}(t_{n});h),
</math>
</div>
where
<div style="text-align: center;">
<math>\mathbf{\phi }(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{f}\left( t_{n},
\mathbf{z}_{n}\right) +\mathbf{f}_{t}\left( t_{n},\mathbf{z}_{n}\right) s)ds
\qquad</math>
</div>
results from the linear approximation, and
<div style="text-align: center;">
<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>
</div>
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}
Line 78 ⟶ 84:
t\right) _{h}</math> is deffined by the recursive expression
<div style="text-align: center;">
<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{.} \qquad \qquad \qquad \qquad (3)</math>
</div>
The Local Linear discretization (3) [[Rate of convergence|converges]] with order 2 to the solution of nonlinear ODEs, but it match the
Line 88 ⟶ 96:
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
<div style="text-align: center;">
<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>
</div>
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},
Line 98 ⟶ 108:
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
<div style="text-align: center;">
<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>
</div>
for all <math>t\in \lbrack t_{k},t_{k+1}]</math>, where
<div style="text-align: center;">
<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 })-%
Line 109 ⟶ 122:
\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>
</div>
The resulting approximation is often called Locally Linearized discretization.
Line 116 ⟶ 130:
Locally Linearized Runge Kutta discretization
<div style="text-align: center;">
<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>
</div>
which is obtained by solving (5) via a s-stage [[Runge–Kutta methods|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>
Line 125 ⟶ 141:
Local Linear Taylor discretization
<div style="text-align: center;">
<small><math display="inline">\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }(t_{n},\mathbf{z}%
_{n};h_{n})+\int_{0}^{h_{n}}e^{\left( h_{n}-s\right) \mathbf{f}_{\mathbf{x}%
Line 132 ⟶ 149:
_{n}\right) \frac{d^{j}\mathbf{x}\left( t\right) }{dt^{j}}\right) \mid _{t=%
\mathbf{z}_{n}}, </math></small>
</div>
which results from the approximation of <math>\mathbf{g}_{n}</math>in (2) by its order-''p'' truncated [[Taylor series|Taylor expansion]].
Line 139 ⟶ 157:
''Linealized Exponential Adams discretization''
<div style="text-align: center;">
<small><math display="inline">\mathbf{z}_{n+1}=\mathbf{z}_{n}+\mathbf{\phi }(t_{n},\mathbf{z}
_{n};h_{n})+h_{n}\sum_{j=1}^{p}\sum_{l=1}^{j}\frac{\gamma _{j+1}}{l}\nabla
Line 146 ⟶ 165:
\begin{array}{c}
-\theta \\
j
\end{array}
\right) d\theta , </math></small>
</div>
which results from the interpolation of <math>\mathbf{g}_{n}</math>in (2) by a [[Hermite polynomials|Hermite polynomial]] of degree ''p'', where <math>\nabla ^{l}\mathbf{g}
_{n}(t_{m},\mathbf{z}_{m})</math> denotes the ''l''-th backward di§erence of <math>\mathbf{g}_{n}(t_{m},\mathbf{z}_{m})</math>.
Line 158 ⟶ 178:
integrals <math>\phi _{j}</math> of the form
<div style="text-align: center;">
<math>\phi _{j}(\mathbf{A},h)=\int\limits_{0}^{h}e^{(h-s)\mathbf{A}}s^{j-1}ds,\qquad j=1,2..., </math>
</div>
where A is an ''d <math>\times</math> d'' matrix. Every numerical implementation <math>\mathbf{y}_{n}</math> of a Local Linear discretization <math>\mathbf{z}_{n}</math> of any order is generically called Local Linearization scheme.
Line 168 ⟶ 190:
approximations for exponential matrix are preferred. For this, a central role is playing by the expression
<div style="text-align: center;">
<math>\sum\nolimits_{i=1}^{l}\phi _{i}(\mathbf{A},h)\mathbf{a}_{i}=\mathbf{L}e^{h
\mathbf{H}}\mathbf{r,} </math>
</div>
where <math>\mathbf{a}_{i}</math> are d-dimensional vectors,
<div style="text-align: center;">
<math>\mathbf{H}= \begin{bmatrix}
\mathbf{A} & \mathbf{v}_{l} & \mathbf{v}_{l-1} & \cdots & \mathbf{v}_{1} \\
Line 183 ⟶ 208:
</math>
</div>
<div style="text-align: center;">
<math>\mathbf{L}=[\mathbf{I}\mathbf{0}_{d\times l)}],\mathbf{r}=[\mathbf{0}
_{1\times (d+l-1)}\quad1]^{\intercal }\quad, and \quad \mathbf{v}_{i}=\mathbf{a}
_{i}(i-1)!
</math>
</div>
If <math>\mathbf{P}_{p,q}(2^{-k}\mathbf{H}h)
Line 194 ⟶ 222:
</math>
<div style="text-align: center;">
<math>\left\vert \sum\nolimits_{i=1}^{l}\phi _{i}(\mathbf{A},h)\mathbf{a}_{i}-
\mathbf{L}\left( \mathbf{\mathbf{P}}_{p,q}(2^{-k}\mathbf{H}h)\right) ^{2^{k}}
\mathbf{r}\right\vert \varpropto h^{p+q+1}.</math>
</div>
If <math>\mathbf{\mathbf{k}}_{m,k}^{p,q}(h,\mathbf{H},\mathbf{r})</math> denotes the ''(m; p; q; k)'' [[Krylov subspace|Krylov-Padé approximation]] of <math>e^{h\mathbf{H}}\mathbf{r}</math>
<div style="text-align: center;">
<math>\left\vert \sum\nolimits_{i=1}^{l}\phi _{i}(\mathbf{A},h)\mathbf{a}_{i}-
\mathbf{L\mathbf{k}}_{m,k}^{p,q}(h,\mathbf{H},\mathbf{r})\right\vert
\varpropto h^{\min {m,p+q+1}}.</math>
</div>
==== Order 2 LL schemes ====
<div style="text-align: center;">
<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,} \qquad \qquad (6)</math>
</div>
where the matrices <math>\mathbf{M}_{n}</math>, '''L''' and '''r''' are deffined as
<div style="text-align: center;">
<math>\mathbf{M}_{n}=
\begin{bmatrix} \mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{y}_{n}) & \mathbf{f}_{t}(t_{n},\mathbf{
Line 218 ⟶ 253:
\end{bmatrix}
\in \mathbb{R}^{(d+2)\times (d+2)},</math>
</div>
<math>\mathbf{L}=\left[
Line 229 ⟶ 265:
\right] </math> with <math>p+q>1</math> . For large systems of ODEs
<div style="text-align: center;">
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{L\mathbf{k}}
_{m_{n},k_{n}}^{p,q}(h_{n},\mathbf{M}_{n},\mathbf{r})\mathbf{,}\qquad with \qquad m_{n}>1. </math>
</div>
==== Order 3 LL-Taylor schemes ====
<div style="text-align: center;">
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{L}_{1}(\mathbf{P}_{p,q}(2^{-k_{n}}
\mathbf{T}_{n}h_{n}))^{2^{k_{n}}}\mathbf{r}_{1}\mathbf{,} \qquad \qquad (7)</math>
</div>
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
<div style="text-align: center;">
<math>\mathbf{T}_{n}=\left[
\begin{array}{cccc}
Line 249 ⟶ 290:
\end{array}%
\right] \in \mathbb{R}^{(d+3)\times (d+3)}, </math>
</div>
<math>\mathbf{L}_{1}=\left[
Line 262 ⟶ 304:
'''I''' the ''d''-dimensional identity matrix, and p + q > 2. For large systems of ODEs.
<div style="text-align: center;">
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{L\mathbf{k}}%
_{m_{n},k_{n}}^{p,q}(h_{n},\mathbf{T}_{n},\mathbf{r})\mathbf{,}\qquad with \qquad m_{n}>2. </math>
</div>
==== Order 4 LL-RK schemes ====
<div style="text-align: center;">
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{u}_{4}+\frac{h_{n}}{6}(2\mathbf{k}%
_{2}+2\mathbf{k}_{3}+\mathbf{k}_{4}), \qquad \qquad (8)</math>
</div>
where
<div style="text-align: center;">
<math>\mathbf{u}_{j}=\mathbf{L}(\mathbf{P}_{p,q}(2^{-\kappa _{j}}\mathbf{M}
_{n}c_{j}h_{n}))^{2^{\kappa _{j}}}\mathbf{r} </math>
</div>
and
<div style="text-align: center;">
<math>\mathbf{k}_{j}=\mathbf{f}\left( t_{n}+c_{j}h_{n},\mathbf{y}_{n}+\mathbf{u}%
_{j}+c_{j}h_{n}\mathbf{k}_{j-1}\right) -\mathbf{f}\left( t_{n},\mathbf{y}%
Line 282 ⟶ 331:
\mathbf{u}_{j}\ -\mathbf{f}_{t}\left( t_{n},\mathbf{y}_{n}\right)
c_{j}h_{n}, </math>
</div>
with <math>\mathbf{k}_{1}\equiv \mathbf{0}, c=\left[
Line 293 ⟶ 343:
==== Locally Linearized Runge-Kutta of Dormand & Prince ====
<div style="text-align: center;">
<math>\mathbf{y}_{n+1}=\mathbf{y}_{n}+\mathbf{u}_{s}+h_{n}\sum_{j=1}^{s}b_{j}
\mathbf{k}_{j}\qquad and \qquad \widehat{\mathbf{y}}_{n+1}=\mathbf{y}
_{n}+\mathbf{u}_{s}+h_{n}\sum_{j=1}^{s}\widehat{b}_{j}\mathbf{k}_{j},
\qquad \qquad (9)</math>
</div>
where ''s'' = 6 is the number of the stages,
<div style="text-align: center;">
<math>\mathbf{k}_{j}=\mathbf{f(}t_{n}+c_{j}h_{n},\mathbf{y}_{n}+\mathbf{u}
_{j}+h_{n}\sum_{i=1}^{s-1}a_{j,i}\mathbf{k}_{i})-\mathbf{f}\left( t_{n},
Line 305 ⟶ 359:
_{n}\right) \mathbf{u}_{j}\ -\mathbf{f}_{t}\left( t_{n},\mathbf{y}
_{n}\right) c_{j}h_{n}, </math>
<div>
with <math>\mathbf{k}_{1}\equiv \mathbf{0}</math>, and <math>a_{j,i}, b_{j}, \widehat{b}_{j} \quad and \quad c_{j}</math>are the [[Dormand–Prince method|Runge-Kutta coefficients of Dormand and Prince]] and p + q > 4. For large systems of ODEs, the vector <math>\mathbf{u}_{j}</math> in the above scheme is replaced by <math>\mathbf{u}
| |||