Content deleted Content added
Lilianmm87 (talk | contribs) |
Lilianmm87 (talk | contribs) agregar new cap |
||
Line 188:
</math> and ''k'' is the smallest integer number such that <math>\left\Vert 2^{-k}\mathbf{H}h\right\Vert \leq \frac{1}{2},
</math>
<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>
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>
<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>
==== Order 2 LL schemes ====
<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>
where the matrices <math>\mathbf{M}_{n}</math>, '''L''' and '''r''' are deffined as
<math>\mathbf{M}_{n}=
\begin{bmatrix} \mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{y}_{n}) & \mathbf{f}_{t}(t_{n},\mathbf{
y}_{n}) & \mathbf{f}(t_{n},\mathbf{y}_{n}) \\
0 & 0 & 1 \\
0 & 0 & 0
\end{bmatrix}
\in \mathbb{R}^{(d+2)\times (d+2)},</math>
<math>\mathbf{L}=\left[
\begin{array}{ll}
\mathbf{I}_{d} & \mathbf{0}_{d\times 2}%
\end{array}%
\right]</math>and <math>\mathbf{r}^{\intercal }=\left[
\begin{array}{ll}
\mathbf{0}_{1\times (d+1)} & 1
\end{array}
\right] </math> with <math>p+q>1</math> . For large systems of ODEs
<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>
==== Order 3 LL-Taylor schemes ====
<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>
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[
\begin{array}{cccc}
\mathbf{f}_{\mathbf{x}}(\mathbf{y}_{n}) & (\mathbf{I}\otimes \mathbf{f}
^{\intercal }(\mathbf{y}_{n}))\mathbf{f}_{\mathbf{xx}}(\mathbf{y}_{n})
\mathbf{f}(\mathbf{y}_{n}) & \mathbf{0} & \mathbf{f}(\mathbf{y}_{n}) \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0%
\end{array}%
\right] \in \mathbb{R}^{(d+3)\times (d+3)}, </math>
<math>\mathbf{L}_{1}=\left[
\begin{array}{ll}
\mathbf{I}_{d} & \mathbf{0}_{d\times 3}
\end{array}
\right] \quad and \quad \mathbf{r}_{1}^{\intercal }=\left[
\begin{array}{ll}
\mathbf{0}_{1\times (d+2)} & 1
\end{array}
\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}}%
_{m_{n},k_{n}}^{p,q}(h_{n},\mathbf{T}_{n},\mathbf{r})\mathbf{,}\qquad with \qquad m_{n}>2. </math>
==== Order 4 LL-RK schemes ====
<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>
where
<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>
and
<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}%
_{n}\right) -\mathbf{f}_{\mathbf{x}}\left( t_{n},\mathbf{y}_{n}\right)
\mathbf{u}_{j}\ -\mathbf{f}_{t}\left( t_{n},\mathbf{y}_{n}\right)
c_{j}h_{n}, </math>
with <math>\mathbf{k}_{1}\equiv \mathbf{0}, c=\left[
\begin{array}{cccc}
0 & \frac{1}{2} & \frac{1}{2} & 1
\end{array}
\right] , </math> and p + q > 3. For large systems of ODEs, the vector <math>\mathbf{u}_{j} </math> in the above �
scheme is replaced by <math>\mathbf{u}_{j}=\mathbf{L\mathbf{k}
}_{m_{j},k_{j}}^{p,q}(c_{j}h_{n},\mathbf{M}_{n},\mathbf{r}).</math>
==== Locally Linearized Runge-Kutta of Dormand & Prince ====
<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>
where ''s'' = 6 is the number of the stages,
<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},
\mathbf{y}_{n}\right) -\mathbf{f}_{\mathbf{x}}\left( t_{n},\mathbf{y}
_{n}\right) \mathbf{u}_{j}\ -\mathbf{f}_{t}\left( t_{n},\mathbf{y}
_{n}\right) c_{j}h_{n}, </math>
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}
_{j}=\mathbf{L\mathbf{k}}_{m_{j},k_{j}}^{p,q}(c_{j}h_{n},\mathbf{M}_{n},
\mathbf{r}).</math>
==== Stability and dynamics ====
By construction the LL and HOLL discretizations inherit the stability and dynamics of the linear ODEs, but
it is not the case of the LL schemes in general. With<math>p\leq q\leq p+2</math>, the LL schemes (6)-(9) are [[Stiff equation|A-stable]].
With ''q = p + 1 or q = p + 2'', the LL schemes (6)-(9) are also [[L-stability|L-stable]]. For linear ODEs, the LL schemes
(6)-(9) converge with order ''p + q''. Moreover, with ''p = q = 6'' and ''mn = d'', all the above described LL schemes yield to the
″exact computation″ (up to the precision of the [[floating-point arithmetic]]) of linear ODEs on
the current personal computers. This includes [[Stiff equation|stiff]] and highly oscillatory linear equations. Moreover, the LL
schemes (6)-(9) are regular for linear ODEs and inherit the [[Symplectic geometry|symplectic structure]] of [[Hamiltonian mechanics|Hamiltonian]] [[Harmonic oscillator|harmonic]]
[[Harmonic oscillator|oscillators]]. These LL schemes are also linearization preserving, and display a better reproduction of the [[Stable manifold|stable]]
[[Stable manifold|and unstable manifolds]] around [[Hyperbolic equilibrium point|hyperbolic equilibrium points]] and [[Limit cycle|periodic orbits]] that [[Numerical methods for ordinary differential equations|other numerical schemes]]
with the same stepsize.
| |||