Rigid system

A rigid system of ordinary differential equations (ODEs) is called (loosely speaking) such a system of ODEs whose numerical solution by explicit methods (for example, Runge-Kutta or Adams methods ) is unsatisfactory due to a sharp increase in the number of calculations (with a small integration step) or due to a sharp increase in the error (the so-called explosion of the error) with an insufficiently small step. Rigid systems are characterized by the fact that for them implicit methods give a better result, usually much better than explicit methods ^[1] .

Content

Formal Definition

Consider the Cauchy problem for an autonomous system of ODEs of the form

{\begin{cases}{\dfrac {d\mathbf {y} (x)}{dx}}=\mathbf {F} (\mathbf {y} (x)),\quad \mathbf {F} (\mathbf {y} )\in C^{p}(\Omega ),\quad \Omega \subset \mathbb {R} ^{m},\\\mathbf {y} (x_{0})=\mathbf {y} _{0}\in \Omega ,\end{cases}}

{\ displaystyle {\ begin {cases} {\ dfrac {d \ mathbf {y} (x)} {dx}} = \ mathbf {F} (\ mathbf {y} (x)), \ quad \ mathbf {F } (\ mathbf {y}) \ in C ^ {p} (\ Omega), \ quad \ Omega \ subset \ mathbb {R} ^ {m}, \\\ mathbf {y} (x_ {0}) = \ mathbf {y} _ {0} \ in \ Omega, \ end {cases}}}

(one)

Where $\mathbf {y} (x)$ ${\ displaystyle \ mathbf {y} (x)}$ - unknown vector function , $\mathbf {F} (\mathbf {y} )$ ${\ displaystyle \ mathbf {F} (\ mathbf {y})}$ Is a given vector function, $x$ ${\ displaystyle x}$ Is an independent variable $\mathbf {y} (x_{0})=\mathbf {y} _{0}$ ${\ displaystyle \ mathbf {y} (x_ {0}) = \ mathbf {y} _ {0}}$ Is the initial condition .

System (1) is called rigid if for any initial values $\mathbf {y} (x_{0})=\mathbf {y} _{0}$ ${\ displaystyle \ mathbf {y} (x_ {0}) = \ mathbf {y} _ {0}}$ on a given segment $[x_{0},\;x_{\mathrm {end} }]$ ${\ displaystyle [x_ {0}, \; x _ {\ mathrm {end}}]}$ belonging to the existence interval of solution (1) , the conditions are satisfied:

maximum modulus of eigenvalues of the Jacobi matrix ( spectral radius $\rho$ ${\ displaystyle \ rho}$ ) is bounded along the solution $\mathbf {y} (x)$ ${\ displaystyle \ mathbf {y} (x)}$ :

0<L\leqslant \rho \left({\frac {\partial \mathbf {y} (x)}{\partial x}}\right)\leqslant \left\|{\frac {\partial \mathbf {y} (x)}{\partial x}}\right\|=\left\|\left.{\frac {\partial K(x+\xi ,\;x)}{\partial \xi }}\right|_{\xi =0}\right\|<+\infty ,\quad x_{0}\leqslant x\leqslant x_{\mathrm {end} };

{\ displaystyle 0 <L \ leqslant \ rho \ left ({\ frac {\ partial \ mathbf {y} (x)} {\ partial x}} \ right) \ leqslant \ left \ | {\ frac {\ partial \ mathbf {y} (x)} {\ partial x}} \ right \ | = \ left \ | \ left. {\ frac {\ partial K (x + \ xi, \; x)} {\ partial \ xi}} \ right | _ {\ xi = 0} \ right \ | <+ \ infty, \ quad x_ {0} \ leqslant x \ leqslant x _ {\ mathrm {end}};}

there are such numbers $\xi _{\mathrm {b} }$ ${\ displaystyle \ xi _ {\ mathrm {b}}}$ , $N$ ${\ displaystyle N}$ , $\nu$ ${\ displaystyle \ nu}$ that satisfy the conditions:

0<\xi _{\mathrm {b} }\ll x_{\mathrm {end} },\quad N\gg 1,\quad 1\leqslant \nu \leqslant p,\quad 0<\xi _{\mathrm {b} }\leqslant x+\xi _{\mathrm {b} }\leqslant x+\xi \leqslant x_{\mathrm {end} };

{\ displaystyle 0 <\ xi _ {\ mathrm {b}} \ ll x _ {\ mathrm {end}}, \ quad N \ gg 1, \ quad 1 \ leqslant \ nu \ leqslant p, \ quad 0 <\ xi _ {\ mathrm {b}} \ leqslant x + \ xi _ {\ mathrm {b}} \ leqslant x + \ xi \ leqslant x _ {\ mathrm {end}};}

The following inequality holds:

\left\|{\frac {\partial ^{\nu }K(x+\xi ,\;x)}{\partial \xi ^{\nu }}}\right\|\leqslant \left({\frac {L}{N}}\right)^{\nu }.

{\ displaystyle \ left \ | {\ frac {\ partial ^ {\ nu} K (x + \ xi, \; x)} {\ partial \ xi ^ {\ nu}}} \ right \ | \ leqslant \ left ( {\ frac {L} {N}} \ right) ^ {\ nu}.}

Here

K(x+\xi ,\;x)=X(x+\xi )X^{-1}(x);

{\ displaystyle K (x + \ xi, \; x) = X (x + \ xi) X ^ {- 1} (x);}

X(x)

{\ displaystyle X (x)}

- the fundamental matrix of the equation in variations for the system (1) ;

\|A\|=\|A\|_{m}=\max _{i}\sum _{j=1}^{m}|a_{ij}|

{\ displaystyle \ | A \ | = \ | A \ | _ {m} = \ max _ {i} \ sum _ {j = 1} ^ {m} | a_ {ij} |}

- matrix

m

{\ displaystyle m}

-norm.

\xi _{\mathrm {b} }

{\ displaystyle \ xi _ {\ mathrm {b}}}

- the so-called length (parameter) of the boundary layer.

Rigid differential ODE systems also include systems for which these conditions are satisfied after scaling the components of the vector $\mathrm {y} (x)$ ${\ displaystyle \ mathrm {y} (x)}$ on every decision.

Since any non-autonomous system of ODEs of order $m$ ${\ displaystyle m}$ can be reduced to autonomous by introducing an additional auxiliary function, then a non-autonomous ODE system is called rigid if the rigid autonomous system of order is equivalent to it $m+1$ ${\ displaystyle m + 1}$ .

Literature

Khairer E., Wanner G. Solution of ordinary differential equations. Rigid and differential algebraic problems / Per. from English - M .: Mir, 1999 .-- 685 p. - ISBN 5-03-003117-0 . .
Curtiss CF, Hirschfelder J. O. Integration of stiff equations // Proceedings of the National Academy of Sciences of the USA. - 1952. - vol. 38 (3). - pp. 235-243.

Links

Notes

↑ Curtiss CF, Hirschfelder J. O. Integration of stiff equations // Proceedings of the National Academy of Sciences of the USA. - 1952. - vol. 38 (3). - pp. 235-243.

[1] Curtiss CF, Hirschfelder J. O. Integration of stiff equations // Proceedings of the National Academy of Sciences of the USA. - 1952. - vol. 38 (3). - pp. 235-243.