Stiff initial value problem

A stiff initial value problem is an initial value problem in the numerics of ordinary differential equations

${\displaystyle y'(t)=f(t,y(t)),\ t\geq t_{0},\quad y(t_{0})=y_{0}}$

in which explicit one-step processes or multi-step processes have considerable difficulties due to their limited stability area. This is the case if the constant in the Lipschitz condition (see Picard-Lindelöf's theorem ) ${\displaystyle L}$

${\displaystyle \|f(t,y_{1})-f(t,y_{2})\|\leq L\|y_{1}-y_{2}\|}$

assumes large values , but the solution is quite smooth. In this case, numerical methods could approximate this solution with relatively large step sizes, but explicit methods are forced to use small step sizes due to the limited stability area. Stiff initial value problems typically occur in the numerical approximation of parabolic partial differential equations after discretization in the spatial domain. One example is the Crank-Nicolson method , in which a finite difference method is used in the location and the implicit trapezoid method is used in the time direction . ${\displaystyle L\gg 1}$ ${\displaystyle y(t)}$

example

The problem is solved with the explicit and implicit Euler method and step size using the linear initial value problem ${\displaystyle h>0}$

${\displaystyle y'(t)=\lambda {\big (}y(t)-2{\big )},\quad y(0)=1,}$

explained with . The exact solution is and for large ones the solution is almost constant, i.e. very smooth. ${\displaystyle \lambda =-4}$ ${\displaystyle y(t)=2-e^{-4t}}$ ${\displaystyle t}$

Explicit Euler method in the example

• The explicit Euler method calculates the approximations with ${\displaystyle y_{0}=1}$

${\displaystyle y_{k+1}=(1+h\lambda )y_{k}-2h\lambda ,\ k=1,2,3,\ldots .}$However, these only provide usable values ​​if the amount of the prefactor is less than one, i.e. here for . For , on the other hand, the product lies outside the stability region that ends at, see Euler's polygon method # Properties . For such steps that are too large, the solutions grow indefinitely, cf. Graphic. ${\displaystyle y_{k}}$ ${\displaystyle |1+h\lambda |<1}$ ${\displaystyle h<1/2}$ ${\displaystyle h>1/2}$ ${\displaystyle h\lambda =-4h}$ ${\displaystyle -2}$

Implicit Euler method in the example

• The implicit Euler method , on the other hand, calculates the approximations on the basis of this ${\displaystyle y_{0}=1}$

${\displaystyle y_{k+1}={\frac {y_{k}-2h\lambda }{1-h\lambda }},\ k=1,2,3,\ldots .}$For every positive step size , here is the prefactor of , the fraction , because is negative. Because the stability region of the implicit Euler method includes the entire left complex half-plane, the method is A-stable . ${\displaystyle h}$ ${\displaystyle y_{k}}$ ${\displaystyle 1/(1-h\lambda )<1}$ ${\displaystyle \lambda =-4}$

• The two diagrams each show the exact solution in blue, an approximate solution with a small step size in green and the approximate solution with in red. ${\displaystyle h=0{,}2}$ ${\displaystyle h=0{,}52}$

In the explicit Euler method , the red approximations keep growing, while these rough approximations also remain close to the exact solution in the implicit Euler method .

Extended stability terms

For a more precise classification of numerical methods for rigid initial value problems, various stability terms have been introduced in the literature, which are usually based on different test equations. This includes the

1. Dahlquist's equation with . Your solutions all go to zero for . ${\displaystyle y'(t)=\lambda y(t)}$ ${\displaystyle \operatorname {Re} (\lambda )<0}$ ${\displaystyle y(t)=e^{\lambda (t-t_{0})}y_{0}}$ ${\displaystyle t\to \infty }$
2. Prothero-Robinson equation with and a smooth function . The solution to this equation is . For very small real parts, all solutions approach the function very quickly . ${\displaystyle y'(t)=\lambda {\big (}y(t)-g(t){\big )}+g'(t)}$ ${\displaystyle \operatorname {Re} (\lambda )<0}$ ${\displaystyle g(t)}$ ${\displaystyle y(t)=g(t)+e^{\lambda (t-t_{0})}(y_{0}-g(t_{0}))}$ ${\displaystyle \operatorname {Re} (\lambda )\ll -1}$ ${\displaystyle g(t)}$
3. The non-linear dissipative equation in which the right-hand side fulfills a one-sided Lipschitz condition. In contrast to the Lipschitz condition above , negative values ​​are also possible for the constant . A consequence of the one-sided Lipschitz condition is that the bound holds for the difference between two solutions of the differential equation , and these therefore keep getting closer and closer for and growing . ${\displaystyle y'(t)=f{\big (}t,y(t){\big )}}$ ${\displaystyle f}$ ${\displaystyle (y-v)^{T}{\big (}f(t,y)-f(t,v){\big )}\leq \mu \|y-v\|_{2}^{2},\quad y,v\in \mathbb {R} ^{n}.}$ ${\displaystyle \mu }$ ${\displaystyle y(t),v(t)}$ ${\displaystyle \|y(t)-v(t)\|_{2}\leq e^{\mu (t-t_{0})}\|y(t_{0})-v(t_{0})\|_{2}}$ ${\displaystyle \mu <0}$ ${\displaystyle t\to \infty }$

In numerical methods it is advantageous if the numerical approximations in test equations behave essentially like the exact solutions. Accordingly, the term calls for

• A stability that approximate solutions approach zero for the first test equations for , ${\displaystyle t\to \infty }$
• B stability that two approximate solutions of the third test equation with do not diverge from each other for . ${\displaystyle \mu =0}$ ${\displaystyle t\to \infty }$

For implicit Runge-Kutta methods , the term algebraic stability is a sufficient criterion for B-stability.

Numerical methods for stiff initial value problems

For stiff initial value problems, implicit methods are more efficient than explicit ones (this can also be seen as a definition of the term "stiff"). Special classes are

• implicit Runge-Kutta method
• Rosenbrock-Wanner method
• BDF process

Since the resolution of the non-linear equation systems requires a lot of effort in the case of implicit methods, linear-implicit one-step methods such as the aforementioned Rosenbrock-Wanner methods (ROW methods) have also been developed.

literature

• E. Hairer, G. Wanner: Solving Ordinary Differential Equations II, Stiff problems , Springer Verlag
• K. Strehmel, R. Weiner, H. Podhaisky: Numerics of ordinary differential equations - non -rigid, rigid and differential-algebraic equations , Springer Spectrum, 2012.