Numerical stability of stiff problem

[matteo86bo]

I have a very complicated stiff system, which can be expressed as:

$x^{\prime}(t)=A(x,y)-B(x,y)$
$y^{\prime}(t)=C(x,y)-D(x,y)$

I decided to solve it with the fixed-point iteration method (http://en.wikipedia.org/wiki/Fixed_point_iteration) but I also have to use adaptive time stepping to iterate the equations. I have decided to chose the time step in the following way:

$dt=\alpha min(\left|\frac{A(x,y)}{x}\right|^{-1},\left|\frac{B(x,y)}{x}\right|^{-1},\left|\frac{C(x,y)}{y}\right|^{-1},\left|\frac{D(x,y)}{y}\right|^{-1},\left|\frac{A(x,y)-B(t,x,y)}{x}\right|^{-1},\left|\frac{C(x,y)-D(x,y)}{y}\right|^{-1})$

with $\alpha$ a constant less than unity.
What do you think of my choice? Do you think there's a better way to define the time step?
Also, I keep $\alpha$ constant but I noticed I have to decresead for some particular values of x and y. What do you think might a way to determin $\alpha$ as a function of x and y?

 

[gtoguy97]

My choice of time step seems reasonable. However, you may need to adjust the value of $\alpha$ depending on the particular values of $x$ and $y$. You could consider using a local error estimator such as the Richardson extrapolation technique (https://en.wikipedia.org/wiki/Richardson_extrapolation) to determine the optimal value of $\alpha$ for each iteration. Alternatively, you might explore more sophisticated adaptive time-stepping methods such as Runge-Kutta or implicit integration techniques.