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Numerical stability of stiff problem

  1. Feb 13, 2013 #1
    I have a very complicated stiff system, which can be expressed as:

    [itex]x^{\prime}(t)=A(x,y)-B(x,y)[/itex]
    [itex]y^{\prime}(t)=C(x,y)-D(x,y)[/itex]

    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:

    [itex]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})[/itex]

    with [itex]\alpha[/itex] 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 [itex]\alpha[/itex] 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 [itex]\alpha[/itex] as a function of x and y?
     
  2. jcsd
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