Actually my doubt is regarding STIFF SYSTEMS. What is the speciality that ode15s possess which ode45 doesn't have so as to use the former one to deal with stiff systems? Is it merely the difference between explicit scheme and implicit scheme which makes them different? Suppose I have an example for the so called 'STIFF SYSTEMS. If I use both solvers ode45 and ode15s to solve it, definitely the system being stiff ode45 either takes long time to give a solution or may not give also. Other than the computation time, can I track some other parameter whose performance given by both the solvers be compared ,which tells me exactly what this stiff property means?
Hi, a system of stiff odes is a system for which classical methods (ode45, etc..) are unstable. Thus, except is the integration step is very small (increasing the CPU time) the solution cannot be obtained exactly. In general the equations contain terms that lead to a rapid variation of the solution.
This is why, implicit methods (ode15s for e.g.) are prefered in this case as they are less sensitive and give better results.
You can refer to the following lecture (attached file)
Hi Nisha,
Take a look at this example with the following ODE:
x' = −1001*x + 3003 − 2000*e^(-t).
If x(0) = 0, the analytical solution is given as
x(t) = -e^(-1001*t) - 2*e^(-t) + 3.
From the solution and our sense of time, you can see the fast exponential term e^(-1001*t), and the slow exponential term e^(-t). When x(t) is plotted, it appears to start at 1, but in fact, there is a rapid transient from x = 0 to 1 that occurs at t < 0.005. After that, the transient diminishes greatly and the slow exponential term e^(-t) takes over gradually.
Physically, if a system contains rapidly and slowly varying components, the ODE is said to be “stiff”. Because the system has rapidly varying components, smaller time step would be required to obtain an accurate solution, and smaller time steps means that the simulation takes longer time.
For your info, Wolfram Mathematica has the "StiffnessSwitching" option implemented in the NDSolve solver that intelligently attempts to detect stiffness at runtime and switch between a non-stiff and a stiff solver as necessary.
http://reference.wolfram.com/language/tutorial/NDSolveStiffnessSwitching.html
L. Shampine and S. Thompson. "Stiff Systems." Scholarpedia. (May 16, 2013) www.scholarpedia.org/article/Stiff_systems.
is a readable and comprehensive account.
Hello Nisha, It looks like this is the major problem you may be encountering. I want to explain in step-by-step process clearly. The LHS of one differential equation say dy/dt = -1000t. This equation is valid if and only if dt is sufficiently close enough to ZERO, here comes the trouble.
We don't know if dt = 0.1 is sufficiently close enough to ZERO to satisfy the above differential equality sign because the definition of the limit is involved here.
So, proper care has to be taken when integrating, not an arbitrary value dt. This is a numerical paradox not a mathematical flaw !!! I hope this further elucidates your understanding !!!
- Badges
- Science topic