I read about stiff and non-stiff differential equations, but I could not understand completely what they are. Are these terms used depending on variation speed of solution? At first glance, can a differential equation be classified as stiff or non-stiff equation?
For linear systems, a system of differential equations is termed stiff if the ratio between the largest and the smallest eigenvalue is large. A stiff system has to treated numerically in a different way than a "normal" system. The wikipedia article on "stiff equation" provides a quick overview. Personally I made good experience with the solvers from Ernst Hairer for systes of linear, stiff differential equations.
For linear systems, a system of differential equations is termed stiff if the ratio between the largest and the smallest eigenvalue is large. A stiff system has to treated numerically in a different way than a "normal" system. The wikipedia article on "stiff equation" provides a quick overview. Personally I made good experience with the solvers from Ernst Hairer for systes of linear, stiff differential equations.
Mr. Dohnal,
Many thanks for your response. You mention about ratio of the eigen values of the equation. What the ratio should be to call an equation stiff? Or is there any ratio?
According to scholarpedia:
Solutions of the differential equation y′(t)=λiy(t), y(0)=1. for Re(λi)
In applications you usually find out if your ODE is stiff by numerically integrating it and watching Runge Kutta fall apart. Then you find the eigenvalues of the linearized DE and realize after the fact you have a stiff system. Then you grumble as you program in your stiff numerical solvers.
As previous answer covered well the theoretical background of stiff system of equations, I want to address the practical point of view.
When solving a stiff system of equations, standard numerical methods are only stable for small time steps. Whereas the same linear solver will perform great (stable + converge) with large time steps when solving a non-stiff linear system. When facing a stiff system of equation the use of a preconditioner (e.g. Incomplete LU factorization) combined with a standard linear solver can improve the stability and the convergence. It basically rearrange the equations in the system of equations to reduce the stiffness.
http://nl.mathworks.com/help/matlab/ref/gmres.html
On top of the great information above, the definition given is not a sure fire for ANY differential equation or system thereof. The paper here probes a bit deeper.
https://people.maths.ox.ac.uk/trefethen/publication/PDF/1993_56.pdf
Is it accurate to use a stiff solver to solve nonlinear ODEs even though the numerical methods used to solve the ODE are numerically stable? The simulation is very slow when using a nonstiff solver..
A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution.
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