1. In the explicit scheme of finite element method, I have found the numerical result of one dimensional unsteady couette flow is unstable and divergent. What is the reason?

2. Also, the explicit scheme of finite volume method, the numerical result of one dimensional unsteady couette flow is oscillated. Why?

ENGWLC

(1). Why not try to find a method which will give you stable solutions?

There are several possible reasons for the behavior you're seeing; you haven't described your problem in sufficient detail that I can accurately diagnose the problem.

However, you definitely want to check what time step size you are using.

It is quite possible that it is too large for the discretization scheme in your simulation.

Explicit schemes (such as forward Euler) are particularly vulnerable to reaching their limits of stability.

Dr Strangelove Thanks. In the explicit scheme by finite element and finite volume, I have used the delta t = 0.002, delta y = 0.001. The numerical result of finite element was unstable and divergent. Finite volume numerical result was oscillate. However, the numerical result of finite difference could obtain 98% accuracy for the exact solution.I do not know the reason.

Can you help?

engwlc,
The values for delta t and for delta y already suggest that you may be exceeding a Courant limit. You should definitely check to see if that is the case.

That fundamental constraint applies to basic hyperbolic equations; but there are other stability limits for parabolic equations, which you might have here if your equation includes a 2nd derivative in space that represents the effect of viscosity.

There are many textbooks that describe these limits, as well as the original papers dating back to (Courant, 1928) and many others as well.

Putting things very roughly, there are 4 important parameters that define, for each scheme, the limits of stability. They are:

• t, the time step size;
• x, the mesh spacing size;
• , the viscosity;
• V, the velocity;

Some books you may find helpful in this regard include

• Difference Methods for Initial-Value Problems, Richtmyer and Morton, 1967;
• Computational Methods for Fluid Flow, Peyret and Taylor, 1985;
• Computational Techniques for Fluid Dynamics, Volume I, Fletcher, 1991.

but I'm sure there are many more that can explain the concepts in much more detail than I would ever be able to reproduce in this forum.

HTH.

Dr Strangelove, Thanks. I have reduced the delt t to obtain the stability and convergence condition in the unsteady couette flow fluid problem.