The 2 "Bibles" of this method are:

DN de G Allen (1954) Relaxation methods in engineering and science. McGraw-Hill. (available at Amazon http://www.amazon.com/Relaxation-Methods-Engineering-Science-Allen/dp/B000OV01LI )

RV Southwell (1956) Relaxation methods in theoretical physics. Oxford. (available at http://www.amazon.com/Relaxation-Methods-Theoretical-Physics-II/dp/B0000CJJ6B 2 volumes)

As you can see, both books are still available and both are extremely readable...excellent publications.

Dear Mr. Nwaigwe,

I have had not a glimpse of what exactly you are talking about, but considering you are dealing with transport-like PDE's (Vlasov-Poisson, Navier-Stokes, etc), you should have a look at non-oscillatory schemes (TVD class, NVD class, FCT, etc).

Then, relaxation would be applied iteratively: 1) calculate tentative future values, 2) check boundedness, 3) apply some limiting rules to the fluxes where boundedness gets violated (again considering you are dealing with transport-like PDEs), 4) go back to 1).

Details may vary from method to method, but they are not far from this.

The more details you give about your problem, more focused will be the help you will get from this forum.

Hth,

Hélio.

There's also a Wikipedia article at https://en.wikipedia.org/wiki/Relaxation_%28iterative_method%29 which gives a rough summary. The 2 textbooks I named above are really the very best sources.

I do not know about relaxation methods in engineering and science, but for human relaxation come to aegean islands in Greece:

Allright, Mr. Drossos, a little humor helps the intelligence.

I think my question is being misunderstood.

I am not talking about numerical relaxation, am talking about modelling relaxation. These are two different things.

Dear Mr. Nwaigwe,

You told us about a PDE, but which one? What kind of physical phenomena are you dealing with?

Without more context, the understanding will go astray and you will get random answers.

Relaxation will mean something in numerical analysis, and another in materials science, and another in physiology, and so on.

Best regards.

If I have an ODE which needs to be solved for a dependent variable u. If there is another constraint that u must not exceed 1 (for instance), then the relaxation strategy can be used to relax this constraint and allow u to exceed values of 1. This is what I mean, not numerical concept. Thanks all.

Dear Mr. Nwaigwe,

Let me try to give you some suggestions of parallel areas, considering you *are* dealing with numbers and, yes, you will need a numerical strategy to deal with them:

1> In texts about optimization methods, there is something called "barrier methods" which turns an unlimited optimization problem into a limited one by the addition of a 'ad hoc' term to the equations which come to effect only when the solution tries to cross the limits. The simulation becomes highly non-linear because of this barrier function.

2> In CFD, one approach to get non-oscillatory solutions is the use of Flux Corrected Transport, which limits the time step and flux approximation order to avoid creating non-physical local extrema.

3> If you are using a ODE, may be you could tweak a little a Runge-Kutta to shrink the time step every time the solution crosses the limits.

4> Try to solve your ODE without the limits and them reinsert them slowly via a control parameter (google for Homotopy, Perturbation, etc).

Again, without more details, the misunderstandings may continue to turn any goodwill useless.

Best regards.

I think Prof. Bortolon's view is more clear. Beyond this, i would like to add something that the relaxation appears differently for the several specific problems. Therefore it would be better to consider the relaxation of the constraints of each variable partnering to the function itself first , then solve the system. Finally, the common intersection of such solutions be the actual solution/ model with maximum  of lower bounds of constraints and minimum of the upper bounds of the constraints also.

The method of integral measurement using for observer realization has been proposed.

Following on your reply re ODEs of 19 days ago. I focus on a very simple form of relaxation by assuming that the logistic equation, du/dt = a*u*(k-u), is related to your problem because its solutions do not exceed k if it is initially below k. Here, a, k > 0 are parameters, k is often known as ‘carrying capacity’. If we add an extra term with time-varying per capita rate v(t), we can be recast the equation as a logistic model with time-varying carrying and a thus new long-term bound on u. Specifically: du/dt = a*u*(k-u)+v(t)*u, with v(t)>0. This can be rewritten as: du/dt = a*u*((k+v(t)/a)-u) = a*u(q(t)-u), where q(t)=k+v(t)/a is a time varying carrying capacity. Unlike the basic logistic, this ODE is not likely to have an exact solution (unless v(t) is special, such as a constant), but numerical solution should be straightforward. There are many options for the function v(t), depending on the specific application. Here is an article exploring the case where q(t) also obeys a logistic equation; in this case, there is a form of relaxation if q increases with time.

P.S Meyer and JH Ausubel, Carrying Capacity: A Model with Logistically Varying Limits, Technological Forecasting and Social Change 61(3):209-214, 1999.

This specific choice of v(t) incorporates a time scale parameter characterising the pace of the relaxation, which may be relevant. This is just a concrete example to illustrate a way of thinking about modelling relaxation; many other basic models and modifications thereof to incorporate relaxation would be possible.