How can you know if a numerical solution is correct?

[maughanster]

Hi PF,

Suppose I numerically solve a nonlinear system of differential equations. How can I know if my solution is correct (if there is no known analytic solution)? What are the standard practices people do? I have a couple of ideas, but I want to know what people are already doing.

Danke!

 

[Ssnow]

I numerically solve a nonlinear system of differential equations.

you must specify in what sense because if it is a system of differential equation the solution is supposed to be a function ... what you mean when you said "I numerically solve a nonlinear system of ...'' ? There is an approximate solution (for example a polynomial ...)? Have you found an evalutation of the solution but not the complete analytic expression ? Another question: there are boundary condition for this system ?
Ssnow

 

[maughanster]

Thanks for the reply!

I don't know a ton about these types of problems. Say I want to numerically itegrate a system of 4 nonlinear ODEs as an IVP. Or what if I have PDES as an IVP and do RK4 and finite differencing. Can I know if my numerical solution is correct? Assume I do not have an approximate solution (i.e. a polynomial)

 

[m4r35n357]

The main method I currently use is to compare my answer with a "better" solution (example search term "Clean Numerical Simulation"). It sounds a bit obvious, but if you do another simulation with higher precision/order and smaller timestep, you can compare the results and see where they are diverging. This can be automated. I use arbitrary precision (MPFR) arithmetic for mitigating roundoff and because I trust INRIA more than I trust Intel to do floating point properly ;) I use the Taylor Series Method to obtain arbitrary order of integration.

Sometimes there are invariants and constants that you can evaluate and monitor. There are always local error estimates, but I tend to solve nonlinear ODEs that get rather degenerate or chaotic and these render local error irrelevant.