Background: I am an ok coder but have not learned to use either of the above programs, both of which are freely available to me at my workplace and both of which I have been told will basically do the job with general numerical methods.
Extra background: The models I'm working with are black holes, so they are singular at at least one boundary (the event horizon), meaning that an adaptive step size method must be used; and given that there are essentially more free parameters to a general solution at infinity than at the event horizon, we sometimes need to use a "shooting" technique where we take 'initial' conditions at (near) both boundaries and try to match them up in the middle.
I can work out/look up the numerical recipes for the methods - but any advice on which programming language is more suitable would be greatly appreciated. Also, show your working ;)
It depends on the complexity of the problem and whether it's one-off or something that will be more long term. For simple to medium complexity and one-off problems, MatLab-for instance-is useful, since it, already, provides a framework in which to set up the problem to be solved.
The price to pay for this convenience is, however, performance. Any such framework involves assumptions about the problems people typically solve with it. Beyond some level of complexity the computer resources handling the framework can't be allocated to the solution itself. One reason Matlab's compiler will be slower than the C compiler is because of the overhead of Matlab.
So for a complex or long term project it's better to invest in writing the code in C and using its libraries (e.g. GNU scientific library) for certain parts.
I would recommend MATLAB ,Matlab is one of very powerful software package for solving numerical problems like ODEs you have as it uses inbuilt Rungekuta codes which gives the most accurate solution.
matlab is ok, you can also put your c or fortan code into your matlab scripts. Nowadays, computation cluster in university also support Matlab well.
C ++ is faster , but working with MATLAB is easier. If you have a very large data, that would be better to choose C , however if the data is not that big, MATLAB is better.
My experience agrees with Seyed above. For ODEs, I've tended to write code initially in MATLAB (4th order Runge-Kutta methods with step size checks). If the code runs quickly enough then there is no need to do anything else, but if it slow, MATLAB code can be easily adapted to c++ which will run essentially the same code, but faster. I've found in the past that FORTRAN is really fast for such brute force computations.
For PDE (finite difference/finite element methods), again I tend to use MATLAB initially. I've found MATLAB to be quite user friendly with regard to parrallelizing code, which can allow computations to be completed. I'm not sure how user friendly this process will be in c++ or FORTRAN though.
Round off errors would always give nightmares.
Choose the Dispersion Relation Preserving (DRP) numerical schemes based on STABILITY ANALYSIS.
Also, anything that can handle QUAD precision is highly recommended to get more accurate solution.
Good luck
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