Daniel, Sean, and Bruce - Thanks to each of you for your detailed and helpful replies. I very much appreciate your help. You've provided me with enough leads and ideas to move forward. At least as far as I'm concerned, please consider this help request closed.

Is an upgrade to Mathematica Starter or Standard likely to improve performance?

No.

1. The home/student/proffesional/... editions of Mathematica are basically equally powerful. There are versions of Mathematica with some extra features that could be used in some cases to make things run faster, but that doesn't apply here. The home edition is for use at home and the student edition are for current students, but they aren't dumbed down at all.

2. A standard Mathematica license gives you 4 subkernels for each kernel (At least I don't believe this has changed. It was true a year ago). This means that if you run something using Parallelize like ParallelMap, the computation will be broken down into 4 subkernels. You'd probably then expect to see 4 cores being used. You can test this by creating some kind of parallel computation with ParallelTable/ParallelMap and see how many cores are being used. To have Parallelize split a computation between 8 cores requeries 8 subkernels for each kernel. You can purchase a license for Mathematica that will give you access to more subkernels. If you wanted to do this, I would contact Wolfram Technical Support and let them know that you want more subkernels for this reason. They can get you into contact with a sales person and explain in more detail.

3. Not everything that runs in parallel or is threaded runs through this subkernel system. For example, Mathematica may use some standard libraries which are already threaded. This is the case, I believe, for some of the libraries it calls for numerical linear algebra for instance. Buying additional subkernels will help you if you have some explicitly parallel code that you've written in the Wolfram Language using Parallelize/ParallelMap/ParallelTable/.... .

4. NDSolve is not a kind of computation that easily lends itself to parallelization. I would not expect to see Improved performance from it on a machine with more cores. That said, it might be possible to write parallelized code to solve your problem. In the meantime, I would suggest trying to improve the speed of your current code. There are a number of resources online which talk about common techniques that are used to improve code speed: http://blog.wolfram.com/2011/12/07/10-tips-for-writing-fast-mathematica-code/

I simply do not know the answers regarding whether we, or others, have useful and recent benchmarks. The links below might be of some use but most, I think, are dated.

http://www.stats.uwo.ca/faculty/aim/epubs/benchmark/mathemat.htm

http://reference.wolfram.com/language/Benchmarking/ref/Benchmark.html

https://homepages.fhv.at/ku/karl/mma.html

http://www.scientificweb.de/mathstef3.html

If at all possible I would suggest posting as concise as possible a version of the slow code, either directly here or, if too large, in an attached notebook. That way you might get useful suggestions on speedup or at least diagnosis as to what might be the cause of the slowness. One thing I can say is that it might well be a problem for which parallelization is simply not a possibility, even at the linear algebra or related low level. But that does not imply there is no other hope for speed improvement, hence the suggestion that you try to post an example.