I am currently using Euler method and it is 4-8 times too slow. Which method will be fastest? I need it to compute Turing's reaction-diffusion system.
Euler's method is the fastest possible single-step, explicit, non-adaptive method for a given fixed step size $h$.
If this is too slow, I would do the following things, in this order.
I'm willing to bet that you can solve your speed problem with step (1) (hint: if you're iterating through 256x256 elements, you don't need to necessarily encapsulate the method in a double loop).