Which method for ODE instead of Euler's?
There are lots of methods. I have usually found a 4th order adaptive step size Runge Kutta method to work fairly well. A Leapfrog method might better preserve invariants over the long term. I'm afraid I don't know enough about reaction-diffusion systems to know how that might change my answer. You might look into our sister sites in scientific computing, computational sciences, etc. for more information. If you're not already familiar with the "numerical recipes" series, that could prove very useful, although it may not have anything exactly suited to your needs.