[eric_t]

No, as the paper says, explicit Euler integration is extremely unstable. Implicit Euler, as the paper uses, is stable but only first order accurate.

If you're dealing with fluids with low diffusion compared to advection, some higher order Runge-Kutta scheme is typically used. For fluids with high diffusion, however, this will give you very high restrictions on the time step. For this cases, a hybrid scheme is often used, where the advection term is integrated explicitly using an Adam-Basforth scheme and the diffusion term is treated semi-implicitly with a Crank-Nicholson scheme. This means you have to solve additional linear equation systems for the velocities for each time step (Helmholtz type equations), but this is still faster since you can use longer time steps.

All of this may sound very complicated, but it all follows the same patterns, so it's really fairly straight forward.