Wave-digital principles can be used advantageously for numerically integrating partial differential equations (PDEs) describing physical systems. The resulting algorithms are massively parallel, fully local, and achieve high robustness by using a multidimensional extension of the inherent physical passivity of the original physical system. As has been shown previously, the approach is applicable also to the nonlinear PDEs of fluid dynamics, and this without requiring large systems of algebraic equations to be inverted. It will be shown that important improvements can be obtained on the one hand by arriving at a substantially better algorithm, and on the other, by systematizing the derivation and refining the earlier way of proceeding.