5.1.5 Going multi-dimensional

Directional splitting or dimensional splitting is simply the technique to apply operator splitting to the spatial derivatives in the Euler equations:

$$\! \frac{\partial }{\partial t} \ensuremath{\mathchoice{\... ...th$\scriptstyle 0$}} {\mbox{\boldmath$\scriptscriptstyle 0$}}}$$ (221)

becomes

$$q_{X,i}^{n,*} = q_i^{n} - \frac{\Delta t}{\Delta x} \left( f \! \left( q_{i+\frac{1}{2}}^n \right) - f \! \left( q_{i-\frac{1}{2}}^n \right) \right)$$

$$q_{X+Y,i}^{n+1} = q_{X,i}^{n,*} - \frac{\Delta t}{\Delta y} \left( f \! \left( q... ...n,*} \right) - f \! \left( q_{X,i-\frac{1}{2}}^{n,*} \right) \right) \enspace .$$ (222)

Directional splitting in general works very well and allows the application of powerful algorithms developed for 1D problems.

However, there are cases when a small amount of additional multi-dimensional tensor viscosity is necessary to damp spurious oscillations (even with PPM scheme).