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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{\...$\scriptstyle 0$}}
{\mbox{\boldmath$\scriptscriptstyle 0$}}}
\end{displaymath} (221)
$\displaystyle q_{X,i}^{n,*}$ $\textstyle =$ $\displaystyle 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)
$\displaystyle q_{X+Y,i}^{n+1}$ $\textstyle =$ $\displaystyle q_{X,i}^{n,*} -
\frac{\Delta t}{\Delta y}   \left( f \! \left( q...
...n,*} \right)
f \! \left( q_{X,i-\frac{1}{2}}^{n,*} \right)
\enspace .$ (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).