4.5 Multidimensional problems

$$\begin{eqnarray*} \frac{\partial \bf u \rm }{\partial t}+ \frac{\partial \bf f... ...ial y}+ \frac{\partial \bf h(u) \rm }{\partial z}= \bf s(u) \rm\end{eqnarray*}$$

• Dimensional splitting
  $$\begin{eqnarray*} {\bf u}^{n+1}_{i,j,k} = {\cal L}_s^{\Delta t/2} {\cal L}_h^{... ...cal L}_f^{\Delta t} {\cal L}_s^{\Delta t/2}{\bf u}^{n}_{i,j,k} \end{eqnarray*}$$
  
  
  
  
  
  

  where ${\cal L}_f$, ${\cal L}_g$ and ${\cal L}_h$ stand for the operators associated, respectively, to the 1D systems (PDEs)

  $$\begin{eqnarray*} \frac{\partial {\bf u}}{\partial t}+\frac{\partial {\bf f(u)}... ...l {\bf u}}{\partial t}+\frac{\partial {\bf h(u)}}{\partial z}=0 \end{eqnarray*}$$
  
  
  
  
  
  
  ${\cal L}_s$ is the one that solves a system of ordinary differential equations (ODEs) of the form
  $$\begin{eqnarray*} \frac{\partial \bf u \rm }{\partial t} = \bf s(u) \rm \end{eqnarray*}$$
  
  
  
  
  
  

• Method of lines
  $$\begin{eqnarray*} \frac{d{\bf u}_{i,j,k}(t) }{d t} & = & -\frac{\hat{\bf f}_{i... ...at{\bf h}_{i,j,k-{1\over 2}}} {\Delta z} + \hat{\bf s}_{i,j,k} \end{eqnarray*}$$
  
  
  
  
  
  
  where the numerical fluxes are given by
  $$\begin{eqnarray*} \hat{\bf f}_{i+{1\over 2},j,k} = {\bf f}({\bf u}_{i-p,j,k}, {\bf u}_{i-p+1,j,k},...,{\bf u}_{i+p,j,k}) \end{eqnarray*}$$
  
  
  
  
  
  
  $$\begin{eqnarray*} \hat{\bf g}_{i,j+{1\over 2},k} = {\bf g}({\bf u}_{i,j-p,k}, {\bf u}_{i,j-p+1,k},...,{\bf u}_{i,j+p,k}) \end{eqnarray*}$$
  
  
  
  
  
  
  $$\begin{eqnarray*} \hat{\bf h}_{i,j,k+{1\over 2}} = {\bf h}({\bf u}_{i,j,k-p}, {\bf u}_{i,j,k-p+1},...,{\bf u}_{i,j,k+p}) \end{eqnarray*}$$