4.2.3 The Roe solver

• Idea: determine the approximate solution by solving a constant coefficient linear system instead of the original nonlinear system

• $\rightarrow$ solve a modified conservation law with flux ${\bf f}= {\bf {\hat A}}{\bf u}$

• The linear problem reads:
  

  $$\frac{\partial {\bf u}}{\partial t} + {\bf {\hat A}}({\bf u}_l,{\bf u}_r) \frac{\partial {\bf u}}{\partial x} = 0$$ (107)

  
  

• to determine ${\bf {\hat A}}$ Roe suggested the following conditions:
  1. ${\bf {\hat A}}({\bf u}_l,{\bf u}_r)({\bf u}_r-{\bf u}_l)= {\bf f}({\bf u}_r)-{\bf f}({\bf u}_l)$
  2. ${\bf {\hat A}}({\bf u}_l,{\bf u}_r)$ is diagonalizable with real eigenvalues
  3. ${\bf {\hat A}}({\bf u}_l,{\bf u}_r) \rightarrow {\bf f}^{'}({\bar {\bf u}})$ smoothly as ${\bf u}_l,{\bf u}_r \rightarrow {\bar {\bf u}}$

• Condition 1 (provided 2 is fulfilled) ensures that if a single discontinuity is located at the interface, then the solution of the linearized problem gives the exact solution to the Riemann problem

• Condition 3 is necessary to recover the linearized algorithm from the non-linear one smoothly

• Once the Roe matrix is obtained for every numerical interface, the numerical fluxes are computed by solving the locally linear system

• Roe's numerical flux is as follows

  
  

  $${\hat{\bf f}}^{\rm ROE} = \frac{1}{2} \left[{\bf f}({\bf u}_... ...}\vert \widetilde{\alpha}^{(p)} \widetilde{\bf r}^{(p)}\right]$$ (108)

  
  
  
  

  $$\widetilde{\alpha}^{(p)} = \widetilde{\bf l}^{(p)} ({\bf u}_{\rm r} - {\bf u}_{\rm l})$$ (109)

  
  

  $\widetilde{\lambda}^{(p)}$, $\widetilde{\bf r}^{(p)}$, $\widetilde{\bf l}^{(p)}$ being the eigenvalues and right and left eigenvectors of $\hat{\bf A}$, respectively ($p$ extends from 1 to the number of equations of the system)