4.2.4 The Marquina solver

References: Donat and Marquina, JCP, 125, 42 (1996)

Donat, Font, Ibáñez and Marquina, JCP, 146, 58 (1998)

• No ``artificial" averaged states at cell interfaces

• Marquina's numerical flux is as follows

  
  

  $${\hat{\bf f}}^M({\bf u}_l,{\bf u}_r)= \sum_{p} \left ( \phi... ...{\bf r}^p({\bf u}_l) + \phi^p_- {\bf r}^p({\bf u}_r) \right )$$ (110)

  
  

  where ${\bf r}^p({\bf u}_l)$, ${\bf r}^p({\bf u}_r)$, are the right (normalized) eigenvectors of the Jacobian matrices ${\bf A}({\bf u}_l)$, ${\bf A}({\bf u}_r)$

• The sided local characteristic variables and fluxes:

  
  

  $$\omega_{l}^p = {\bf l}^p({\bf u}_l) \cdot {\bf u}_l \quad \phi_{l}^p = {\bf l}^p({\bf u}_l) \cdot {\bf f}({\bf u}_l)$$ (111)

  $$\omega_{r}^p = {\bf l}^p({\bf u}_r) \cdot {\bf u}_r \quad \phi_{r}^p = {\bf l}^p({\bf u}_r) \cdot {\bf f}({\bf u}_r)$$ (112)

  
  

  where ${\bf l}^p({\bf u}_l)$, ${\bf l}^p({\bf u}_r)$, are the (normalized) left eigenvectors

• If eigenvalue $${\lambda_k({\bf u})}$$ does not change sign in $[{\bf u}_l,{\bf u}_r]$, then (upwind)
  
  If $${ \lambda_k({\bf u}_l)> 0}$$ then
  $${\phi_+^k= \phi_l^k}$$
  $${\phi_-^k= 0 }$$
  else
  $${\phi_+^k= 0 }$$
  $${\phi_-^k= \phi_r^k}$$
  endif
  
  else (entropy-satisfying local-Lax-Friedrichs)
  
  $${ \alpha_k= \max_{{\bf u} \in \Gamma ( {\bf u}_l ,{\bf u}_r)} \vert\lambda_k({\bf u})\vert}$$ $${\phi_{\pm}^k=.5(\phi_{l,r}^k\pm\alpha_k \omega_{l,r}^k)}$$
  
  endif

  where $\Gamma( {\bf u}_l,{\bf u}_r)$ is a curve in phase space connecting ${\bf u}_l$ and ${\bf u}_r$

• For any hyperbolic system where the fields are either genuinely nonlinear or linearly degenerate, we can test the possible sign changes of $\lambda_k({\bf u})$ by checking the sign of $${\lambda_k({\bf u}_l)\cdot\lambda_k({\bf u}_r)}$$

• $\alpha_k$ can also be determined as
  
  $$\alpha_k= \max\{\vert\lambda_k({\bf u}_l)\vert,\vert\lambda_k({\bf u}_r)\vert\}.$$
  
  

• Interesting properties:
  1. Designed for general hyperbolic systems of conservation laws
  2. When applied to constant-coefficient one-dimensional systems yields the exact solution to the Riemann problem
  3. Able to handle ultrarelativistic flows with great accuracy
  4. The dissipation of the scheme eliminates undesired numerical ``pathologies":
     • overheating in shock reflections
     • long wavelength noise behind slowly moving shocks
     • ``carbuncle" formation in hypersonic flow past blunt bodies

• Extended to relativistic hydrodynamics in Donat, Font, Ibáñez and Marquina, JCP, 146, 58 (1998)