4.3.1 Godunov-type schemes

4.3.1 Godunov-type schemes

4.3.1 Godunov-type schemes

Godunov's idea: solve a separate Riemann problem at each cell boundary (Fig. 32).
Figure 32: Many Riemann problems.
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RSA: three steps:

* Reconstruct $\rho(x)$, $\rho\ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle v$}} {\mbox{\boldmath... ...{\mbox{\boldmath$\scriptstyle v$}} {\mbox{\boldmath$\scriptscriptstyle v$}}}(x)$, $\rho e_{\rm ik}(x)$ from $\rho_i$, $\rho\ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle v$}} {\mbox{\boldmath... ... {\mbox{\boldmath$\scriptstyle v$}} {\mbox{\boldmath$\scriptscriptstyle v$}}}_i$, ${\rho e_{\rm ik}}_i$ (constant values within cell).
* Solve the Riemann problems for $\Delta t$ (and compute fluxes across cell boundaries).
* Average $\rho(x)$, $\rho\ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle v$}} {\mbox{\boldmath... ...{\mbox{\boldmath$\scriptstyle v$}} {\mbox{\boldmath$\scriptscriptstyle v$}}}(x)$, $\rho e_{\rm ik}(x)$ to get $\rho_i$, $\rho\ensuremath{\mathchoice{\mbox{\boldmath$\displaystyle v$}} {\mbox{\boldmath... ... {\mbox{\boldmath$\scriptstyle v$}} {\mbox{\boldmath$\scriptscriptstyle v$}}}_i$, ${\rho e_{\rm ik}}_i$ (apply the conservative update formula).

The concept is very useful, but the scheme is too diffusive in its original form.