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# The Godunov Method for Euler Equations, using EXACT Riemann Solver

## Big Picture

- Define a piece-wise function of initial data

- Solve the IVP for the conservation laws, but with modified (discretized) initial data to get the solution for the next time level

- This produces local Riemann Problems (RPs), centered on intercell boundaries.

- Solve these local RP's, and use solution to compute fluxes

- Use these fluxes to update cells.

## The method

### Finding Pstar

Pstar is the solution for pressure in the "star region", which is set up at the intercell boundaries by the left and right cells involved in the local Riemann problems. This region is where the contact discontinuity lies, flanked on either side by a given combination of 2 shock or rarefaction waves. The first step at each intercell boundary is to solve for pstar. Pstar is found by Newton's iterative scheme, with the routine as it was in the Exact Riemann solver described before.

In conservative form, the Godunov method is written:

where

is the numerical flux, which = the physical flux evaluated along the intercell boundary using the exact solution to the local Riemann Problem (LRP).

The time-step, is constrained by the CFL condition, which uses the max wave speed on the grid. The constraint used in the God. code here allows any given wave on the grid to travel a full delta_x in a time-step. In allowing this, we achieve a more efficient time-marching scheme, but with the assumption that wave interactions do not lead to wave accelerations.

## The code

Here is an html version of the main program, with encompassing module, and problem.data file.

## Debugging

## Results

### Attachments (14)

- GodEulRho.png (19.0 KB ) - added by 12 years ago.
- GodEulP.png (21.1 KB ) - added by 12 years ago.
- GodEulU.png (22.3 KB ) - added by 12 years ago.
- God.2.png (50.2 KB ) - added by 12 years ago.
- God.png (50.2 KB ) - added by 12 years ago.
- LoopSR.png (44.1 KB ) - added by 12 years ago.
- program_out.png (28.7 KB ) - added by 12 years ago.
- God_density_init.png (12.8 KB ) - added by 12 years ago.
- God_density.png (21.5 KB ) - added by 12 years ago.
- GodDebug.png (72.1 KB ) - added by 11 years ago.
- test.png (5.8 KB ) - added by 11 years ago.
- test4.png (8.7 KB ) - added by 11 years ago.
- ExactTest4.png (29.1 KB ) - added by 11 years ago.
- GodTest4.png (37.9 KB ) - added by 11 years ago.

Download all attachments as: .zip

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