Changes between Version 11 and Version 12 of u/ehansen/buildcode
- Timestamp:
- 10/31/11 19:13:53 (13 years ago)
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u/ehansen/buildcode
v11 v12 21 21 [[BR]] 22 22 == Discretisation == 23 Discretisation is the process of taking a domain and dividing it into cells. Each cell contains data corresponding to a specific problem initialization thus creating many Riemann problems. In other words, each neighboring pair of cells is now a "local" Riemann problem. Given two neighboring cells, the left cell is the left data state, the right cell is the right data state, and the intercell boundary is the discontinuity position. Now each local Riemann problem can be solved and that solution can be used to calculate the fluxes required to update the cells to the next time step.23 Discretisation is the process of taking a domain and dividing it into cells. The "mesh size" [[latex($\Delta x$)]] is defined as the domain length divided by the number of cells. Each cell contains data corresponding to a specific problem initialization thus creating many Riemann problems. In other words, each neighboring pair of cells is now a "local" Riemann problem. Given two neighboring cells, the left cell is the left data state, the right cell is the right data state, and the intercell boundary is the discontinuity position. Now each local Riemann problem can be solved and that solution can be used to calculate the fluxes required to update the cells to the next time step. 24 24 [[BR]] 25 25 == Godunov's First Order Upwind Scheme == … … 52 52 53 53 [[latex($U_{i}^{n+1} = U_{i}^{n} + \frac{\Delta t}{\Delta x}[F(U_{i-\frac{1}{2}}(0))-F(U_{i+\frac{1}{2}}(0))] \hspace{1 in} (9)$)]] 54 55 In summary, the necessary quantities to update a cell are: its initial quantities, the mesh size, the time step size, and the fluxes through its boundaries. Initial quantities are known, the mesh size is chosen by the user, and the fluxes are calculated from solving the Riemann problem. However, in order to calculate the fluxes for the first and last cell we need more information. 56 [[BR]] 57 == Boundary Conditions == 58 [[BR]] 59 == The CFL Condition == 54 60 [[BR]] 55 61 == Program Outline ==