# 2.4.1. One-dimensionnal example

This example is the one contained in the Fortran package provided for this course. The length of the domain is L (variable Long_Physique). The total number of cells is NX.

The mesh is depected in Figure 2.4

> Figure 2.4 Definition of the 1D mesh used in the Fortran package. The usefulness of ghost cells will de detailed in Boundary Conditions

Each cell is defined by :

- Its length (volume) \(\Delta x = \frac{L}{NX}\)
- Two faces numbered 2 (right) and 4 (left)
- For each face the normal vector components. Here (1,0) and (-1,0) for the right (2) and left (4) face, respectively
- For each face, the surface: here equal to 1.

## 2.4.1.1. Numbering

The numbering of the cell is labeled with index i which varies from 0 to NX+1. There are two-fictitious cells (0 and NX+1) used for the treatment of boundary conditions. The conservative vector U must be defined in these cells to make the calculation of the flux at face 4 of cell 1 (at face 2 of cell NX) possible.

# 2.4.2. Two-dimensionnal mesh

Here is shown what the final 2D mesh should look like at the end of this course.

Two lengths are required Lx in the x direction and Ly in the y direction. Cells are labeled with the couple (i,j) and identically to 1D mesh i varies from 0 to NX+1 and j from 0 to NY+1.

The mesh is depicted in Figure 2.4

> Figure 2.5 A typical2D mesh expected at the end of the course

Each cell is defined by :

- Its surface (volume) \(\Delta x * \Delta y\)
- Four faces numbered 1 to 4 (See Figure 2.3)
- For each face \(\vec{n}_k\), the normal vector.
- For each face, its length (surface): here equal to \(\Delta x\) or \(\Delta y\)