3.2.1 Numerical schemes

Wide variety of finite difference methods to solve equation ``i" in (72) (the scalar advection equation)

Finite difference methods are based on a discretization of the $x-t$ plane defined by the discrete mesh points $(x_j,t^n)$

$$x_j = (j-1/2)\Delta x, j=1,2,\ldots$$ (74)

$$t^n = n\Delta t, n=0,1,2,\ldots$$ (75)

where $\Delta x$ and $\Delta t$ are, respectively, the cell width and the time step

A finite difference scheme is a time-marching procedure which permits to obtain approximations to the solution in the mesh points, $w_j^{n+1}$, from the approximations in previous time steps $w_j^{n}$