Ngô Quốc Anh

April 25, 2010

Explicit One-Step Schemes for the Advection Equation: The upwind scheme

Filed under: Giải tích 9 (MA5265), PDEs — Ngô Quốc Anh @ 1:56

Let us consider a very simple class of schemes solving the convection equations entitled the upwind schemes. Precisely, let us consider

\displaystyle u_t+cu_x=0, \quad x \in \mathbb R, t>0

together with the following initial data

\displaystyle u(x,0)=u_0(x), \quad x\in \mathbb R.

It is well-known that the above problem has a unique solution given by u(x, t ) = u_0(x - c t ), which is a right traveling wave of speed c.

Since the upwind schemes are well-known and appear in most of numerical analysis’s textbooks, what I am trying to do here is to give some basic ideas together with their motivation.

First of all, the original upwind schemes are just explicit one-step schemes. Upwind schemes use an adaptive or solution-sensitive finite difference stencil to numerically simulate more properly the direction of propagation of information in a flow field. The upwind schemes attempt to discretize hyperbolic partial differential equations by using differencing biased in the direction determined by the sign of the characteristic speeds. Historically, the origin of upwind methods can be traced back to the work of Courant, Isaacson, and Reeves who proposed the CIR method.

Discretization. Like the finite difference methods, we need to use a a grid with points (x_j , t_n) defined by

x_j=\pm j\Delta x, j \geqslant 0, \quad t^n = \pm n\Delta t, n \geqslant 0.

for some spatial grid size \Delta x and time step \Delta t.


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