Upwind Scheme - First-order Upwind Scheme

First-order Upwind Scheme

The simplest upwind scheme possible is the first-order upwind scheme. It is given by

\quad (1) \qquad \frac{u_i^{n+1} - u_i^n}{\Delta t} + a \frac{u_i^n - u_{i-1}^n}{\Delta x} = 0 \quad \text{for} \quad a > 0
\quad (2) \qquad \frac{u_i^{n+1} - u_i^n}{\Delta t} + a \frac{u_{i+1}^n - u_i^n}{\Delta x} = 0 \quad \text{for} \quad a < 0

Defining

\qquad \qquad a^+ = \text{max}(a,0)\,, \qquad a^- = \text{min}(a,0)

and

\qquad \qquad u_x^- = \frac{u_i^{n} - u_{i-1}^{n}}{\Delta x}\,, \qquad u_x^+ = \frac{u_{i+1}^{n} - u_{i}^{n}}{\Delta x}

the two conditional equations (1) and (2) can be combined and written in a compact form as

\quad (3) \qquad u_i^{n+1} = u_i^n - \Delta t \left

Equation (3) is a general way of writing any upwind-type schemes. The upwind scheme is stable if the following Courant–Friedrichs–Lewy condition (CFL) condition is satisfied.

\qquad \qquad c = \left| \frac{a\Delta t}{\Delta x} \right| \le 1 .

A Taylor series analysis of the upwind scheme discussed above will show that it is first-order accurate in space and time. The first-order upwind scheme introduces severe numerical diffusion in the solution where large gradients exist.

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