First Order Upwind

The Upwind scheme is a common remedy for the problem of convection in the finite volume method. It is based on the upstream velocity direction and thus inherits the upstream value of the dependent variable $\varphi$. A simple convection equation in one dimension is

\begin{align} \frac{\partial\phi}{\partial t} = \frac{\mathrm{d}u\phi}{\mathrm{d}x} \end{align}

Using a control volume integration, we have

\begin{align} \int_{\mathrm{w}}^{\mathrm{e}} \int_{t_n}^{t_{n+1}} \frac{\partial\phi}{\partial t} \mathrm{d} t \mathrm{d}x = \int_{t_n}^{t_{n+1}} \int_{\mathrm{w}}^{\mathrm{e}} \frac{\mathrm{d}u \phi}{\mathrm{d}x} \mathrm{d}x \mathrm{d} t \end{align}

By assuming first backward euler integration in time, we have

\begin{align} \int_{\mathrm{w}}^{\mathrm{e}} \left(\phi_P^{n+1} - \phi_P^n\right) \mathrm{d}x = \int_{t_n}^{t_{n+1}} \left[ \left(u \phi\right)^{\mathrm{e}} - \left(u \phi\right)^{\mathrm{w}} \right] \mathrm{d} t \end{align}


\begin{align} \int_{\mathrm{w}}^{\mathrm{e}} \int_{t_n}^{t_{n+1}} \frac{\partial\phi}{\partial t} \mathrm{d} t \mathrm{d}x = \end{align}

assuming that the value of $\phi_P$ prevails over the control volume, the spatial integral

\begin{align} \int_{\mathrm{w}}^{\mathrm{e}} \left(\phi_P^{n+1} - \phi_P^n\right) \mathrm{d}x = \left(\phi_P^{n+1} - \phi_P^n\right) \Delta x \end{align}

Furthermore, we assume that the velocity u