To derive the equivalent differential equation for the first-order upwind scheme in the context of a 1D pure convection flow model, we start with the one-dimensional convection equation given by:

$\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0$

where $u(x, t)$ is the dependent variable (e.g., concentration, temperature), $c$ is the convection speed, and $x$ is the spatial coordinate.

Derivation of the Upwind Scheme

For the first-order upwind scheme, we discretize the spatial domain using a grid with a spacing of $\Delta x$ and a time step of $\Delta t$. The upwind discretization assumes that we are moving in the positive $x$-direction, thus we can express the derivative $\frac{\partial u}{\partial x}$ using the values from the grid:

$\frac{\partial u}{\partial x} \approx \frac{u_i - u_{i-1}}{\Delta x}$

Here, $u_i$ represents the value at grid point $i$, and $u_{i-1}$ is the value at the previous grid point.

Substituting this into our convection equation, we have:

$\frac{\partial u}{\partial t} + c \frac{u_i - u_{i-1}}{\Delta x} = 0$

Now, we can express the time derivative $\frac{\partial u}{\partial t}$ using a forward difference:

$\frac{\partial u}{\partial t} \approx \frac{u_i^{n+1} - u_i^n}{\Delta t}$

where $u_i^n$ is the value at time level $n$.

Combining these equations, we can rewrite the convection equation as:

$\frac{u_i^{n+1} - u_i^n}{\Delta t} + c \frac{u_i^n - u_{i-1}^n}{\Delta x} = 0$

Rearranging this gives us the update formula for the first-order upwind scheme:

$u_i^{n+1} = u_i^n - \frac{c \Delta t}{\Delta x} (u_i^n - u_{i-1}^n)$

Letting $\lambda = \frac{c \Delta t}{\Delta x}$, we have:

$u_i^{n+1} = u_i^n - \lambda (u_i^n - u_{i-1}^n)$

Stability Analysis

To discuss the stability of this numerical scheme, we employ the von Neumann stability analysis. We assume a solution of the form:

$u_i^n = \xi^n e^{ikx_i}$

where $\xi$ is the growth factor, $k$ is the wave number, and $x_i = i \Delta x$.

Substituting this into our update equation gives:

$\xi e^{ikx_i} = e^{ikx_i} \left( 1 - \lambda (1 - e^{-ik\Delta x}) \right)$

Cancelling $e^{ikx_i}$ and rearranging, we find:

$\xi = 1 - \lambda (1 - e^{-ik\Delta x})$

For stability, we require $|\xi| \leq 1$. Analyzing this, we find that:

$|\xi| = |1 - \lambda (1 - e^{-ik\Delta x})|$

The term $e^{-ik\Delta x}$ oscillates and does not contribute to growth or decay in magnitude. Thus, the critical factor for stability comes from $\lambda$. The necessary condition for stability generally is:

$\lambda \leq 1$

Thus, for the upwind scheme to be stable, we require:

$\frac{c \Delta t}{\Delta x} \leq 1 \quad \text{or} \quad \Delta t \leq \frac{\Delta x}{c}$

Summary

In summary, the first-order upwind scheme for the 1D pure convection model leads to a specific update formula with a stability condition dependent on the CFL (Courant-Friedrichs-Lewy) condition. The scheme is conditionally stable for $\lambda \leq 1$.

Further Questions

1. What are the implications of violating the CFL condition in numerical simulations?
2. How would the scheme change for convection in a negative direction?
3. Can you explain the significance of the upwind scheme compared to central difference schemes?
4. What modifications can be made to improve the accuracy of the upwind scheme?
5. How would you apply this analysis to multi-dimensional convection problems?

Tip

Always ensure that the time step and spatial discretization are appropriately chosen to maintain stability in numerical simulations.