CFL Criterion Calculation for 1D Advection

The problem requires determining the maximum permissible time-step ($\Delta t$) for the numerical solution of the 1-D advection equation, $\frac{\partial \Phi}{\partial t} + c \frac{\partial \Phi}{\partial x} = 0$, using the Courant-Friedrichs-Lewy (CFL) criterion.

Understanding the CFL Criterion

The CFL criterion is a necessary condition for the stability of certain numerical methods used for solving partial differential equations. For the 1-D advection equation, the condition is often expressed as:

• $C$ represents the Courant number.
• $c$ is the speed of the wave (advection velocity).
• $\Delta t$ is the time-step size.
• $\Delta x$ is the spatial grid resolution.
• $C_{max}$ is the maximum Courant number allowed for stability, typically assumed to be $1$ for simple explicit schemes like the first-order upwind method.

We will use $C_{max} = 1$.

Applying the CFL Condition

The given parameters are:

• Advection velocity, $c = 300$ m s$^{-1}$.
• Spatial resolution, $\Delta x = 3$ km.

First, ensure consistent units by converting the spatial resolution from kilometers to meters:

To find the maximum time-step ($\Delta t_{max}$), we rearrange the CFL formula:

Substituting the known values (with $C_{max} = 1$):

Performing the calculation:

Result

The maximum time-step allowed by the CFL criterion for the given spatial resolution and advection speed is 10 seconds.