06 - 1D Advection (schemes & stability)

This exercise investigates the one-dimensional advection equation (pure transport, no diffusion) and compares several time–space discretizations on two initial conditions (Gaussian vs. block anomaly). The case highlights numerical stability, diffusion, and dispersion and how they depend on the scheme and the Courant number.

The main objectives are:

1. Formulate the 1D advection equation and the Courant–Friedrichs–Lewy (CFL) stability constraint.
2. Implement and compare multiple schemes:
   • FTCS (Forward Time–Centered Space) — note: unstable for pure advection,
   • Upwind (1st order), Downwind, Lax–Friedrichs,
   • Leapfrog (SLF), Semi-Lagrangian, and a tracer method.
3. Apply periodic boundary conditions using ghost cells and verify mass/shape transport.
4. Assess numerical diffusion/dispersion by contrasting results for a Gaussian (smooth) vs. block (sharp) profile.
5. Visualize the time evolution and simple diagnostics (e.g., peak amplitude).

An example animation of the evolving profile using the upwind scheme and the semi-lagrangian scheme is shown in Figure 1 and Figure 2, respectively.

Figure 1. Advection of Gaussian temperature anomaly using the upwind scheme.

Figure 2. Advection of Gaussian temperature anomaly using the semi-lagrangian scheme.