Basic Driver

All of the numerical methods for advection and hydro follow the same general form, so the basic driver to evolve for a time \(T\) is:

• setup the grid

• set initial conditions

• evolve while \(t < T\):

  • fill boundary conditions

  • compute the timestep

  • evolve for a single timestep

    • construct interface states

    • evaluate fluxes

    • do conservative update

  • t = t + dt

  • output

Controlling the Timestep

Recall that for stability, explicit methods are limited such that the information cannot move more than one zone per timestep—this is the Courant-Friedrichs-Lewy condition.

For advection, we can express this as:

\[\Delta t = C \frac{\Delta x}{|u|}\]

where \(C \le 1\) is a dimensional number called the Courant number.

When writting an advection solver, the user will typically specify the number of zones, which gives \(\Delta x\), and the Courant number, and then we compute the timestep.

Note

It might be tempting to just set \(C = 1\), but in practice we usually do something like \(C = 0.8\). The hydrodynamics equations are non-linear and stability analysis is done on the linear equations, so we need to allow ourselves a little headroom.