advection_basics.rst

Advection solvers

The linear advection equation:

a_t + ua_x + va_y = 0

provides a good basis for understanding the methods used for compressible hydrodynamics. Chapter 4 of the notes summarizes the numerical methods for advection that we implement in pyro.

pyro has several solvers for linear advection, which solve the equation with different spatial and temporal intergration schemes.

advection solver

:pyadvection implements the directionally unsplit corner transport upwind algorithm colella:1990 with piecewise linear reconstruction. This is an overall second-order accurate method, with timesteps restricted by

$$\Delta t < \min \left \{ \frac{\Delta x}{|u|}, \frac{\Delta y}{|v|} \right \}$$

The parameters for this solver are:

advection_fv4 solver

:pyadvection_fv4 uses a fourth-order accurate finite-volume method with RK4 time integration, following the ideas in mccorquodalecolella. It can be thought of as a method-of-lines integration, and as such has a slightly more restrictive timestep:

$$\Delta t \lesssim \left [ \frac{|u|}{\Delta x} + \frac{|v|}{\Delta y} \right ]^{-1}$$

The main complexity comes from needing to average the flux over the faces of the zones to achieve 4th order accuracy spatially.

The parameters for this solver are:

advection_nonuniform solver

:pyadvection_nonuniform models advection with a non-uniform velocity field. This is used to implement the slotted disk problem from ZALESAK1979335. The basic method is similar to the algorithm used by the main advection solver.

The paramters for this solver are:

advection_rk solver

:pyadvection_rk uses a method of lines time-integration approach with piecewise linear spatial reconstruction for linear advection. This is overall second-order accurate, so it represents a simpler algorithm than the advection_fv4 method (in particular, we can treat cell-centers and cell-averages as the same, to second order).

The parameter for this solver are:

advection_weno solver

:pyadvection_weno uses a WENO reconstruction and method of lines time-integration

The main parameters that affect this solver are:

General ideas

The main use for the advection solver is to understand how Godunov techniques work for hyperbolic problems. These same ideas will be used in the compressible and incompressible solvers. This video shows graphically how the basic advection algorithm works, consisting of reconstruction, evolution, and averaging steps:

Examples

smooth

The smooth problem initializes a Gaussian profile and advects it with u = v = 1 through periodic boundaries for a period. The result is that the final state should be identical to the initial state—any disagreement is our numerical error. This is run as:

./pyro.py advection smooth inputs.smooth

By varying the resolution and comparing to the analytic solution, we can measure the convergence rate of the method. The smooth_error.py script in analysis/ will compare an output file to the analytic solution for this problem.

The points above are the L2-norm of the absolute error for the smooth advection problem after 1 period with CFL=0.8, for both the advection and advection_fv4 solvers. The dashed and dotted lines show ideal scaling. We see that we achieve nearly 2nd order convergence for the advection solver and 4th order convergence with the advection_fv4 solver. Departures from perfect scaling are likely due to the use of limiters.

tophat

The tophat problem initializes a circle in the center of the domain with value 1, and 0 outside. This has very steep jumps, and the limiters will kick in strongly here.

Exercises

The best way to learn these methods is to play with them yourself. The exercises below are suggestions for explorations and features to add to the advection solver.

Explorations

• Test the convergence of the solver for a variety of initial conditions (tophat hat will differ from the smooth case because of limiting). Test with limiting on and off, and also test with the slopes set to 0 (this will reduce it down to a piecewise constant reconstruction method).
• Run without any limiting and look for oscillations and under and overshoots (does the advected quantity go negative in the tophat problem?)

Extensions

• Implement a dimensionally split version of the advection algorithm. How does the solution compare between the unsplit and split versions? Look at the amount of overshoot and undershoot, for example.

• Research the inviscid Burger's equation—this looks like the advection equation, but now the quantity being advected is the velocity itself, so this is a non-linear equation. It is very straightforward to modify this solver to solve Burger's equation (the main things that need to change are the Riemann solver and the fluxes, and the computation of the timestep).

  The neat thing about Burger's equation is that it admits shocks and rarefactions, so some very interesting flow problems can be setup.