2. The Lax problem

The Sod input becomes the Lax problem by changing two states and the final time; then the face flux changes from WENO flux splitting to a Riemann solver. Both runs are checked against the exact solution.

From Sod to Lax

Copy src/tests/flume/regression/sod-x/input.ini and change these keys:

[IO]
output_basename = lax

[initial_conditions_region_1]   ; x < 0.5
r = 0.445
u = 0.698
p = 3.528

[initial_conditions_region_2]   ; x > 0.5
r = 0.5
u = 0.0
p = 0.571

[time]
time_max = 0.13

Unlike Sod, the left state moves ($u=0.698$), and its contact is stronger, so the dissipation of the face flux shows in the error. Run and check it:

mpirun -np 2 ../../exe/adam_flume_cpu input.ini > log.txt
../../exe/.regression-venv/bin/python ../../src/tests/flume/verification/sod/sod_oracle.py input.ini .

/lax: axis x, it 245, cells 200, L1(rho) = 1.063839e-02

A Riemann solver at the faces

scheme_space = weno splits the flux and reconstructs the two halves. weno-riemann instead interpolates the face states with WENO, solves a Riemann problem at every face, and adds a 6th-order flux correction gated by the WENO weights. Change scheme_space and add the three keys of the Riemann flux:

[numerics]
scheme_space             = weno-riemann
reconstruction_variables = characteristic
riemann_solver           = hllc      ; llf, hll or hllc
flux_correction          = 6th
flux_correction_sensor   = weno
reflux                   = .true.

/lax: axis x, it 245, cells 200, L1(rho) = 1.080670e-02

Both numbers are the ones of the Lax verification page ($1.064\cdot {10}^{-2}$ splitting, $1.081\cdot {10}^{-2}$ HLLC). Lax is the one problem where splitting beats every Riemann solver, by 1.6%; the page shows why, and gives LLF and HLL too.

What you learned

• A Riemann problem is two region sections and a final time.
• The face flux is a [numerics] choice: weno (flux splitting) or weno-riemann with a solver, a correction and its sensor.
• The verification pages record the numbers your runs must reproduce.

Previous: 1. Sod. Next: 3. A blast on a static AMR grid.