1D Riemann tests
The 1D L&W tests consist of single Riemann problems with known solutions, and an interacting two shock test with no analytical solution, but which may be used to compare low and highresolution results.
NOTE: The final time for the Blast problem was omitted by L&W, but has been determined from previous published results and is presented here.
The code and configuration files plus output for Fyris to run the 1D Riemann problems will be available soon.
The 6 simple tests
These are specified as follows:
 adiabatic index: Gamma = 1.4
 grid domain: 0.0 < x < 1.0
 all cells with central coordinates < x0 are set to the Left State, all cells with central coordinates > x0 are set to the Right State (x0 lies on an exact cell boundary)
 tests are run from time = 0 until time = T
 left and right boundaries are free or natural boundaries that replicate the adjacent grid values.
Left State  

Test  Density  Pressure  Velocity 
1  1.0000  1.0000  0.7500 
2  1.0000  0.4000  2.0000 
3a  1.0000  1000.0000  19.59745 
4  5.9992  460.894  19.5975 
5  1.4000  1.0000  0.0000 
6  1.4000  1.0000  0.1000 
Right State  Grid  

Test  Density  Pressure  Velocity  x0  T  Cells 
1  0.1250  0.1000  0.0000  0.3000  0.2000  100 
2  1.0000  0.4000  2.0000  0.5000  0.1500  100 
3a  1.0000  0.0100  19.59745  0.8000  0.0120  200 
4  5.9924  46.0950  6.1963  0.4000  0.0350  200 
5  1.0000  1.0000  0.0000  0.5000  2.0000  100 
6  1.0000  1.0000  0.1000  0.5000  2.0000  100 
The 1D Noh Test
 adiabatic index: Gamma = 5/3
 grid domain: 0.0 < x < 1.0
 run until time = 1.000
 free boundaries
Left State  Right State  Grid  

Test  Density  Pressure  Velocity  Density  Pressure  Velocity  x0  T  Cells 
1  1.0000  1.0e6  1.0000  1.0000  1.0e6  1.0000  0.5000  1.000  100 
The 1D Peak Test
 adiabatic index: Gamma = 1.4
 grid domain: 0.1 < x < 0.6
 run until time = 0.0039
 free boundaries
Left State  Right State  Grid  

Test  Density  Pressure  Velocity  Density  Pressure  Velocity  x0  T  Cells 
1  0.1261192  782.92899  8.9047029  6.591493  3.1544874  2.2654207  0.500  0.0039  800 
The 1D WoodwardCollela blast wave test
 adiabatic index: Gamma = 1.4
 all cells with central coordinates < x0_left are set to the Left State, all cells with central coordinates > x0_right are set to the Right State, other cells are set to the middle state.
 left and right boundaries are reflecting boundaries
 tests are run from time = 0 until time = 0.038
 grid domain: x = 0 > 1
Left State  Middle State  Right State  

Test  Density  Pressure  Velocity  Density  Pressure  Velocity  Density  Pressure  Velocity 
1  1.000  1000.0  0.000  1.000  0.0100  0.000  1.000  100.00  0.000 
Grid  

Test  x0_left  x0_Right  T  Cells 
1  0.100  0.900  0.038  400/2000 
Results

The models are run and compared with the exact solution, or the 2000 cell model for the blast wave test.

The exact solutions are integrated over each cell to get cell average values at the same resolution as the test models.

For the exact solutions the cells were subsampled by a factor of 5 and averaged, consistent with the factor between the 400 and 2000 cell blast tests.

L1 norms between the models and the integrated exact solutions are computed as percentages.

All L1 norms were performed on the density, except for Test 2, where it was on Specific Energy = P/(d*(gamma1.0))

The exact solutions were computed with the exact Riemann solver code riemann.c (see also the Riemann page).

The exact solutions are included the riemann.c download as a set of text files.

All Fyris Alpha 1D tests were run with a Courant number of 0.8, with an initial step with a Courant number of 0.64 (0.8x0.8).
The Fyris Alpha L1 norms for the tests 16, Noh, Peak, and Blast are shown below in a table that includes the results from Liska and Wendroff. Each is the L1 norm as a percentage for density, except Test 2 which is internal energy, and Peak which is the velocity L1 norm. Blast is given as the L1 density norm percentage w.r.t. the 2000 cell solution Blast density, rebinned by a factor of 5. To compute the other L1 norms, the exact solution was evaluated in 5 locations across each cell and averaged, allowing for solutions to have boundaries part way through cells, and remaining consistent with the rebinning used for the Blast L1 norm.
The results are similar to PPM and VH1, which use related algorithms, although Fyris does significantly better with the strong shock Noh problem. This is reflected in the 2D tests where Fyris also performs the difficult 2D version of the Noh problem very well, using standard settings, and with sufficient stability to derive an L1 norm for density and pressure for the 2D case. The smallest error code for each test is shown in red.
Code

1

2

3a

4

5

6

Noh

Peak

Blast


Fyris

1.0

9.8

3.6

1.3

0.0

0.3

0.95

0.8

5.3

CFLFh

1.5

10.2

10.3

2.7

0.7

0.8

1.9

1.9


JT

1.3

6.4

8.1

2.3

0.6

0.6

1.7

1.1


LL

1.3

31.3

5.2

2.4

0.5

0.7

1.5

0.8


CLAW

0.8

fail

3.1

1.7

0

0.4

1.3

fail


WAFT

0.7

21.9

2.6

1.4

0

0.3

2.8

1


WENO

1.3

23.7

9.2

2.2

0

0.4

2.0

2.4


PPM

0.5

6.3

9.4

1.1

0

0.1

4.6

1.3


VH1

0.9

9.6

3.7

1.3

0

0.3

1.5

0.8


Plots comparing the exact and computed models follow. Finite models are shown as yellow dots, exact solutions as red line and red square dots.