1D convergence: Sod shock
The 1D convergence tests consist of single Riemann problem with a known solution, the standard Sod shock. The solution on 100, 200, 400 and 800 cell resolution. The L1 density norm over the whole grid and in 3 defined subregions are compared.
The code and configuration files plus output for Fyris to run the 1D Riemann Sod problem will be available soon.
The Sod test
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)
 left and right boundaries are free or natural boundaries that replicate the adjacent grid values
 tests are run from time = 0 until time = T = 0.2
Left State  

Test  Density  Pressure  Velocity 
sod100  1.0000  1.0000  0.000 
sod200  1.0000  1.0000  0.000 
sod400  1.0000  1.0000  0.000 
sod800  1.0000  1.0000  0.000 
Right State  Grid  

Test  Density  Pressure  Velocity  x0  T  Cells 
sod100  0.1250  0.10000  0.0000  0.5000  0.20  100 
sod200  0.1250  0.10000  0.0000  0.5000  0.20  200 
sod400  0.1250  0.10000  0.0000  0.5000  0.20  400 
sod800  0.1250  0.10000  0.0000  0.5000  0.20  800 
Results

The models are run and compared with the exact solution.

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.

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

All L1 norms were performed on the density.

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).
Four regions are defined for L1 norm analysis:

Region 0 is the whole 0.0 < x < 1.0

Region 1 is the interval 0.20 < x < 0.30 the head of the R wave

Region 2 is 0.49 < x < 0.68, between the CD and the R wave

Region 3 is 0.69 < x < 0.85, between the CD and the S wave
L1 Norm Errors, as percentages
Code

sod100

sod200

sod400

sod800

Order 12

Order 24

Order 48


R0

0.76  0.43  0.20  0.09  0.83  1.13  1.05 
R1

0.62  0.31  0.15  0.08  1.01  1.01  1.00 
R2

0.86  0.27  0.05  0.02  1.67  2.32  1.71 
R3

1.67  0.69  0.25  0.10  1.28  1.48  1.36 
Plots comparing the exact and computed models follow. Finite models are shown as yellow dots, exact solutions as red line and red square dots.
Symbols used in the region plots, sod100 green, sod800 yellow, exact solution red line only.