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1D Convergence Test: 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 sub-regions are compared.

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			Right State					Grid
Test	Density	Pressure	Velocity	Density	Pressure	Velocity	x0	T	Cells
sod100	1.0000	1.0000	0.000	0.1250	0.10000	0.0000	0.5000	0.20	100
sod200	1.0000	1.0000	0.000	0.1250	0.10000	0.0000	0.5000	0.20	200
sod400	1.0000	1.0000	0.000	0.1250	0.10000	0.0000	0.5000	0.20	400
sod800	1.0000	1.0000	0.000	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 were computed with an 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).

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 1-2	Order 2-4	Order 4-8
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.

R0, sod100 Density

R0, sod800 Density

R1

R2

R3