riemann2d.f90 File Reference

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Functions/Subroutines
program	riemann2d
subroutine	makeconfig (Sim, config, ic)
subroutine	initdata (Mesh, Physics, Timedisc, ic)
	Set initial conditions. More...

Function/Subroutine Documentation

initdata()

subroutine riemann2d::initdata	(	class(mesh_base), intent(in)	Mesh,
		class(physics_base), intent(in)	Physics,
		class(timedisc_base), intent(inout)	Timedisc,
		integer, intent(in)	ic
	)

Set initial conditions.

Definition at line 679 of file riemann2d.f90.

makeconfig()

subroutine riemann2d::makeconfig	(	class(fosite), intent(inout)	Sim,
		type(dict_typ), pointer	config,
		integer, intent(in)	ic
	)

Definition at line 542 of file riemann2d.f90.

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riemann2d()

program riemann2d

Test:

19 different 2D Riemann problems for the inviscid Euler equations

Author

Tobias Illenseer

Test description

Numerical solutions of the inviscid Euler equations in 2D for the 19 tests proposed by Schulz-Rinne et al. [43] . These 2D Riemann problems were also used by several other authors ([29] , [27] , [31] ) and became a standard collection of tests for compressible flow solvers.
The test examines the shock/contact discontinuity and rarefraction wave interaction in two dimensions subdividing a quadratic computational domain into four equally sized quadrants. The computations were carried out on the unit square \( \left[-0.5,0.5 \right]\times \left[ -0.5,0.5 \right] \) centered on the origin of a planar cartesian grid. In all tests we assume an ideal gas equation of state with a constant ratio of specific heats (adiabatic index \(\gamma \)).

Simulation parameters
adiabatic index	\( \gamma = 1.4 \)
reconstruction	conservative
limiter	monocent
diffusivity	\( \theta = 1.2 \)
numerical flux functions	KT (Kurganov-Tadmor)
boundary conditions	absorbing

Initial conditions

The initial setup provides a distinct and constant state vector in each quadrant. To distinguish the different configurations we use the numbering scheme given in [27] .

Quadrant	numbers
2	1
3	4

The panel below shows the results for the density obtained for the 19 different test cases. The resolution was always set to \( 400 \times 400 \). The diagrams were also genereted in SVG format. Thus if you want to zoom in and study the details klick on the link below.

Configuration No. 1

Configuration No. 5

Configuration No. 2

Configuration No. 6

Configuration No. 3

Configuration No. 7

Configuration No. 4

Configuration No. 8

Configuration No. 9

Configuration No. 15

Configuration No. 10

Configuration No. 16

Configuration No. 11

Configuration No. 17

Configuration No. 12

Configuration No. 18

Configuration No. 13

Configuration No. 19

Configuration No. 14

Configuration No. 1 \anchor conf01

The initial condition yields four rarefactions along the interfaces of the four quadrants.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.0	0.0	1.0
2	0.5197	-0.7259	0.0	0.4
3	0.1072	-0.7259	-1.4045	0.0439
4	0.2579	0.0	-1.4045	0.15

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_01_contour.svg)

Configuration No. 2 \anchor conf02

The initial condition yields four rarefactions along the interfaces of the four quadrants.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.0	0.0	1.0
2	0.5197	-0.7259	0.0	0.4
3	1.0	-0.7259	-0.7259	1.0
4	0.5197	0.0	-0.7259	0.4

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_02_contour.svg)

Configuration No. 3 \anchor conf03

The initial condition yields four shocks along the interfaces of the four quadrants.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.5	0.0	0.0	1.5
2	0.5323	1.2060	0.0	0.3
3	0.1380	1.2060	1.2060	0.029
4	0.5323	0.0	1.2060	0.3

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_03_contour.svg)

Configuration No. 4 \anchor conf04

The initial condition yields four shocks along the interfaces of the four quadrants.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.1	0.0	0.0	1.1
2	0.5065	0.8939	0.0	0.35
3	1.1	0.8939	0.8939	1.1
4	0.5065	0.0	0.8939	0.35

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_04_contour.svg)

Configuration No. 5 \anchor conf05

The initial condition yields four contacts along the interfaces of the four quadrants.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	-0.75	-0.5	1.0
2	2.0	-0.75	0.5	1.0
3	1.0	0.75	0.5	1.0
4	3.0	0.75	-0.5	1.0

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_05_contour.svg)

Configuration No. 6 \anchor conf06

The initial condition yields four contacts along the interfaces of the four quadrants.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.75	-0.5	1.0
2	2.0	0.75	0.5	1.0
3	1.0	-0.75	0.5	1.0
4	3.0	-0.75	-0.5	1.0

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_06_contour.svg)

Configuration No. 7 \anchor conf07

The initial condition contains two rarefactions at the upper and the right interfaces and two contacts at the lower and left interfaces.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.1	0.1	1.0
2	0.5197	-0.6259	0.1	0.4
3	0.8	0.1	0.1	0.4
4	0.5197	0.1	-0.6259	0.4

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_07_contour.svg)

Configuration No. 8 \anchor conf08

The initial condition ontains two rarefactions at the upper and the right interfaces and two contacts at the lower and left interfaces.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	0.5197	0.1	0.1	0.4
2	1.0	-0.6259	0.1	1.0
3	0.8	0.1	0.1	1.0
4	1.0	0.1	-0.6259	1.0

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_08_contour.svg)

Configuration No. 9 \anchor conf09

The initial condition ycontains two rarefactions at the left and right interfaces and two contacts at the upper and lower interfaces.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.0	0.3	1.0
2	1.0	0.0	-0.3	1.0
3	1.0390	0.0	-0.8133	0.4
4	0.5197	0.0	-0.4259	0.4

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_09_contour.svg)

Configuration No. 10 \anchor conf10

The initial condition contains two rarefactions at the left and right interfaces and two contacts at the upper and lower interfaces.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.0	0.4297	1.0
2	0.5	0.0	0.6076	1.0
3	0.2281	0.0	-0.6076	0.3333
4	0.4562	0.0	-0.4297	0.3333

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_10_contour.svg)

Configuration No. 11 \anchor conf11

The initial condition contains two shocks at the upper and right interfaces and two contacts at the lower and left interfaces.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.1	0.0	1.0
2	0.5313	0.8276	0.0	0.4
3	0.8	0.1	0.0	0.4
4	0.5313	0.1	0.7276	0.4

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_11_contour.svg)

Configuration No. 12 \anchor conf12

The initial condition contains two shocks at the upper and right interfaces and two contacts at the lower and left interfaces.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.0	0.0	0.4
2	0.5313	0.7276	0.0	1.0
3	0.8	0.0	0.0	1.0
4	1.0	0.0	0.7276	1.0

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_12_contour.svg)

Configuration No. 13 \anchor conf13

The initial condition contains two shocks at the left and right interfaces and two contacts at the upper and lower interfaces.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.0	-0.3	1.0
2	2.0	0.0	0.3	1.0
3	1.0625	0.0	0.8145	0.4
4	0.5313	0.0	0.4276	0.4

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_13_contour.svg)

Configuration No. 14 \anchor conf14

The initial condition contains two shocks at the left and right interfaces and two contacts at the upper and lower interfaces.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	2.0	0.0	-0.5606	8.0
2	1.0	0.0	-1.2172	8.0
3	0.4736	0.0	1.2172	2.6667
4	0.9474	0.0	1.1606	2.6667

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_14_contour.svg)

Configuration No. 15 \anchor conf15

The initial condition contains two contacts at the lower and left interfaces, a rarefaction at the upper and shock at the right interface.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.1	-0.3	1.0
2	0.5197	-0.6259	-0.3	0.4
3	0.8	0.1	-0.3	0.4
4	0.5313	0.1	0.4276	0.4

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_15_contour.svg)

Configuration No. 16 \anchor conf16

The initial condition contains two contacts at the lower and left interfaces, a rarefaction at the upper and shock at the right interface.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	0.5313	0.1	0.1	0.4
2	1.02222	-0.6179	0.1	1.0
3	0.8	0.1	0.1	1.0
4	1.0	0.1	0.8276	1.0

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_16_contour.svg)

Configuration No. 17 \anchor conf17

The initial conditioncontains two contacts at the upper and lower interfaces, a shock at the left and rarefaction at the right interface.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.0	-0.4	1.0
2	2.0	0.0	-0.3	1.0
3	1.0625	0.0	0.2145	0.4
4	0.5197	0.0	-1.1259	0.4

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_17_contour.svg)

Configuration No. 18 \anchor conf18

The initial condition contains two contacts at the upper and lower interfaces, a shock at the left and rarefaction at the right interface.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.0	1.0	1.0
2	2.0	0.0	-0.3	1.0
3	1.0625	0.0	0.2145	0.4
4	0.5197	0.0	0.2741	0.4

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_18_contour.svg)

Configuration No. 19 \anchor conf19

The initial condition contains two contacts at the upper and lower interfaces, a shock at the left and rarefaction at the right interface.

Quadrant	Density	X-Velocity	Y-Velocity	Pressure
1	1.0	0.0	0.3	1.0
2	2.0	0.0	-0.3	1.0
3	1.0625	0.0	0.2145	0.4
4	0.5197	0.0	-0.4259	0.4

[view SVG] (http://www.astrophysik.uni-kiel.de/fosite/riemann2d_19_contour.svg)

References

• [43] C. W. Schulz-Rinne et al.: Numerical Solution of the Riemann Problem for Gas Dynamics, SIAM J. Sci. Comp. 14 (1993), 1394-1414
• [29] P. Lax, X.-D. Liu: Solution of Two-dimensional Riemann Problems of Gas Dynamics by Positive Schemes, SIAM J. Sci. Comp. 19 (1998), 319-340
• [27] A. Kurganov, E. Tadmor: Solution of Two-Dimensional Riemann Problems for Gas Dynamics without Riemann Problem Solvers, NMPDE 18 (2002), 561-588

Definition at line 465 of file riemann2d.f90.

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