For more info on these animations see   Composite schemes for conservation laws

Composite Schemes - Riemann Problems for 2D Gas Dynamics
 

The initial data consist of a single constant state in each of four quadrants of the x - y plane. The problem is solved in the x-y region (0,1) x (0,1) and the four quadrants are given by dividing this region by two lines x = 1/2, y = 1/2 . We will use the subscripts ll,lr,ul,ur to denote lower-left, lower-right, upper-left and upper-right quadrants respectively.
 


Configuation 1:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (0.4, 0.5197, -0.7259, 0)                Vur = (1, 1, 0, 0)
Vll = (0.0439, 1.1072, -0,7259, -1.4045)     Vlr =  (0.15, 0. 2579, 0, -1.4045)

The result of this example up to t = 0.2 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (72 kB MPEG)      Contour plot (304 kB GIF) - this can play any browser

Surface plot (216 kB MPEG)        Surface plot (672 kB GIF) - this can play any browser 


Configuation 2:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (0.4, 0.5197, -0.7259, 0) Vur = (1, 1, 0, 0)
Vll = (1, 1, -0,7259, -0,7259)  Vlr =  (0.4, 0. 5197, 0, -0.7259)

The result of this example up to t = 0.2 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (87 kB MPEG)      Contour plot (355 kB GIF) - this can play any browser

Surface plot (194 kB MPEG)        Surface plot (623 kB GIF) - this can play any browser 


Configuation 3:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (0.3, 0.5323, 1.206, 0)             Vur = (1.5, 1.5, 0, 0)
Vll = (0.029, 0.138, 1.206, 1.206)          Vlr =  (0.3, 0.5323, 0, 1.206)

The result of this example up to t = 0.3 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (67 kB MPEG)      Contour plot (311 kB GIF) - this can play any browser

Surface plot (313 kB MPEG)        Surface plot (1 243 kB GIF) - this can play any browser 


Configuation 4:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (0.35, 0.5065, 0.8939, 0)  Vur = (1.1, 1.1, 0, 0)
Vll = (1.1, 1.1, 0.8939, 0.8939)       Vlr =  (0.35, 0. 5065, 0, 0.8939)

The result of this example up to t = 0.25 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot (79 kB MPEG)       Contour plot (286 kB GIF) - this can play any browser

Surface plot (266 kB MPEG)        Surface plot (729 kB GIF) - this can play any browser 



Configuation 5:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 2, -0.75, 0.5)  Vur = (1, 1, -0.75, -0.5)
Vll = (1, 1, 0.75, 0.5)          Vlr =  (1, 3, 0.75, -0.5)

The result of this example up to t = 0.23 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (66 kB MPEG)      Contour plot (323 kB GIF) - this can play any browser

Surface plot (246 kB MPEG)        Surface plot (820 kB GIF) - this can play any browser 


Configuation 6:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 2, 0.75, 0.5)      Vur = (1, 1, 0.75, -0.5)
Vll = (1, 1, -0.75, 0.5)          Vlr =  (1, 3, -0.75, 0.5)

The result of this example up to t = 0.3 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (120 kB MPEG)      Contour plot (523 kB GIF) - this can play any browser

Surface plot (274 kB MPEG)        Surface plot (910 kB GIF) - this can play any browser


Configuation 7:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (0.4, 0.5197, -0.6259, 0.1)  Vur = (1, 1, 0.1, 0.1)
Vll = (0.4, 0.8, 0.1, 0.1)                 Vlr =  (0.4, 0.5197, 0.1, -0.6259)

The result of this example up to t = 0.25 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (99 kB MPEG)      Contour plot (371 kB GIF) - this can play any browser

Surface plot (282 kB MPEG)       Surface plot (718 kB GIF) - this can play any browser 


Configuation 8:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 1, -0.6259, 0.1)  Vur = (0.4, 0.5197, 0.1, 0.1)
Vll = (1, 0.8, 0.1, 0.1)          Vlr =  (1, 1, 0.1, -0.6259)

The result of this example up to t = 0.25 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (111 kB MPEG)      Contour plot (414 kB GIF) - this can play any browser

Surface plot (259 kB MPEG)        Surface plot (827 kB GIF) - this can play any browser 


Configuation 9:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 2, 0, -0.3)                      Vur = (1, 1, 0, 0.3)
Vll = (0.4, 1.039, 0, -0.8133)          Vlr =  (0.4, 0.5197, 0, -0.4259)

The result of this example up to t = 0.3 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (92 kB MPEG)      Contour plot (420 kB GIF) - this can play any browser

Surface plot (287 kB MPEG)       Surface plot (1 046 kB GIF) - this can play any browser 


Configuation 10:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 0.5, 0, 0.6076)                      Vur = (1, 1, 0, 0.4297)
Vll = (0.3333, 0.2281, 0, -0.6076)          Vlr = (0.3333, 0.4562, 0, -0.4297)

The result of this example up to t = 0.15 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (39 kB MPEG)      Contour plot (184 kB GIF) - this can play any browser

Surface plot (154 kB MPEG)        Surface plot (582 kB GIF) - this can play any browser 


Configuation 11:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (0.4, 0.5313, 0.8276, 0)  Vur = (1, 1, 0.1, 0)
Vll = (0.4, 0.8, 0.1, 0)               Vlr =  (0.4, 0.5313, 0.1, 0.7276)

The result of this example up to t = 0.3 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (97 kB MPEG)      Contour plot (423 kB GIF) - this can play any browser

Surface plot (321 kB MPEG)        Surface plot (1 042 kB GIF) - this can play any browser 


Configuation 12:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 1, 0.7276, 0) Vur = (0.4, 0.5313, 0, 0)
Vll = (1, 0.8, 0, 0)           Vlr =  (1, 1, 0, 0.7276)

The result of this example up to t = 0.25 calculated by the CFLF8 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot (79 kB MPEG)         Contour plot (361 kB GIF) - this can play any browser

Surface plot (245 kB MPEG)        Surface plot (680 kB GIF) - this can play any browser 



Configuation 13:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 2, 0, 0.3)                        Vur = (1, 1, 0, -0.3)
Vll = (0.4, 1.0625, 0, 0.8145)          Vlr =  (0.4, 0.5313, 0.1, 0.4276)

The result of this example up to t = 0.3 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (73 kB MPEG)      Contour plot (347 kB GIF) - this can play any browser

Surface plot (278 kB MPEG)        Surface plot (958 kB GIF) - this can play any browser 


Configuation 14:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (8, 1, 0, -1.2172)                     Vur = (8, 2, 0, -0.5606)
Vll = (2.6667, 0.4736, 0, 1.2172)          Vlr =  (2.6667, 0.9474, 0, 1.1606)

The result of this example up to t = 0.1 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (27 kB MPEG)      Contour plot (118 kB GIF) - this can play any browser

Surface plot (119 kB MPEG)        Surface plot (365 kB GIF) - this can play any browser 


Configuation 15:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (0.4, 0.5197, -0.6259, -0.3)    Vur = (1, 1, 0.1, -0.3)
Vll = (0.4, 0.8, 0.1, -0.3)                   Vlr =  (0.4, 0.5313, 0.1, 0.4276)

The result of this example up to t = 0.2 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (56 kB MPEG)       Contour plot (277 kB GIF) - this can play any browser

Surface plot (203 kB MPEG)        Surface plot (705 kB GIF) - this can play any browser


Configuation 16:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 1.0222, -0.6179, 0.1)    Vur = (0.4, 0.5313, 0.1, 0.1)
Vll = (1, 0.8, 0.1, 0.1)                   Vlr =  (1, 1, 0.1, 0.8276)

The result of this example up to t = 0.2 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (72 kB MPEG)       Contour plot (294 kB GIF) - this can play any browser

Surface plot (238 kB MPEG)        Surface plot (850 kB GIF) - this can play any browser


Configuation 17:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 2, 0, -0.3)                      Vur = (1, 1, 0, -0.4)
Vll = (0.4, 1.0625, 0, 0.2145)          Vlr =  (0.4, 0.5197, 0, -1.1259)

The result of this example up to t = 0.3 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (80 kB MPEG)       Contour plot (253 kB GIF) - this can play any browser

Surface plot (441 kB MPEG)        Surface plot (961 kB GIF) - this can play any browser


Configuation 18:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 2, 0, -0.3)                      Vur = (1, 1, 0, 1)
Vll = (0.4, 1.0625, 0, 0.2145)          Vlr =  (0.4, 0.5197, 0, 0.2741)

The result of this example up to t = 0.2 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (53 kB MPEG)       Contour plot (203 kB GIF) - this can play any browser

Surface plot (263 kB MPEG)        Surface plot (713 kB GIF) - this can play any browser


Configuation 19:

The initial conditions for V = (p,$\rho$, u, v) in the four quadrants are:
Vul = (1, 2, 0, -0.3)                      Vur = (1, 1, 0, 0.3)
Vll = (0.4, 1.0625, 0, 0.2145)          Vlr =  (0.4, 0.5197, 0, -0.4259)

The result of this example up to t = 0.3 calculated by the CFLF4 method is presented in the following animations including contour and surface plots.

Animations:

Contour plot  (91 kB MPEG)        Contour plot (414 kB GIF) - this can play any browser

Surface plot (283 kB MPEG)        Surface plot (1 019 kB GIF) - this can play any browser


 

For more info on these animations see   Composite schemes for conservation laws

 

last update:   Dec.18, 1998

vachal@galileo.fjfi.cvut.cz