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.
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
The initial conditions for V = (p,,
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
last update: Dec.18, 1998