High order well-balanced WENO scheme for the gas dynamics ...
WENO Tests
description
Transcript of WENO Tests
![Page 1: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/1.jpg)
WENO Tests
Z.G. Xiong
![Page 2: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/2.jpg)
• Fundamental Principle
• WENO Tests Weighted Essentially Non-Oscillatory
![Page 3: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/3.jpg)
Fundamental Principle
• HD Equations(Euler Equations):
0)()2
(
0
0
2
vPv
edt
d
Pdt
vd
vdt
d
![Page 4: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/4.jpg)
• Governing Equations (Conservation Equations):
![Page 5: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/5.jpg)
• Discrete Methods:
• finite difference
![Page 6: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/6.jpg)
• Shock Treatment :
• Shock Capture
![Page 7: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/7.jpg)
• WENO History:
• TVD(TVB)—ENO—WENO
![Page 8: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/8.jpg)
• WENO Technology:• space discretization:
![Page 9: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/9.jpg)
• flux splitting:
![Page 10: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/10.jpg)
• Time discretization:
Runge-Kutta method
![Page 11: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/11.jpg)
• Time step:
![Page 12: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/12.jpg)
WENO Tests
• Shock Tube Tests • Problem:
ρ1= 1.5p1= 1.0
ρ2= 1.0p2= 0.2
![Page 13: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/13.jpg)
• Simulation:
Grid: 64
polytropic index : 1.4
movie:
comparison:
(Feng et al, 2004)
![Page 14: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/14.jpg)
• Sedov Tests• Problem: E0
ρ0
![Page 15: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/15.jpg)
• analytic solution (Landau & Lifshitz):
shock evolution:
15.1,3/5
)/( 510
20
tERs
![Page 16: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/16.jpg)
• physical quantity profile:
Density profile:
![Page 17: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/17.jpg)
• Simulation:
initial condition:
boundary condition:
equivalent extrapolation
grid: 128 x128 x128
0
0
100.2
10
0
0
3240
510
p
v
cmg
ergE
![Page 18: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/18.jpg)
• Results: 1. shock evolution
movie:
a. xyz system
b. rθφ system
![Page 19: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/19.jpg)
evolution law:
a. xyz system
b. rθφ system
![Page 20: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/20.jpg)
2. density profile:
simulation result:
![Page 21: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/21.jpg)
simulation result vs analytic result:
![Page 22: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/22.jpg)
• New problem:
• Smoothing technique
![Page 23: WENO Tests](https://reader036.fdocuments.net/reader036/viewer/2022070404/56813a71550346895da26958/html5/thumbnails/23.jpg)
• Thank You!