docExampleCases.md

Example problems

• Scalar problems
• Compressible flow (homogeneous Euler equations)
• Atmospheric flow (Euler equations with gravity source term)

The Python scripts to generate the cases below must be launched from the software main directory (me@myPc: SomeFolder/CAELUM/$).

Scalar problems

• Setting up a 1D linear transport case: caseLinear python3 python/caseLinear.py

  

• Setting up a 1D Burgers problem: caseBurgers python3 python/caseBurgers.py

  

• Setting up a 3D linear transport case: caseLinear3D python3 python/caseLinear3D.py

  

Compressible flow

• Setting up a 1D Riemann Problem for the Euler equations (sod shock problem) [1]: caseRP python3 python/caseRP.py

  

• Setting up a 2D Riemann Problem for the Euler equations [2]: caseRP2D python3 python/caseRP2D.py

  

• Setting up a 3D Riemann Problem for the Euler equations [3]: caseRP3D python3 python/caseRP3D.py

  

• Advanced case: the shock-bubble test case [4]: caseShockBub python3 python/caseShockBub.py

  

• Advanced case: shock-cylinder interaction [5]: caseShockCyl caseShockCyl_plots python3 python/caseShockCyl.py; python3 python/caseShockCyl_plots.py;

  

• Advanced case: shock-induced turbulent mixing (Richtmeyer Meshkov instability): caseRM3D caseRM3D_plots python3 python/caseRM3D.py; python3 python/caseRM3D_plots.py;

  

Atmospheric flow (Euler equations with gravity source term)

• Setting up a 2D atmospheric simulation case (colliding thermals test case) [6]: caseCollidingBub python3 python/caseCollidingBub.py

  

• Setting up a 3D atmospheric simulation case (colliding thermals test case): caseCollidingBub3D caseCollidingBub3D_plots python3 python/caseCollidingBub.py

  

References and sources for the cases above:

[1] Sod, G. A. (1978). A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws", IJournal of Computational Physics, 1978, 27 (1), pp. 1-31.

[2] Lax, P. D., & Liu, X. D. (1998). Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM Journal on Scientific Computing, 19(2), 319-340.

[3] Nils Hoppe, Nico Fleischmann, Benedikt Biller, Stefan Adami, Nikolaus A. Adams, A systematic analysis of three-dimensional Riemann problems for verification of compressible-flow solvers, Computers & Fluids, Volume 278, 2024, 106298

[4] Mandli, K.T., Ahmadia, A.J., Berger, M.J., Calhoun, D.A., George, D.L.,Hadjimichael, Y., Ketcheson, D.I., Lemoine, G.I., LeVeque, R.J., 2016.Clawpack: building an open source ecosystem for solving hyperbolic PDEs. PeerJ Computer Science. doi:10.7717/peerj-cs.

[5] Langseth, J.O. (2001). 3D Visualization of Shock Waves Using Volume Rendering. In: Toro, E.F. (eds) Godunov Methods. Springer, New York, NY.

[6] Norman, M. R. (2021). A high-order WENO-limited finite-volume algorithm for atmospheric flow using the ADER-differential transform time discretization. Quarterly Journal of the Royal Meteorological Society, 147(736), 1661-1690.