Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

doi:10.1007/s42967-021-00159-w

Abstract

Abstract: We develop error-control based time integration algorithms for compressible fluid dynamics (CFD) applications and show that they are efficient and robust in both the accuracy-limited and stability-limited regime. Focusing on discontinuous spectral element semidiscretizations, we design new controllers for existing methods and for some new embedded Runge-Kutta pairs. We demonstrate the importance of choosing adequate controller parameters and provide a means to obtain these in practice. We compare a wide range of error-control-based methods, along with the common approach in which step size control is based on the Courant-Friedrichs-Lewy (CFL) number. The optimized methods give improved performance and naturally adopt a step size close to the maximum stable CFL number at loose tolerances, while additionally providing control of the temporal error at tighter tolerances. The numerical examples include challenging industrial CFD applications.

Key words: Explicit Runge-Kutta methods, Step size control, Compressible Euler equations, Compressible Navier-Stokes equations, hp-adaptive spatial discretizations

CLC Number:

Hendrik Ranocha, Lisandro Dalcin, Matteo Parsani, David I. Ketcheson. Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1191-1228.

TrendMD

References

1. Abhyankar, S., Brown, J., Constantinescu, E.M., Ghosh, D., Smith, B.F., Zhang, H.:PETSc/TS:a modern scalable ODE/DAE solver library (2018). arXiv:1806.01437[math.NA.]
2. Al Jahdali, R., Boukharfane, R., Dalcin, L., Parsani, M.:Optimized explicit Runge-Kutta schemes for entropy stable discontinuous collocated methods applied to the Euler and Navier-Stokes equations. In:AIAA Scitech 2021 Forum, p. 0633 (2021). https://doi.org/10.2514/6.2021-0633
3. Almquist, M., Dunham, E.M.:Elastic wave propagation in anisotropic solids using energy-stable finite differences with weakly enforced boundary and interface conditions (2020). arXiv:2003.12811[math.NA.]
4. Arévalo, C., Söderlind, G., Hadjimichael, Y., Fekete, I.:Local error estimation and step size control in adaptive linear multistep methods. Numer. Algorithm 86, 537-563 (2021). https://doi.org/10.1007/s11075-020-00900-1
5. Baggag, A., Atkins, H., Keyes, D.:Parallel implementation of the discontinuous Galerkin method. Tech. Rep. NASA/CR-1999-209546, NASA, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton VA United States (1999)
6. Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Dener, A., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., May, D.A., McInnes, L.C., Mills, R.T., Munson, T., Rupp, K., Sanan, P., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.:PETSc users manual. Tech. Rep. ANL-95/11-Revision 3.13, Argonne National Laboratory (2020)
7. Berzins, M.:Temporal error control for convection-dominated equations in two space dimensions. SIAM J. Sci. Comput. 16(3), 558-580 (1995)
8. Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.:Julia:a fresh approach to numerical computing. SIAM Rev. 59(1), 65-98 (2017). https://doi.org/10.1137/141000671. arXiv:1411.1607[cs.MS]
9. Bogacki, P., Shampine, L.F.:A 3(2) pair of Runge-Kutta formulas. Appl. Math. Lett. 2(4), 321-325 (1989). https://doi.org/10.1016/0893-9659(89)90079-7
10. Bogacki, P., Shampine, L.F.:An efficient Runge-Kutta (4,5) pair. Comput. Math. Appl. 32(6), 15-28 (1996). https://doi.org/10.1016/0898-1221(96)00141-1
11. Buscariolo, F.F., Hoessler, J., Moxey, D., Jassim, A., Gouder, K., Basley, J., Murai, Y., Assi, G.R.S., Sherwin, S.J.:Spectral/hp element simulation of flow past a Formula One front wing:validation against experiments (2019). http://arxiv.org/abs/1909.06701v1
12. Butcher, J.C.:Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2016). https://doi.org/10.1002/9781119121534
13. Carpenter, M.H., Fisher, T.C., Nielsen, E.J., Frankel, S.H.:Entropy stable spectral collocation schemes for the Navier-Stokes equations:discontinuous interfaces. SIAM J. Sci. Comput. 36(5), B835-B867 (2014). https://doi.org/10.1137/130932193
14. Carpenter, M.H., Parsani, M., Fisher, T.C., Nielsen, E.J.:Towards an entropy stable spectral element framework for computational fluid dynamics. In:54th AIAA Aerospace Sciences Meeting. American Institute of Aeronautics and Astronautics (2016). https://doi.org/10.2514/6.2016-1058
15. Chan, J., Fernández, D.C.D.R., Carpenter, M.H.:Efficient entropy stable Gauss collocation methods. SIAM J. Sci. Comput. 41(5), A2938-A2966 (2019). https://doi.org/10.1137/18M1209234
16. Christopher, L.R.:The NASA juncture flow test as a model for effective CFD/experimental collaboration. In:2018 Applied Aerodynamics Conference. American Institute of Aeronautics and Astronautics (2018). https://doi.org/10.2514/6.2018-3319
17. Citro, V., Giannetti, F., Sierra, J.:Optimal explicit Runge-Kutta methods for compressible Navier-Stokes equations. Appl. Numer. Math. 152, 511-526 (2020). https://doi.org/10.1016/j.apnum.2019.11.005
18. Conde, S., Fekete, I., Shadid, J.N.:Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge-Kutta methods (2018). arXiv:1806.08693[math.NA.]
19. Dormand, J.R., Prince, P.J.:A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6(1), 19-26 (1980). https://doi.org/10.1016/0771-050X(80)90013-3
20. Fernández, D.C.D.R., Carpenter, M.H., Dalcin, L., Zampini, S., Parsani, M.:Entropy stable h/p-nonconforming discretization with the summation-by-parts property for the compressible Euler and Navier-Stokes equations. SN Partial Differ. Equ. Appl. 1(2), 1-54 (2020). https://doi.org/10.1007/s42985-020-00009-z
21. Figueroa, A., Jackiewicz, Z., Löhner, R.:Explicit two-step Runge-Kutta methods for computational fluid dynamics solvers. Int. J. Numer. Methods Fluids 93(2), 429-444 (2021). https://doi.org/10.1002/fld.4890
22. Fisher, T.C., Carpenter, M.H.:High-order entropy stable finite difference schemes for nonlinear conservation laws:finite domains. J. Comput. Phys. 252, 518-557 (2013). https://doi.org/10.1016/j.jcp.2013.06.014
23. Gassner, G.J., Winters, A.R., Kopriva, D.A.:Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. J. Comput. Phys. 327, 39-66 (2016). https://doi.org/10.1016/j.jcp.2016.09.013
24. Gottlieb, S., Ketcheson, D.I.:Time discretization techniques. In:Abgrall, R., Shu, C.-W. (eds) Handbook of Numerical Analysis, vol. 17, pp. 549-583. Elsevier (2016). https://doi.org/10.1016/bs.hna.2016.08.001
25. Gustafsson, K.:Control theoretic techniques for stepsize selection in explicit Runge-Kutta methods. ACM Trans. Math. Softw. (TOMS) 17(4), 533-554 (1991). https://doi.org/10.1145/210232.210242
26. Gustafsson, K., Lundh, M., Söderlind, G.:A PI stepsize control for the numerical solution of ordinary differential equations. BIT Numer. Math. 28(2), 270-287 (1988). https://doi.org/10.1007/BF01934091
27. Hadri, B., Parsani, M., Hutchinson, M., Heinecke, A., Dalcin, L., Keyes, D.:Performance study of sustained petascale direct numerical simulation on Cray XC40 systems. Concurrency Computat. Pract. Exper. 32(20), e5725 (2020). https://doi.org/10.1002/cpe.5725
28. Hairer, E., Nørsett, S.P., Wanner, G.:Solving Ordinary Differential Equations I:Nonstiff Problems, Springer Series in Computational Mathematics, vol. 8. Springer-Verlag, Berlin/Heidelberg (2008). https://doi.org/10.1007/978-3-540-78862-1
29. Hairer, E., Wanner, G.:Solving Ordinary Differential Equations II:Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, vol. 14. Springer-Verlag, Berlin/Heidelberg (2010). https://doi.org/10.1007/978-3-642-05221-7
30. Hall, G., Higham, D.J.:Analysis of stepsize selection schemes for Runge-Kutta codes. IMA J. Numer. Anal. 8(3), 305-310 (1988). https://doi.org/10.1093/imanum/8.3.305
31. Higham, D.J., Hall, G.:Embedded Runge-Kutta formulae with stable equilibrium states. J. Comput. Appl. Math. 29(1), 25-33 (1990). https://doi.org/10.1016/0377-0427(90)90192-3
32. Hutchinson, M., Heinecke, A., Pabst, H., Henry, G., Parsani, M., Keyes, D.:Efficiency of high order spectral element methods on petascale architectures. In:Kunkel, J., Balaji, P., Dongarra, J. (eds) High Performance Computing. ISC High Performance 2016. Lecture Notes in Computer Science, vol. 9697. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-41321-1_23
33. Iyer, P.S., Malik, M.R.:Wall-modeled LES of the NASA juncture flow experiment. In:AIAA Scitech 2020 Forum, pp. 1-23 (2020). https://doi.org/10.2514/6.2020-1307
34. Karniadakis, G.E., Sherwin, S.:Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press, Oxford (2013). https://doi.org/10.1093/acprof:oso/9780198528692.001.0001
35. Kennedy, C.A., Carpenter, M.H.:Fourth order 2N-storage Runge-Kutta schemes. Technical Memorandum NASA-TM-109112, NASA, NASA Langley Research Center, Hampton VA 23681-0001, United States (1994)
36. Kennedy, C.A., Carpenter, M.H.:Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44(1/2), 139-181 (2003). https://doi.org/10.1016/S0168-9274(02)00138-1
37. Kennedy, C.A., Carpenter, M.H., Lewis, R.M.:Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations. Appl. Numer. Math. 35(3), 177-219 (2000). https://doi.org/10.1016/S0168-9274(99)00141-5
38. Ketcheson, D.I., Ahmadia, A.J.:Optimal stability polynomials for numerical integration of initial value problems. Commun. Appl. Math. Comput. Sci. 7(2), 247-271 (2013). https://doi.org/10.2140/camcos.2012.7.247
39. Ketcheson, D.I.:Highly efficient strong stability-preserving Runge-Kutta methods with low-storage implementations. SIAM J. Sci. Comput. 30(4), 2113-2136 (2008). https://doi.org/10.1137/07070485X
40. Ketcheson, D.I.:Runge-Kutta methods with minimum storage implementations. J. Comput. Phys. 229(5), 1763-1773 (2010). https://doi.org/10.1016/j.jcp.2009.11.006
41. Ketcheson, D.I.:Relaxation Runge-Kutta methods:conservation and stability for inner-product norms. SIAM J. Numer. Anal. 57(6), 2850-2870 (2019). https://doi.org/10.1137/19M1263662. arXiv:1905.09847[math.NA.]
42. Ketcheson, D.I., Parsani, M., Grant, Z.J., Ahmadia, A., Ranocha, H.:RK-Opt:a package for the design of numerical ODE solvers. J. Open Source Softw. 5(54), 2514 (2020). https://doi.org/10.21105/joss.02514. https://github.com/ketch/RK-Opt
43. Ketcheson, D.I., Ranocha, H., Parsani, M., bin Waheed, U., Hadjimichael, Y.:NodePy:a package for the analysis of numerical ODE solvers. J. Open Source Softw. 5(55), 2515 (2020). https://doi.org/10.21105/joss.02515. https://github.com/ketch/nodepy
44. Knepley, M.G., Karpeev, D.A.:Mesh algorithms for PDE with Sieve I:mesh distribution. Sci. Program. 17(3), 215-230 (2009). https://doi.org/10.3233/SPR-2009-0249
45. Kopriva, D.A.:Implementing Spectral Methods for Partial Differential Equations:Algorithms for Scientists and Engineers. Springer Science & Business Media, New York (2009). https://doi.org/10.1007/978-90-481-2261-5
46. Kopriva, D.A., Jimenez, E.:An assessment of the efficiency of nodal discontinuous Galerkin spectral element methods. In:Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, pp. 223-235. Springer, Berlin/Heidelberg (2013). https://doi.org/10.1007/978-3-642-33221-0_13
47. Kraaijevanger, J.F.B.M.:Contractivity of Runge-Kutta methods. BIT Numer. Math. 31(3), 482-528 (1991). https://doi.org/10.1007/BF01933264
48. Kubatko, E.J., Dawson, C., Westerink, J.J.:Time step restrictions for Runge-Kutta discontinuous Galerkin methods on triangular grids. J. Comput. Phys. 227(23), 9697-9710 (2008)
49. Langseth, J.O., LeVeque, R.J.:A wave propagation method for three-dimensional hyperbolic conservation laws. J. Comput. Phys. 165(1), 126-166 (2000)
50. Langtry, R.B., Kuntz, M., Menter, F.R.:Drag prediction of engine-airframe interference effects with CFX-5. J. Aircr. 42(6), 1523-1529 (2005)
51. LeVeque, R.J.:Finite Difference Methods for Ordinary and Partial Differential Equations:Steady-State and Time-Dependent Problems. SIAM, Philadelphia, PA, USA (2007)
52. Mogensen, P.K., Riseth, A.N.:Optim:a mathematical optimization package for Julia. J. Open Source Softw. 3(24), 615 (2018). https://doi.org/10.21105/joss.00615
53. Montijano, J.I., Rández, L., Ketcheson, D.I.:Low-storage FSAL embedded pairs of Runge-Kutta methods (2020) (In preparation)
54. O'Reilly, O., Lundquist, T., Dunham, E.M., Nordström, J.:Energy stable and high-order-accurate finite difference methods on staggered grids. J. Comput. Phys. 346, 572-589 (2017)
55. Parsani, M., Boukharfane, R., Nolasco, I.R., Fernández, D.C.D.R., Zampini, S., Hadri, B., Dalcin, L.:High-order accurate entropy-stable discontinuous collocated Galerkin methods with the summation-by-parts property for compressible CFD frameworks:scalable SSDC algorithms and flow solver. J. Comput. Phys. 424, 109844 (2021). https://doi.org/10.1016/j.jcp.2020.109844
56. Parsani, M., Carpenter, M.H., Nielsen, E.J.:Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier-Stokes equations. J. Comput. Phys. 290, 132-138 (2015). https://doi.org/10.1016/j.jcp.2015.02.042
57. Parsani, M., Carpenter, M.H., Nielsen, E.J.:Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. J. Comput. Phys. 292, 88-113 (2015). https://doi.org/10.1016/j.jcp.2015.03.026
58. Parsani, M., Ketcheson, D.I., Deconinck, W.:Optimized low-order explicit Runge-Kutta schemes for the high-order spectral difference method. In:Proceedings of the 11th Finnish Mechanics Days, pp. 49-54. University of Oulu, Department of Mechanical Engineering (2012). http://hdl.handle.net/10754/333613
59. Parsani, M., Ketcheson, D.I., Deconinck, W.:Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM J. Sci. Comput. 35(2), A957-A986 (2013). https://doi.org/10.1137/120885899
60. Pegrum, J.:Experimental study of the vortex system generated by a Formula 1 front wing. Ph.D. thesis, Imperial College London (2007)
61. Prince, P.J., Dormand, J.R.:High order embedded Runge-Kutta formulae. J. Comput. Appl. Math. 7(1), 67-75 (1981). https://doi.org/10.1016/0771-050X(81)90010-3
62. Rackauckas, C., Nie, Q.:DifferentialEquations.jl-a performant and feature-rich ecosystem for solving differential equations in Julia. J. Open Res. Softw. 5(1), 15 (2017). https://doi.org/10.5334/jors.151
63. Ranocha, H., Dalcin, L., Parsani, M.:Fully-discrete explicit locally entropy-stable schemes for the compressible Euler and Navier-Stokes equations. Comput. Math. Appl. 80(5), 1343-1359 (2020). https://doi.org/10.1016/j.camwa.2020.06.016.. arXiv:2003.08831[math.NA.]
64. Ranocha, H., Dalcin, L., Parsani, M., Ketcheson, D.I.:Coefficients of optimized low-storage Runge-Kutta methods with automatic step size control for spectral element methods applied to compressible computational fluid dynamics. 2021. https://github.com/ranocha/Optimized-RK-CFD, https://doi.org/10.5281/zenodo.4671927
65. Ranocha, H., Lóczi, L., Ketcheson, D.I.:General relaxation methods for initial-value problems with application to multistep schemes. Numer. Math. 146, 875-906 (2020). https://doi.org/10.1007/s00211-020-01158-4. arXiv:2003.03012[math.NA.]
66. Ranocha, H., Sayyari, M., Dalcin, L., Parsani, M., Ketcheson, D.I.:Relaxation Runge-Kutta methods:fully-discrete explicit entropy-stable schemes for the compressible Euler and Navier-Stokes equations. SIAM J. Sci. Comput. 42(2), A612-A638 (2020). https://doi.org/10.1137/19M1263480.. arXiv:1905.09129[math.NA.]
67. Rojas, D., Boukharfane, R., Dalcin, L., Fernández, D.C.D.R., Ranocha, H., Keyes, D.E., Parsani, M.:On the robustness and performance of entropy stable discontinuous collocation methods. J. Comput. Phys. 426, 109891 (2021). https://doi.org/10.1016/j.jcp.2020.109891. arXiv:1911.10966[math.NA.]
68. Rumsey, C.L., Morrison, J.H.:Goals and status of the NASA juncture flow experiment, p. STO-MP-AVT-246., NATO (2016)
69. Shu, C.-W.:Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Final Report NASA/CR-97-206253, NASA, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton VA United States (1997)
70. Shu, C.-W., Osher, S.:Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439-471 (1988). https://doi.org/10.1016/0021-9991(88)90177-5
71. Sjögreen, B., Yee, H.:High order entropy conservative central schemes for wide ranges of compressible gas dynamics and MHD flows. J. Comput. Phys. 364, 153-185 (2018). https://doi.org/10.1016/j.jcp.2018.02.003
72. Sjögreen, B., Yee, H.C., Kotov, D.:Skew-symmetric splitting and stability of high order central schemes. In:Journal of Physics:Conference Series, vol. 837, p. 012019. IOP Publishing (2017). https://doi.org/10.1088/1742-6596/837/1/012019
73. Söderlind, G.:Automatic control and adaptive time-stepping. Numer. Algorithms 31(1/2/3/4), 281-310 (2002). https://doi.org/10.1023/A:1021160023092
74. Söderlind, G.:Digital filters in adaptive time-stepping. ACM Trans. Math. Softw. (TOMS) 29(1), 1-26 (2003). https://doi.org/10.1145/641876.641877
75. Söderlind, G.:Time-step selection algorithms:adaptivity, control, and signal processing. Appl. Numer. Math. 56(3/4), 488-502 (2006). https://doi.org/10.1016/j.apnum.2005.04.026
76. Söderlind, G., Wang, L.:Adaptive time-stepping and computational stability. J. Comput. Appl. Math. 185(2), 225-243 (2006). https://doi.org/10.1016/j.cam.2005.03.008
77. Tsitouras, C.:Runge-Kutta pairs of order 5(4) satisfying only the first column simplifying assumption. Comput. Math. Appl. 62(2), 770-775 (2011). https://doi.org/10.1016/j.camwa.2011.06.002
78. Vincent, P.E, Castonguay, P., Jameson, A.:A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput. 47, 50-72 (2011). https://doi.org/10.1007/s10915-010-9420-z
79. Vincent, P., Witherden, F., Vermeire, B., Park, J.S., Iyer, A.:Towards green aviation with Python at petascale. In:SC'16:Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pp. 1-11. IEEE Press (2016)
80. Ware, J., Berzins, M.:Adaptive finite volume methods for time-dependent PDEs. In:Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations, pp. 417-430. Springer (1995). https://doi.org/10.1007/978-1-4612-4248-2_20

[1]	Hendrik Ranocha, Gregor J. Gassner. Preventing Pressure Oscillations Does Not Fix Local Linear Stability Issues of Entropy-Based Split-Form High-Order Schemes [J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 880-903.
[2]	F. L. Romeo, M. Dumbser, M. Tavelli. A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations [J]. Communications on Applied Mathematics and Computation, 2021, 3(4): 607-647.