Abstract
In this article, we endeavour to find a fast solver for finite volume discretizations for compressible unsteady viscous flows. Thereby, we concentrate on comparing the efficiency of important classes of time integration schemes, namely time adaptive Rosenbrock, singly diagonally implicit (SDIRK) and explicit first stage singly diagonally implicit Runge-Kutta (ESDIRK) methods. To make the comparison fair, efficient equation system solvers need to be chosen and a smart choice of tolerances is needed. This is determined from the tolerance TOL that steers time adaptivity. For implicit Runge-Kutta methods, the solver is given by preconditioned inexact Jacobian-free Newton-Krylov (JFNK) and for Rosenbrock, it is preconditioned Jacobian-free GMRES. To specify the tolerances in there, we suggest a simple strategy of using TOL/100 that is a good compromise between stability and computational effort. Numerical experiments for different test cases show that the fourth order Rosenbrock method RODASP and the fourth order ESDIRK method ESDIRK4 are best for fine tolerances, with RODASP being the most robust scheme.
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References
Alexander, R.: Design and implementation of DIRK integrators for stiff systems. Appl. Numer. Math. 46(1), 1–17 (2003)
Bijl, H., Carpenter, M.H., Vatsa, V.N., Kennedy, C.A.: Implicit time integration schemes for the unsteady compressible Navier-Stokes equations: Laminar flow. J. Comp. Phys. 179, 313–329 (2002)
Birken, P.: Numerical simulation of tunnel fires using preconditioned finite volume schemes. ZAMP 59, 416–433 (2008)
Birken, P.: Numerical Methods for the Unsteady Compressible Navier-Stokes Equations. Habilitation Thesis, University of Kassel (2012)
Birken, P.: Solving nonlinear systems inside implicit time integration schemes for unsteady viscous flows. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds.) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, pp. 57–71. Springer (2013)
Colonius, T., Lele, S.K.: Computational aeroacoustics: progress on nonlinear problems of sound generation. Prog. Aerosp. Sci. 40(6), 345–416 (2004). http://linkinghub.elsevier.com/retrieve/pii/S0376042104000570
Cox, J.S., Rumsey, C.L., Brentner, K.S., Younis, B.A.: Computation of vortex shedding and radiated sound for a circular cylinder. Theor. Comput. Fluid Dyn. 12(4), 233–253 (1998)
Dembo, R., Eisenstat, R., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)
Eisenstat, S.C., Walker, H.F.: Choosing the forcing terms in an inexact newton method. SIAM J. Sci. Comput. 17(1), 16–32 (1996)
Farhat, C.: CFD-based nonlinear computational aeroelasticity. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics: Fluids, chap. 13, vol. 3, pp. 459–480. Wiley (2004)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, Series in edn. Springer, Berlin (2004)
Hindmarsh, A.C., Brown, P.N., Grant, K.E., Lee, S.L., Serban, R., Shumaker, D.E., Woodward, C.S.: SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers. ACM TOMS 31(3), 363–396 (2005)
Jameson, A.: Aerodynamics. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics: Fluids, chap. 11, vol. 3, pp. 325–406. Wiley (2004)
John, V., Rang, J.: Adaptive time step control for the incompressible Navier-Stokes equations. Comp. Meth. Appl. Mech. Eng. 199, 514–524 (2010)
Jothiprasad, G., Mavriplis, D.J., Caughey, D.A.: Higher-order time integration schemes for the unsteady Navier-Stokes equations on unstructured meshes. J. Comp. Phys. 191, 542–566 (2003)
Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)
Kennedy, C.A., Carpenter, M.H.: Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Num. Math. 44, 139–181 (2003)
Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: A survey of approaches and applications. J. Comp. Phys. 193, 357–397 (2004)
Kværnø, A.: Singly diagonally implicit Runge-Kutta methods with an explicit first stage. BIT Numer. Math. 44(3), 489–502 (2004)
Lior, N.: The cooling process in gas quenching. J. Mater. Process. Technol. 155–156, 1881–1888 (2004)
Lucas, P., Bijl, H., Van Zuijlen, A.H.: Efficient unsteady high Reynolds number flow computations on unstructured grids. Comput. Fluids 39(2), 271–282 (2010)
Meister, A., Sonar, T.: Finite-volume schemes for compressible fluid flow. Surv. Math. Ind. 8, 1–36 (1998)
Oßwald, K., Siegmund, A., Birken, P., Hannemann, V., Meister, A.: L2Roe: A low dissipation version of Roe’s approximate Riemann solver for low Mach numbers. Int. J. Numer. Methods Fluids (2015). doi:10.1002/fld.4175
Pierce, N.A., Giles, M.B.: Preconditioned multigrid methods for compressible flow calculations on stretched meshes. Tech. rep. (1997)
Qin, N., Ludlow, D.K., Shaw, S.T.: A matrix-free preconditioned Newton/GMRES method for unsteady Navier-Stokes solutions. Int. J. Num. Meth. Fluids 33, 223–248 (2000)
Rang, J.: A new stiffly accurate Rosenbrock-Wanner method for solving the incompressible Navier-Stokes equations. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds.) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, Notes on Numerical Fluid Mechanics and Multidisciplinary Design vol. 120, pp. 301–315. Springer (2013)
Rang, J.: An analysis of the Prothero–Robinson example for constructing new DIRK and ROW methods. J. Comput. Appl. Math. 262, 105–114 (2014)
Rang, J.: Improved traditional Rosenbrock-Wanner methods for stiff ODEs and DAEs. J. Comput. Appl. Math. (2015)
Rang, J., Angermann, L.: New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1. BIT 45, 761–787 (2005)
Reisner, J., Mousseau, V., Wyszogrodzki, A., Knoll, D.A.: A fully implicit hurricane model with physics-based preconditioning. Mon. Weather Rev. 133, 1003–1022 (2005)
Reisner, J., Wyszogrodzki, A., Mousseau, V., Knoll, D.: An efficient physics-based preconditioner for the fully implicit solution of small-scale thermally driven atmospheric flows. J. Comp. Phys. 189(1), 30–44 (2003). doi:10.1016/S0021-9991(03)00198-0. http://linkinghub.elsevier.com/retrieve/pii/S0021999103001980
Reitsma, F., Strydom, G., de Haas, J.B.M., Ivanov, K., Tyobeka, B., Mphahlele, R., Downar, T.J., Seker, V., Gougar, H.D., Da Cruz, D.F., Sikik, U.E.: The PBMR steadystate and coupled kinetics core thermal-hydraulics benchmark test problems. Nucl. Eng. Des. 236(5–6), 657–668 (2006)
Schmitt, B.A., Weiner, R.: Matrix-free W-methods using a multiple Arnoldi iteration. Appl. Num. Math. 18, 307–320 (1995)
Shampine, L.F.: Numerical Solution of Ordinary Differential Equations. Springer (1994)
Silva, R.S., Almeida, R.C., Galeão, A.C.: A preconditioner freeze strategy for numerical solution of compressible flows. Commun. Numer. Methods Eng. 19(3), 197–203 (2003)
Skvortsov, L.M.: Diagonally implicit Runge—Kutta methods for differential algebraic equations of indices two and three. Comput. Math. Math. Phys. 50(6), 993–1005 (2010)
Söderlind, G.: Digital filters in adaptive time–stepping. ACM Trans. Math. Softw., 1–26 (2003)
Söderlind, G., Wang, L.: Adaptive time-stepping and computational stability. J. Comp. Appl. Math. 185, 225–243 (2006). doi:10.1016/j.cam.2005.03.008
Söderlind, G., Wang, L.: Evaluating numerical ODE/DAE methods, algorithms and software. J. Comp. Appl. Math. 185, 244–260 (2006). doi:10.1016/j.cam.2005.03.009
St-Cyr, A., Neckels, D.: A fully implicit jacobian-free high-order discontinuous Galerkin Mesoscale flow solver, pp. 243–252. Springer, Berlin, Heidelberg (2009)
Steinebach, G.: Order-reduction of ROW-methods for DAEs and method of lines applications, vol. Preprint 1741. Technische Universität Darmstadt (1995)
Strehmel, K., Weiner, R.: Linear-implizite Runge-Kutta-Methoden und ihre Anwendung. Teubner, Stuttgart (1992)
Van Zuijlen, A.H.: Fluid-Structure Interaction Simulations - Efficient Higher Order Time Integration of Partitioned Systems. Ph.D. thesis, Delft University of Technology (2006)
Wang, L., Mavriplis, D.J.: Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations. J. Comp. Phys. 225, 1994–2015 (2007)
Weidig, U., Saba, N., Steinhoff, K.: Massivumformprodukte mit funktional gradierten Eigenschaften durch eine differenzielle thermo-mechanische Prozessführung. WT-Online, pp. 745–752 (2007)
Weiner, R., Schmitt, B.A.: Order Results for Krylov-W-Methods. Computing 61, 69–89 (1998)
Weiner, R., Schmitt, B.A., Podhaisky, H.: ROWMAP a ROW-code with Krylov techniques for large stiff ODEs. Appl. Num. Math. 25, 303–319 (1997)
Williams, R., Burrage, K., Cameron, I., Kerr, M.: A four-stage index 2 Diagonally Implicit Runge–Kutta method. Appl. Numer. Math. 40(3), 415–432 (2002)
Zahle, F., Soerensen, N.N., Johansen, J.: Wind turbine rotor-tower interaction using an incompressible overset grid method. Wind Energy 12, 594–619 (2009). doi:10.1002/we.327
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Communicated by: Silas Alben
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Blom, D.S., Birken, P., Bijl, H. et al. A comparison of Rosenbrock and ESDIRK methods combined with iterative solvers for unsteady compressible flows. Adv Comput Math 42, 1401–1426 (2016). https://doi.org/10.1007/s10444-016-9468-x
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DOI: https://doi.org/10.1007/s10444-016-9468-x
Keywords
- Rosenbrock methods
- Navier-Stokes equations
- ESDIRK
- Jacobian-free Newton-Krylov
- Unsteady flows
- Time adaptivity