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Up to sixth-order accurate A-stable implicit schemes applied to the discontinuous Galerkin discretized Navier-Stokes equations. (English) Zbl 1349.76247
Summary: In this paper a high-order implicit multi-step method, known in the literature as Two Implicit Advanced Step-point (TIAS) method, has been implemented in a high-order Discontinuous Galerkin (DG) solver for the unsteady Euler and Navier-Stokes equations. Application of the absolute stability condition to this class of multi-step multi-stage time discretization methods allowed to determine formulae coefficients which ensure A-stability up to order 6. The stability properties of such schemes have been verified by considering linear model problems. The dispersion and dissipation errors introduced by TIAS method have been investigated by looking at the analytical solution of the oscillation equation. The performance of the high-order accurate, both in space and time, TIAS-DG scheme has been evaluated by computing three test cases: an isentropic convecting vortex under two different testing conditions and a laminar vortex shedding behind a circular cylinder. To illustrate the effectiveness and the advantages of the proposed high-order time discretization, the results of the fourth- and sixth-order accurate TIAS schemes have been compared with the results obtained using the standard second-order accurate Backward Differentiation Formula, BDF2, and the five stage fourth-order accurate Strong Stability Preserving Runge-Kutta scheme, SSPRK4.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76Nxx Compressible fluids and gas dynamics
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[1] Cockburn, B.; Karniadakis, G. E.; Shu, C.-W., (Cockburn, B.; Karniadakis, G. E.; Shu, C.-W., Discontinuous Galerkin Methods: Theory, Computation, and Applications, (2000), Springer Berlin)
[2] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131, 267-279, (1997) · Zbl 0871.76040
[3] Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 5, 1749-1779, (2002) · Zbl 1008.65080
[4] Luo, H.; Baum, J. D.; Lohner, R., A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids, J. Comput. Phys., 211, 767-783, (2006) · Zbl 1138.76408
[5] Cockburn, B., Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws, J. Comput. Appl. Math., 128, 187-204, (2001) · Zbl 0974.65092
[6] Bassi, F.; Crivellini, A.; Rebay, S.; Savini, M., Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-ω turbulence model equations, Comput. Fluids, 34, 507-540, (2005) · Zbl 1138.76043
[7] Bassi, F.; Crivellini, A.; Di Pietro, D. A.; Rebay, S., An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows, Comput. Fluids, 36, 10, 1529-1546, (2007) · Zbl 1194.76102
[8] Nigro, A.; Renda, S.; De Bartolo1o, C.; Hartmann, R.; Bassi, F., A high-order accurate discontinuous Galerkin finite element method for laminar low Mach number flows, Internat. J. Numer. Methods Fluids, 72, 1, 43-68, (2013)
[9] John, V.; Rang, J., Adaptive time step control for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 199, 514-524, (2010) · Zbl 1227.76048
[10] St-Cyr, A.; Neckels, D., A fully implicit Jacobian-free high-order discontinuous Galerkin mesoscale flow solver, (Proceedings of the 9th International Conference on Computational Science, (2009), Springer Berlin, Heidelberg), 243-252
[11] Wang, L.; Mavriplis, D. J., Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations, J. Comput. Phys., 225, 1994-2015, (2007) · Zbl 1343.76022
[12] Dolejsi, V.; Kus, P., Adaptive backward difference formula—discontinuous Galerkin finite element method for the solution of conservation laws, Int. J. Numer. Methods Eng., 73, 12, 1739-1766, (2008) · Zbl 1159.76349
[13] Persson, P.-O.; Peraire, J., Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier-Stokes equations, SIAM J. Sci. Comput., 30, 2709-2733, (2008) · Zbl 1362.76052
[14] Kanevsky, A.; Carpenter, M. H.; Gottlieb, D.; Hesthaven, J. S., Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes, J. Comput. Phys., 225, 1753-1781, (2007) · Zbl 1123.65097
[15] Birken, P.; Gassner, G.; Haas, M.; Munz, C.-D., Preconditioning for modal discontinuous Galerkin methods for unsteady 3D Navier-Stokes equations, J. Comput. Phys., 240, 20-35, (2013) · Zbl 1426.76520
[16] Birken, P.; Gassner, G.; Haas, M.; Munz, C.-D., Efficient time integration for discontinuous Galerkin method for unsteady 3D Navier-Stokes equations, (ECCOMAS 2012, Vienna, Austria, September 10-14, (2012)), 10-14
[17] Lorcher, F.; Gassner, G.; Munz, C.-D., A discontinuous Galerkin scheme based on a space-time expansion. I. inviscid compressible flow in one space dimension, J. Sci. Comput., 32, 175-199, (2007) · Zbl 1143.76047
[18] Psihoyios, G., Advanced step-point methods for the solution of initial value problems, (29/12/1995), University of London - Imperial College of Science and Technology, PhD thesis · Zbl 1165.65364
[19] Psihoyios, G. Y.; Cash, J. R., A stability result for general linear methods with characteristic function having real poles only, BIT Numer. Math., 38, 3, 612-617, (1998) · Zbl 0915.65095
[20] Cash, J. R., On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae, Numer. Math., 34, 235-246, (1980) · Zbl 0411.65040
[21] Cash, J. R., The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae, Comput. Math. Appl., 9, 5, 645-657, (1983) · Zbl 0526.65052
[22] Psihoyios, G., A general formula for the stability functions of a group of implicit advanced step-point (IAS) methods, Math. Comput. Modelling, 46, 1-2, 214-224, (2007) · Zbl 1132.65074
[23] Cash, J. R., Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and daes, J. Comput. Appl. Math., 125, 117-130, (2000) · Zbl 0971.65063
[24] Hindmarsh, A.; Brown, P.; Grant, K.; Lee, S.; Serban, R.; Shumaker, D.; Wooward, C., SUNDIALS: suite of nonlinear and differential/algebraic equation solvers, ACM Trans. Math. Softw., 31, 3, 363-396, (2005) · Zbl 1136.65329
[25] Nigro, A.; Ghidoni, A.; Bassi, F.; Rebay, S., Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows, Int. J. Numer. Methods Fluids, (2014), to be published
[26] Bassi, F.; Rebay, S., GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations, (Lecture Notes in Computational Science and Engineering, vol. 11, (2000), Springer-Verlag Berlin), 197-208 · Zbl 0989.76040
[27] Curtis, C. F.; Hirschfelder, J. O., Integration of stiff equations, Proc. Natl. Acad. Sci. USA, 38, 235-243, (1952) · Zbl 0046.13602
[28] Spiteri, R. J.; Ruuth, S. J., A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J. Numer. Anal., 40, 2, 469-491, (2002) · Zbl 1020.65064
[29] Bassi, F.; Rebay, S.; Mariotti, G.; Pedinotti, S.; Savini, M., A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, (Decuypere, R.; Dibelius, G., Proceeding of the 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics, Antwerpen, Belgium, March 5-7, 1997, (1997), Technologisch Instituut), 99-108
[30] Bassi, F.; Rebay, S., A high order discontinuous Galerkin method for compressible turbulent flows, (Cockburn, B.; Karniadakis, G.; Shu, C.-W., Discontinuous Galerkin Methods: Theory, Computation and Applications, vol. 11, (2000), Springer), 77-88 · Zbl 0991.76039
[31] Tirani, R.; Paracelli, C., An algorithm for starting multistep methods, Comput. Math. Appl., 45, 123-129, (2003) · Zbl 1035.65077
[32] Balay, S.; Buschelman, K.; Gropp, W. D.; Kaushik, D.; Knepley, M. G.; Curfman Mclnneses, L.; Smith, B. F.; Zhang, H., Petsc web page, (2001)
[33] Ober, C. C.; Shadid, J. N., Studies on the accuracy of time-integration methods for the radiation-diffusion equations, J. Comput. Phys., 195, 743-772, (2004) · Zbl 1053.65082
[34] Lambert, J. D., Computational methods in ordinary differential equations, (1973), John Wiley · Zbl 0258.65069
[35] Wolfram Research, Inc., Mathematica, version 7.0, (2008), Champaign, IL
[36] Norsett, S. P.; Wolfbrandt, A., Attainable orders of rational approximations to the exponential function with only real poles, BIT, 17, 200-208, (1977) · Zbl 0361.41011
[37] Takacs, L. L., A two-step scheme for the advection equation with minimized dissipation and dispersion errors, Mon. Weather Rev., 111, 455-467, (1980)
[38] Durran, D. R., The third-order Adams-bashfort method: an attractive alternative to leapfrog time differencing, Mon. Weather Rev., 119, 702-720, (1990)
[39] Hu, C.; Shu, C. W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 97-127, (1999) · Zbl 0926.65090
[40] Yee, H. C.; Sandham, N. D.; Djomehri, M. J., Low dissipative high order shock-capturing methods using characteristic-based filters, J. Comput. Phys., 150, 199-238, (1999) · Zbl 0936.76060
[41] Wang, L.; Mavriplis, D. J., Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations, J. Comput. Phys., 225, 1994-2015, (2007) · Zbl 1343.76022
[43] Bijl, H.; Carpenter, M. H.; Vatsa, V. N.; Kennedy, C. A., Implicit time integration schemes for the unsteady compressible navierstokes equations: laminar flow, J. Comput. Phys., 179, 313-329, (2002) · Zbl 1060.76079
[44] Carpenter, M. H.; Kennedy, C. A.; Bijl, H.; Viken, S. A.; Vatsa, V. N., Fourth-order Runge-Kutta schemes for fluid mechanics applications, J. Sci. Comput., 25, 1, 175-194, (2005) · Zbl 1203.76112
[45] Roshko, A., On the development of turbulent wakes from vortex streets, (1953), NACA TN-2913
[46] Geuzaine, C.; Remacle, J. F., Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Eng., 79, 11, 1309-1331, (2009) · Zbl 1176.74181
[47] Liang, C.; Premasuthan, S.; Jameson, A., High-order accurate simulation of low-Mach laminar flow past two side-by-side cylinders using spectral difference method, Comput. Struct., 87, 812-827, (2009)
[48] Meneghini, J. R.; Saltara, F.; Siqueira, C. L.R.; Ferrari, J. A., Numerical simulation of flow interference between two circular cylinders in tandem and side-by-side arrangements, J. Fluids Struct., 15, 327-350, (2001)
[49] Roy, C. J., Review of code and solution verification procedures for computational simulation, J. Comput. Phys., 205, 131-156, (2005) · Zbl 1072.65118
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