A stabilized FEM formulation with discontinuity-capturing for solving Burgers’-type equations at high Reynolds numbers. (English) Zbl 1511.76051
Summary: This computational study is concerned with the numerical solutions of Burgers’-type equations at high Reynolds numbers. The high Reynolds numbers drive the nonlinearity to play an essential role and the equations to become more convection-dominated, which causes the solutions obtained with the standard numerical methods to involve spurious oscillations. To overcome this challenge, the Galerkin finite element formulation is stabilized by using the streamline-upwind/Petrov-Galerkin method. The stabilized formulation is further supplemented with YZ\(\beta\) shock-capturing to achieve better solution profiles around strong gradients. The nonlinear equation systems arising from the space and time discretizations are solved by using the Newton-Raphson (N-R) method at each time step. The resulting linearized equation systems are solved with the BiCGStab technique combined with ILU preconditioning at each N-R iteration. A comprehensive set of test examples is provided to demonstrate the robustness of the proposed formulation and the techniques used.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
Keywords:
Burgers’ equation; high Reynolds number; finite elements; SUPG; YZ \(\beta\) shock-capturingReferences:
| [1] | Bateman, H., Some recent researches on the motion of fluids, Mon. Weather Rev., 43, 163-170 (1915) |
| [2] | Burgers, J. M., A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, 171-199 (1948), Elsevier |
| [3] | Frisch, U.; Bec, J., Burgulence, (David, F.; Lesieur, M.; Yaglom, A., New Trends in Turbulence - Turbulence: Nouveaux Aspects. New Trends in Turbulence - Turbulence: Nouveaux Aspects, Les Houches - Ecole d’Ete de Physique Theorique (2001), Springer Berlin Heidelberg), 341-383 · Zbl 1309.76114 |
| [4] | Bonkile, M. P.; Awasthi, A.; Lakshmi, C.; Mukundan, V.; Aswin, V. S., A systematic literature review of Burgers’ equation with recent advances, Pramana, 90, 6, 69 (2018) |
| [5] | Hopf, E., The partial differential equation \(u_t + u u_x = \mu u_{x x}\), Commun. Pure Appl. Math., 3, 3, 201-230 (1950) · Zbl 0039.10403 |
| [6] | Cole, J. D., On a quasi-linear parabolic equation occurring in aerodynamics, Q. Appl. Math., 9, 3, 225-236 (1951) · Zbl 0043.09902 |
| [7] | Liao, W., A fourth-order finite-difference method for solving the system of two-dimensional Burgers’ equations, Int. J. Numer. Methods Fluids, 64, 5, 565-590 (2009) · Zbl 1377.65108 |
| [8] | Kutluay, S.; Bahadir, A. R.; Özdeş, A., Numerical solution of one-dimensional Burgers equation: explicit and exact-explicit finite difference methods, J. Comput. Appl. Math., 103, 2, 251-261 (1999) · Zbl 0942.65094 |
| [9] | Srivastava, V. K.; Awasthi, M. K.; Singh, S., An implicit logarithmic finite-difference technique for two dimensional coupled viscous Burgers’ equation, AIP Adv., 3, 12, 122105 (2013) |
| [10] | Bhatt, H. P.; Khaliq, A. Q.M., Fourth-order compact schemes for the numerical simulation of coupled Burgers’ equation, Comput. Phys. Commun., 200, 117-138 (2016) · Zbl 1351.35167 |
| [11] | Hassanien, I. A.; Salama, A. A.; Hosham, H. A., Fourth-order finite difference method for solving Burgers’ equation, Appl. Math. Comput., 170, 2, 781-800 (2005) · Zbl 1084.65078 |
| [12] | Aksan, E. N., A numerical solution of Burgers’ equation by finite element method constructed on the method of discretization in time, Appl. Math. Comput., 170, 2, 895-904 (2005) · Zbl 1103.65104 |
| [13] | Dogan, A., A Galerkin finite element approach to Burgers’ equation, Appl. Math. Comput., 157, 2, 331-346 (2004) · Zbl 1054.65103 |
| [14] | Arminjon, P.; Beauchamp, C., A finite element method for Burgers’ equation in hydrodynamics, Int. J. Numer. Methods Eng., 12, 3, 415-428 (1978) · Zbl 0388.76041 |
| [15] | Ladeia, C. A.; Romeiro, N. M.L., Numerical solutions of the 1D convection-diffusion-reaction and the Burgers equation using implicit multi-stage and finite element methods, Integral Methods in Science and Engineering, 205-216 (2013), Springer New York · Zbl 1279.65107 |
| [16] | Elgindy, K. T.; Karasözen, B., High-order integral nodal discontinuous Gegenbauer-Galerkin method for solving viscous Burgers’ equation, Int. J. Comput. Math., 96, 10, 2039-2078 (2018) · Zbl 07474758 |
| [17] | Chai, Y.; Ouyang, J., Appropriate stabilized Galerkin approaches for solving two-dimensional coupled Burgers’ equations at high Reynolds numbers, Comput. Math. Appl., 79, 5, 1287-1301 (2020) · Zbl 1443.65198 |
| [18] | Chino, E.; Tosaka, N., Dual reciprocity boundary element analysis of time-independent Burger’s equation, Eng. Anal. Bound. Elem., 21, 3, 261-270 (1998) · Zbl 0954.76058 |
| [19] | Bossy, M.; Fezoui, L.; Piperno, S., Comparison of a stochastic particle method and a finite volume deterministic method applied to Burgers equation, Monte Carlo Methods Appl., 3, 2, 113-140 (1997) · Zbl 0962.65075 |
| [20] | Gao, Y.; Le, L.-H.; Shi, B. C., Numerical solution of Burgers’ equation by lattice Boltzmann method, Appl. Math. Comput., 219, 14, 7685-7692 (2013) · Zbl 1291.65313 |
| [21] | Lai, H.; Ma, C., A new lattice Boltzmann model for solving the coupled viscous Burgers’ equation, Physica A, 395, 445-457 (2014) · Zbl 1395.76070 |
| [22] | Li, Q.; Chai, Z.; Shi, B., Lattice Boltzmann models for two-dimensional coupled Burgers’ equations, Comput. Math. Appl., 75, 3, 864-875 (2018) · Zbl 1409.65083 |
| [23] | Jiwari, R., A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation, Comput. Phys. Commun., 183, 11, 2413-2423 (2012) · Zbl 1302.35337 |
| [24] | Jiwari, R., A hybrid numerical scheme for the numerical solution of the Burgers’ equation, Comput. Phys. Commun., 188, 59-67 (2015) · Zbl 1344.65082 |
| [25] | Zhu, H.; Shu, H.; Ding, M., Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method, Comput. Math. Appl., 60, 3, 840-848 (2010) · Zbl 1201.65190 |
| [26] | Bouhamidi, A.; Hached, M.; Jbilou, K., A meshless RBF method for computing a numerical solution of unsteady Burgers’-type equations, Comput. Math. Appl., 68, 3, 238-256 (2014) · Zbl 1369.65125 |
| [27] | Zhang, X. H.; Ouyang, J.; Zhang, L., Element-free characteristic Galerkin method for Burgers’ equation, Eng. Anal. Bound. Elem., 33, 3, 356-362 (2009) · Zbl 1244.76097 |
| [28] | Hughes, T. J.R.; Brooks, A. N., A multi-dimensional upwind scheme with no crosswind diffusion, (Hughes, T. J.R., Finite Element Methods for Convection Dominated Flows, AMD-vol. 34 (1979), ASME: ASME New York), 19-35 · Zbl 0423.76067 |
| [29] | Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput Methods Appl Mech Eng, 32, 199-259 (1982) · Zbl 0497.76041 |
| [30] | Tezduyar, T. E., Stabilized finite element formulations for incompressible flow computations, Adv. Appl. Mech., 28, 1-44 (1992) · Zbl 0747.76069 |
| [31] | Tezduyar, T. E.; Hughes, T. J.R., Development of Time-Accurate Finite Element Techniques for First-Order Hyperbolic Systems with Particular Emphasis on the Compressible Euler Equations, NASA Technical Report (1982), NASA |
| [32] | Reno, Nevada |
| [33] | Hughes, T. J.R.; Tezduyar, T. E., Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations, Comput. Methods Appl. Mech. Eng., 45, 217-284 (1984) · Zbl 0542.76093 |
| [34] | Hughes, T. J.R.; Franca, L. P.; Mallet, M., A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems, Comput. Methods Appl. Mech. Eng., 63, 97-112 (1987) · Zbl 0635.76066 |
| [35] | Le Beau, G. J.; Tezduyar, T. E., Finite element computation of compressible flows with the SUPG formulation, Advances in Finite Element Analysis in Fluid Dynamics, FED-vol. 123, 21-27 (1991), ASME: ASME New York |
| [36] | Tezduyar, T. E.; Park, Y. J., Discontinuity capturing finite element formulations for nonlinear convection-diffusion-reaction equations, Comput. Methods Appl. Mech. Eng., 59, 307-325 (1986) · Zbl 0593.76096 |
| [37] | Tezduyar, T. E., Determination of the stabilization and shock-capturing parameters in SUPG formulation of compressible flows, Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004 (CD-ROM), Jyvaskyla, Finland (2004) |
| [38] | Tezduyar, T. E., Finite element methods for fluid dynamics with moving boundaries and interfaces, (Stein, E.; Borst, R. D.; Hughes, T. J.R., Encyclopedia of Computational Mechanics, Volume 3: Fluids (2004), Wiley) |
| [39] | Tezduyar, T. E., Finite elements in fluids: stabilized formulations and moving boundaries and interfaces, Comput. Fluids, 36, 191-206 (2007) · Zbl 1177.76202 |
| [40] | Tezduyar, T. E., Computation of moving boundaries and interfaces and stabilization parameters, Int. J. Numer. Methods Fluids, 43, 555-575 (2003) · Zbl 1032.76605 |
| [41] | Rispoli, F.; Corsini, A.; Tezduyar, T. E., Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD), Comput. Fluids, 36, 121-126 (2007) · Zbl 1181.76098 |
| [42] | Tezduyar, T. E.; Senga, M., Stabilization and shock-capturing parameters in SUPG formulation of compressible flows, Comput. Methods Appl. Mech. Eng., 195, 1621-1632 (2006) · Zbl 1122.76061 |
| [43] | Tezduyar, T. E.; Senga, M., SUPG finite element computation of inviscid supersonic flows with YZ \(\beta\) shock-capturing, Comput. Fluids, 36, 147-159 (2007) · Zbl 1127.76029 |
| [44] | Tezduyar, T. E.; Senga, M.; Vicker, D., Computation of inviscid supersonic flows around cylinders and spheres with the SUPG formulation and YZ \(\beta\) shock-capturing, Comput. Mech., 38, 469-481 (2006) · Zbl 1176.76077 |
| [45] | Shakib, F., Finite Element Analysis of the Compressible Euler and Navier-Stokes Equations (1988), Department of Mechanical Engineering, Stanford University, Ph.D. thesis |
| [46] | xx-xx. In press |
| [47] | Abali, B. E., Computational Reality: Solving Nonlinear and Coupled Problems in Continuum Mechanics, Advanced Structured Materials, vol. 55 (2016), Springer Singapore |
| [48] | Logg, A.; Mardal, K.-A.; Wells, G., Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, Lecture Notes in Computational Science and Engineering, vol. 84 (2012), Springer-Verlag Berlin Heidelberg · Zbl 1247.65105 |
| [49] | Alnæs, M. S.; Logg, A.; Ølgaard, K. B.; Rognes, M. E.; Wells, G. N., Unified form language: a domain-specific language for weak formulations of partial differential equations, ACM Trans. Math. Softw., 40, 2, 1-37 (2014) · Zbl 1308.65175 |
| [50] | Gowrisankar, S.; Natesan, S., An efficient robust numerical method for singularly perturbed Burgers’ equation, Appl. Math. Comput., 346, 385-394 (2019) · Zbl 1429.65182 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
