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The standard upwind compact difference schemes for incompressible flow simulations. (English) Zbl 1351.76163
Summary: Compact difference schemes have been used extensively for solving the incompressible Navier-Stokes equations. However, the earlier formulations of the schemes are of central type (called central compact schemes, CCS), which are dispersive and susceptible to numerical instability. To enhance stability of CCS, the optimal upwind compact schemes (OUCS) are developed recently by adding high order dissipative terms to CCS. In this paper, it is found that OUCS are essentially not of the upwind type because they do not use upwind-biased but central type of stencils. Furthermore, OUCS are not the most optimal since orders of accuracy of OUCS are at least one order lower than the maximum achievable orders. New upwind compact schemes (called standard upwind compact schemes, SUCS) are developed in this paper. In contrast to OUCS, SUCS are constructed based completely on upwind-biased stencils and hence can gain adequate numerical dissipation with no need for introducing optimization calculations. Furthermore, SUCS can achieve the maximum achievable orders of accuracy and hence be more compact than OUCS. More importantly, SUCS have prominent advantages on combining the stable and high resolution properties which are demonstrated from the global spectral analyses and typical numerical experiments.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Fu, D. X.; Ma, Y. W., A high order accurate difference scheme for complex flow fields, J. Comput. Phys., 134, 1-15, (1997) · Zbl 0882.76054
[2] Sun, H. W.; Li, L. Z., A CCD-ADI method for unsteady convection-diffusion equations, Comput. Phys. Commun., 185, 790-797, (2014) · Zbl 1360.35193
[3] Zhou, K.; Ni, S. H.; Tian, Z. F., Exponential high-order compact scheme on nonuniform grids for the steady MHD duct flow problems with high Hartmann numbers, Comput. Phys. Commun., 196, 194-211, (2015)
[4] Xie, S. S.; Li, G. X.; Yi, S., Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schrödinger equation, Comput. Methods Appl. Mech. Eng., 198, 1052-1060, (2009) · Zbl 1229.81011
[5] Mohebbi, A.; Abbaszadeh, M.; Dehghan, M., Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives, Comput. Methods Appl. Mech. Eng., 264, 163-177, (2013) · Zbl 1286.76014
[6] Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 16-42, (1992) · Zbl 0759.65006
[7] Chu, P. C.; Fan, C., A three-point combined compact difference scheme, J. Comput. Phys., 140, 370-399, (1998) · Zbl 0923.65071
[8] Chu, P. C.; Fan, C., A three-point sixth-order nonuniform combined compact difference scheme, J. Comput. Phys., 148, 663-674, (1999) · Zbl 0930.65116
[9] Shukla, R. K.; Zhong, X., Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation, J. Comput. Phys., 204, 404-429, (2005) · Zbl 1067.65088
[10] Sengupta, T. K.; Lakshmanan, V.; Vijay, V. V.S. N., A new combined stable and dispersion relation preserving compact scheme for non-periodic problems, J. Comput. Phys., 228, 3048-3071, (2009) · Zbl 1282.76142
[11] Sengupta, T. K.; Vijay, V. V.S. N.; Bhaumik, S., Further improvement and analysis of CCD scheme: dissipation discretization and de-aliasing properties, J. Comput. Phys., 228, 6150-6168, (2009) · Zbl 1173.76034
[12] Zhong, X., High-order finite-difference schemes for numerical solution of hypersonic boundary-layer transition, J. Comput. Phys., 144, 662-709, (1998) · Zbl 0935.76066
[13] Sengupta, T. K.; Ganeriwal, G.; De, S., Analysis of central and upwind compact schemes, J. Comput. Phys., 192, 677-694, (2003) · Zbl 1038.65082
[14] De, A. K.; Eswaran, V., Analysis of a new high resolution upwind compact scheme, J. Comput. Phys., 218, 398-416, (2006) · Zbl 1103.65092
[15] Bhumkar, Y. G.; Sheu, T. W.H.; Sengupta, T. K., A dispersion relation preserving optimized upwind compact difference scheme for high accuracy flow simulations, J. Comput. Phys., 278, 378-399, (2014) · Zbl 1349.76434
[16] Sengupta, T. K.; Dipankar, A.; Sagaut, P., Error dynamics: beyond von Neumann analysis, J. Comput. Phys., 226, 1211-1218, (2007) · Zbl 1125.65337
[17] Chen, W.; Chen, J. C.; Lo, E. Y., An interpolation based finite difference method on non-uniform grid for solving Navier-Stokes equations, Comput. Fluids, 101, 273-290, (2014) · Zbl 1391.76468
[18] Bell, J. B.; Colella, P.; Glaz, H. M., A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85, 257-283, (1989) · Zbl 0681.76030
[19] Brown, D. L.; Cortez, R.; Minion, M. L., Accurate projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 168, 464-499, (2001) · Zbl 1153.76339
[20] Liu, J., Open and traction boundary conditions for the incompressible Navier-Stokes equations, J. Comput. Phys., 228, 7250-7267, (2009) · Zbl 1386.76114
[21] Liu, J.-G.; Liu, J.; Pego, R. L., Stable and accurate pressure approximation for unsteady incompressible viscous flow, J. Comput. Phys., 229, 3428-3453, (2010) · Zbl 1307.76029
[22] Shirokoff, D.; Rosales, R. R., An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary, J. Comput. Phys., 230, 8619-8646, (2011) · Zbl 1426.76489
[23] Dong, S.; Karniadakis, G. E.; Chryssostomidis, C., A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains, J. Comput. Phys., 261, 83-105, (2014) · Zbl 1349.76569
[24] Vreman, A. W., The projection method for the incompressible Navier-Stokes equations: the pressure near a no-slip wall, J. Comput. Phys., 263, 353-374, (2014) · Zbl 1349.76547
[25] Shankar, P. N.; Deshpande, M. D., Fluid mechanics in the driven cavity, Annu. Rev. Fluid Mech., 32, 93-136, (2000) · Zbl 0988.76006
[26] Boersma, B. J., A 6th order staggered compact finite difference method for the incompressible Navier-Stokes and scalar transport equations, J. Comput. Phys., 230, 4940-4954, (2011) · Zbl 1416.76172
[27] Ghia, U.; Ghia, K. N.; Shin, C. T., High-re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48, 387-411, (1982) · Zbl 0511.76031
[28] Wahba, E. M., Steady flow simulations inside a driven cavity up to Reynolds number 35,000, Comput. Fluids, 66, 85-97, (2012) · Zbl 1365.76203
[29] Auteri, F.; Parolini, N.; Quartapelle, L., Numerical investigation on the stability of singular driven cavity flow, J. Comput. Phys., 183, 1-25, (2002) · Zbl 1021.76040
[30] Bruneau, C.-H.; Saad, M., The 2D lid-driven cavity problem revisited, Comput. Fluids, 35, 326-348, (2006) · Zbl 1099.76043
[31] Sengupta, T. K.; Sengupta, A., A new alternating bi-diagonal compact scheme for non-uniform grids, J. Comput. Phys., 310, 1-25, (2016) · Zbl 1349.76139
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