A compact fourth-order spatial discretisation applied to the Navier-Stokes equations

Abstract

Modern direct and large eddy simulation of turbulent and transition flows requires accurate solution of the Navier--Stokes equations. High accuracy is achieved using a high order discretisation. The standard high order approach for local methods, such as finite-difference or finite-volume, produces large computational molecules and thus introduces complexity in the boundary treatment and parallelisation. Existing compact schemes need to invert a matrix system, which increases the computational cost, and are restricted to application on non-uniform grids. The fourth-order compact scheme proposed here iteratively applies a low order compact method to achieve higher accuracy. The scheme allows for a simple application of boundary conditions, can be applied on a non-uniform grid and allows a standard parallelisation approach to be used. The scheme is implemented and tested in an unsteady finite-difference heat equation solver and benchmarked against the analytical solution to validate the order of accuracy. It is also included in a full fractional-step Navier-Stokes solver and validated for the lid-driven cavity problem. References

• A. J . Chorin. Numerical solution of the Navier–Stokes equations. Math. Comput., 22:745–762, 1968.doi:10.1090/S0025-5718-1968-0242392-2
• J. H. Ferziger and M. Peric. Computational methods for fluid dynamics, pp. 42–46, 49. Springer, 1999. doi:10.1007/978-3-642-56026-2
• F. Gibou and R. Fedkiw. A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem. J. Comput. Phys. 202:577–601, 2005. doi:10.1016/j.jcp.2004.07.018
• S. K. Lele. Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103:16–42, 1992. doi:10.1016/0021-9991(92)90324-R
• J. Zhao, W. Dai, and T. Niu. Fourth-order compact schemes of a heat conduction problem with Neumann boundary conditions. Numer. Meth. Part. D. E. 23:949–959, 2007. doi:10.1002/num.20200
• S. Laizet and E. Lamballais. High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy. J. Comput. Phys. 228(16):5989–6015, 2009. doi:10.1016/j.jcp.2009.05.010
• P. C. Chu, C. Fan. A three-point combined compact difference scheme. J. Comput. Phys.140:370–399, 1998. doi:10.1006/jcph.1998.5899
• J. D. Hoffman. Numerical methods for engineers and scientists, pp. 271. CRC Press, 2001. https://www.crcpress.com/Numerical-Methods-for-Engineers-and-Scientists-Second-Edition/Hoffman-Hoffman-Frankel/p/book/9780824704438
• L. D. Kudryavtsev and M. K. Samarin. Lagrange interpolation formula. Encyclopedia of mathematics, 2016. http://www.encyclopediaofmath.org/index.php?title=Lagrange_interpolation_formula&oldid=17497.
• H. J. Crowder and C. Dalton. Errors in the use of non uniform mesh systems. J. Comput. Phys. 7:32–45, 1971. doi:10.1016/0021-9991(71)90047-7
• E. Kalnay de Rivas. On the use of non-uniform grids in finite difference equations. J. Comput. Phys. 10:202–210, 1972. doi:10.1016/0021-9991(72)90060-5
• A. E. P. Veldman and K. Rinzema. Playing with non-uniform grids. J. Eng. Math. 26:119–130, 1992. doi:10.1007/BF00043231
• P. E. O. Buelow, S. Venkateswaran, and C. L. Merkle. Effect of grid aspect ratio on convergence. AIAA J. 32:2401–2408, 1994. doi:10.2514/3.12306
• J. Crank and P. Nicolson. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc. Camb. Phil. Soc. 43:50–67, 1947. doi:10.1007/BF02127704
• J. G. Charney, R. Fjortoft, and J. von Neumann. Numerical integration of the barotropic vorticity equation. Tellus, 2:237–254, 1950. doi:10.1111/j.2153-3490.1950.tb00336.x
• O. Botella and R. Peyret. Benchmark spectral results on the lid-driven cavity flow. Comput. Fluid, 27:421–433, 1998. doi:10.1016/S0045-7930(98)00002-4
• G. B. Deng, J. Piquet, P. Queutey, and M. Visonneau. Incompressible flow calculations with a consistent physical interpolation finite volume approach. Comput. Fluids, 23:1029–1047, 1994. doi:10.1016/0045-7930(94)90003-5
• C.-H. Brunei and C. Jouron. An efficient scheme for solving steady incompressible Navier–Stokes equations, J. Comput. Phys. 89:389–413, 1990. doi:10.1016/0021-9991(90)90149-U
• U. Ghia, K. N. Ghia, and C. T. Shin. High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J. Comput. Phys. 48:387–411, 1982. doi:10.1016/0021-9991(82)90058-4
• R. Peyret and T. D. Taylor. Computational methods for fluid flow. Springer, 1983. doi:10.1007/978-3-642-85952-6
• S. W. Armfield and R. Street. The fractional-step method for the Navier–Stokes equations on staggered grids: The accuracy of three variations. J. Comput. Phys. 153:660–665, 1999. doi:10.1006/jcph.1999.6275
• S. W. Armfield and R. Street. Fractional step methods for the Navier–Stokes equations on non-staggered grids, ANZIAM J. 42:134–156, 2000. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/593
• S. W. Armfield and R. Street. An analysis and comparison of the time accuracy of fractional-step methods for the Navier–Stokes equations on staggered grids, Int. J. Numer. Meth. Fluids, 38:255–282, 2002. doi:10.1002/fld.217