Skip to main content
SpringerLink
Log in

GePUP: Generic Projection and Unconstrained PPE for Fourth-Order Solutions of the Incompressible Navier–Stokes Equations with No-Slip Boundary Conditions

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

A generic projection maps one vector to another such that their difference is a gradient field and the projected vector does not have to be solenoidal. Via a commutator of Laplacian and the generic projection, the projected velocity is formulated as the sole evolutionary variable with the incompressibility constraint enforced by a pressure Poisson equation so that the dissipation of velocity divergence is governed by a heat equation. Different from previous projection methods, the GePUP formulation treats the time integrator as a black box. This prominent advantage is illustrated by straightforward formations of a semi-implicit time-stepping scheme and another explicit time-stepping scheme. Apart from its stability, the GePUP schemes have an optimal efficiency in that within each time step the solution is advanced by solving a sequence of linear systems with geometric multigrid. A key component of the GePUP schemes is a fourth-order discrete projection for no-penetration domains. Results of numerical tests in two and three dimensions demonstrate that the GePUP schemes are fourth-order accurate both in time and in space. To facilitate efficiency comparison to other methods, a simple formula is introduced. Systematic arguments and timing results show that the GePUP schemes could be vastly superior over lower-order methods in terms of efficiency and accuracy. In some cases, the GePUP schemes running on the author’s personal desktop would be faster than a second-order method running on the fastest supercomputer in the world! This paper contains enough details so that one can reproduce the numerical results by following the exposition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. For the INSE, a numerical method benefits from this requirement in that the accuracy of the computed velocity is largely decoupled from that of the computed pressure gradient.

  2. Equation (14e) in [57] contains an error; it should be \( \mathbf {F}\left\langle \varphi ,\psi \right\rangle _{{\mathbf {i}}+\frac{1}{2}{\mathbf {e}}^d} = \frac{1}{h^{{\textsc {D}}-1}} \int _{_{{\mathcal {F}}_{{\mathbf {i}}+\frac{1}{2}{\mathbf {e}}^d}}} \varphi \psi + O(h^4)\). The author regrets the oversight of dropping the constant \(\frac{1}{h^{{\textsc {D}}-1}}\) in the RHS.

  3. Due to the transverse gradient (49), the ghost cells diagonal to domain corners also need to be filled. For this purpose, the off-diagonal ghost cells are filled first, their values are then plugged into (53) to calculate those of the diagonal ghost cells.

  4. In other words, each cell face only has the single velocity component normal to it. The face-averaged velocity here is the same as that on staggered grids. In comparison, each face in (Cnv-3) has all the components of velocity averaged over it.

  5. The coefficients \(-\frac{11}{10h}\) and \(-\frac{11}{10h}\) can be easily derived from (46) and (54).

  6. This constant is the average of these Neumann boundary values. For all numerical experiments in Sect. 6, no relative changes of \(\left\langle \frac{\partial q}{\partial n} \right\rangle _{{\mathbf {j}}+ \frac{1}{2}{\mathbf {e}}^n}\) are greater than 0.005.

  7. If \(\nabla p\) is needed, \(\mathbf {a}^*\) will have to be extrapolated by (56) to the boundary to obtain the Neumann boundary condition for p.

  8. The main arguments are (i) the velocity is at best fourth-order accurate, (ii) the Neumann boundary condition of \(\nabla q\) in (34d) cannot be calculated to fourth-order accurate near the boundary due to the lop-sided ghost-filling formulas in Sect. 3.3 and the O(1 / h) coefficients in (44) and (45), and (iii) the error in calculating the boundary condition causes an error of the same order in the solution, since the Green’s function defined by \(G''(x)=0,\ x\in [0,1]\); \(G'(0)=1, G(1)=0\) is \(G(x)=x-1\).

  9. Two motivations for doing so could be an even higher temporal accuracy or a better stability for very stiff problems.

  10. Following [32, Sec. A.5], the accuracy of numerical results are measured by the p-norm of a grid function as \(\Vert \psi \Vert _p = \left( h^{{\textsc {D}}}\sum _{{\mathbf {i}}\subseteq \varOmega } |\psi _{{\mathbf {i}}}|^p \right) ^{1/p}\). If \(\psi =O(h^3)\) on \(O(\frac{1}{h^{{\textsc {D}}-1}})\) cells and \(\psi =O(h^4)\) on \(O(\frac{1}{h^{{\textsc {D}}}})\) cells, then \(\Vert \psi \Vert _1 = h^{{\textsc {D}}} O(h^4) O(\frac{1}{h^{{\textsc {D}}}}) + h^{{\textsc {D}}} O(h^3) O(\frac{1}{h^{{\textsc {D}}-1}}) = O(h^4)\) and \(\Vert \psi \Vert _2 = \left( h^{{\textsc {D}}} O(h^8) O(\frac{1}{h^{{\textsc {D}}}}) + h^{{\textsc {D}}} O(h^6) O(\frac{1}{h^{{\textsc {D}}-1}})\right) ^{1/2} = O(h^{3.5})\).

  11. The thresholds of relative convergence in Fig. 5 are set according to the condition numbers of multigrid solvers.

  12. The Chombo code is compiled with the switches “OPT=HIGH DEBUG=FALSE MPI=TRUE.”

  13. This fact supports the methodology of efficiency comparison in [57], where the number of multigrid solves is used to measure the CPU time.

  14. As shown in Table 7, MCG has not reached its asymptotic convergence rates on the finest grid for the 3D viscous-box test with Re = \(10^4\). Nonetheless, it is assumed that the convergence rates of MCG upon the very next grid refinement will reach 2. This assumption favors the performance of MCG and makes the values of the estimated speedup smaller. This footnote also applies to Tables 10, 11, and 12.

  15. With the assumption of perfect scaling, this upper bound is very loose. The load balancing problem for high performance computing is notoriously difficult. For practical problems, it is often the case that the computing time can not be reduced any more once the number of processes reach tens of thousands, letting alone millions of them.

  16. When a derivative has to be calculated from the results of the primary variables, the order of accuracy of a method and the grid size for a specific problem needs to be balanced to avoid excessive lost of information from catastrophic cancellations. The hp-adaptivity in finite element method is a good example.

References

  1. Adams, M., Colella, P., Graves, D.T., Johnson, J., Keen, N., McCorquodale, T.J.L.D.F.M.P., Schwartz, D.M.P., Sternberg, T., Straalen, B.V.: Chombo Software Package for AMR Applications—Design Document. Technical report, Lawrence Berkeley National Laboratory. LBNL-6616E (2014)

  2. Almgren, A., Aspden, A., Bell, J.B., Minion, M.L.: On the use of higher-order projection methods for incompressible turbulent flow. SIAM J. Sci. Comput. 35(1), B25–B42 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alonso-Mallo, I.: Runge–Kutta methods without order reduction for linear initial boundary value problems. Numer. Math. 91, 577–603 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baek, H., Karniadakis, G.E.: Sub-iteration leads to accuracy and stability enhancements of semi-implicit schemes for the Navier–Stokes equations. J. Comput. Phys. 230(12), 4384–4402 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bell, J.B., Colella, P., Howell, L.H.: An efficient second order projection method for viscous incompressible flow. In: AIAA 10th Computational Fluid Dynamics Conference, pp. 360–367 (1991)

  8. Boscarino, S., Pareschi, L., Russo, G.: Implicit-explicit Runge–Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit. SIAM J. Sci. Comput. 35(1), A22–A51 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brandt, A.: Algebraic multigrid theory: the symmetric case. Appl. Math. Comput. 19(1–4), 23–56 (1986). doi:10.1016/0096-3003(86)90095-0

    MathSciNet  MATH  Google Scholar 

  10. Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial. SIAM, 2nd edn. (2000). ISBN:0-89871-462-1

  11. Brown, D.L., Cortez, R., Minion, M.L.: Accurate projection methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 168(2), 464–499 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chombo.: Software for adaptive solutions of partial differential equations (April 2014). http://seesar.lbl.gov/ANAG/software.html. Version 3.2

  13. Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22(104), 745–762 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cozzi, E., Pego, R.L.: On optimal estimates for the Laplace–Leray commutator in planar domains with corners. Proc. Am. Math. Soc. 139(5), 1691–1706 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. E, W., Liu, J.-G.: Gauge method for viscous incompressible flows. Comm. Math. Sci. 1(2), 317 (2003)

  16. Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: High order strong stability preserving time discretizations. J. Sci. Comput. 38(3), 251–289 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gresho, P.M., Sani, R.L.: On presssure boundary conditions for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 7, 1115–1145 (1987)

    Google Scholar 

  18. Griffith, B.E.: An accurate and efficient method for the incompressible Navier–Stokes equations using the projection method as a preconditioner. J. Comput. Phys. 228(20), 7565–7595 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guermond, J.L., Migeon, C., Pineau, G., Quartapelle, L.: Start-up flows in a three-dimensional rectangular driven cavity of aspect ratio 1:1:2 at Re = 1000. J. Fluid. Mech. 450, 169–199 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195(44–47), 6011–6045 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gullbrand, J.: An Evaluation of a Conservative Fourth Order DNS Code in Turbulent Channel. Technical report, Center for Turbulence Research, Stanford University (2000)

  22. Henshaw, W.D.: A fourth-order accurate method for the incompressible Navier–Stokes equations on overlapping grids. J. Comput. Phys. 113, 13–25 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hokpunna, A., Manhart, M.: Compact fourth-order finite volume method for numerical solutions. J. Comput. Phys. 229, 7545–7570 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Johnston, H., Liu, J.-G.: Accurate, stable and efficient Navier–Stokes solvers based on explicit treatment of the pressure term. J. Comput. Phys. 199(1), 221–259 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Karniadakis, G., Israeli, M., Orszag, S.: High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97(2), 414–443 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kassam, A.-K., Trefethen, L.: Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26(4), 1214–1233 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kennedy, C.A., Carpenter, M.H.: Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl. Numer. Math. 44, 139–181 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59(2), 308–323 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kleiser, L., Schumann, U.: Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows. In: Proceedings of the Third GAMM—Conference on Numerical Methods in Fluid Mechanics, vol 2 of Notes on Numerical Fluid Mechanics, pp. 165–173. Springer (1980). ISBN:978-3-528-08076-1

  30. Knikker, R.: Study of a staggered fourth-order compact scheme for unsteady incompressible viscous flows. Int. J. Numer. Methods Fluids 59(10), 1063–1092 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Leriche, E., Perchat, E., Labrosse, G., Deville, M.: Numerical evaluation of the accuracy and stability properties of high-order direct Stokes solvers with or without temporal splitting. J. Sci. Comput. 26(1), 25–43 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. LeVeque, R.J.: Finite Difference Methods for Ordinary Differential Equations. SIAM, Philadelphia (2007). ISBN 978-0-898716-29-0

  33. Liu, J.-G., Liu, J., Pego, R.L.: Stability and convergence of efficient Navier–Stokes solvers via a commutator estimate. Commun. Pure Appl. Math. 60, 1443–1487 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, J.-G., Liu, J., Pego, R.L.: Stable and accurate pressure approximation for unsteady incompressible viscous flow. J. Comput. Phys. 229(9), 3428–3453 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Martin, D.F., Colella, P., Graves, D.: A cell-centered adaptive projection method for the incompressible Navier–Stokes equations in three dimensions. J. Comput. Phys. 227, 1863–1886 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. McCorquodale, P., Colella, P.: A high-order finite-volume method for hyperbolic conservation laws on locally-refined grids. Commun. Appl. Math. Comput. Sci. 6(1), 1–25 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Meuer, H., Strohmaier, E., Dongarra, J., Simon, H.: Top 500 supercomputer sites. (April 2012). http://www.top500.org

  38. Minion, M.L.: Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1(3), 471–500 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Morinishi, Y., Lund, T.S., Vasilyev, O.V., Moin, P.: Fully conservative higher order finite difference schemes for incompressible flow. J. Comput. Phys. 143(1), 90–124 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ohlsson, J., Schlatter, P., Fischer, P., Henningson, D.: Direct numerical simulation of separated flow in a three-dimensional diffuser. J. Fluid Mech. 650, 307–318 (2010)

    Article  MATH  Google Scholar 

  41. Ohlsson, J., Schlatter, P., Fischer, P.F., Henningson, D.S.: Stabilization of the spectral-element method in turbulent flow simulations. In: Hesthaven, J.S., Ronquist, E.M. (eds.) Spectral and High Order Methods for Partial Differential Equations, volume 76 of Lecture Notes in Computational Science and Engineering, pp. 449–458. Springer, Berlin (2011)

  42. Orszag, S.A., Israeli, M., Deville, M.O.: Boundary conditions for incompressible flows. J. Sci. Comput. 1(1), 75–111 (1986)

    Article  MATH  Google Scholar 

  43. Pereira, J.M.C., Kobayashi, M.H., Pereira, J.C.F.: A fourth-order-accurate finite volume compact method for the incompressible Navier-Stokes solutions. J. Comput. Phys. 167(1), 217–243 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  44. Portero, L., Jorge, J.C., Bujanda, B.: Avoiding order reduction of fractional step Runge–Kutta discretizations for linear time dependent coefficient parabolic problems. Appl. Numer. Math. 48, 409–424 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sanderse, B., Koren, B.: Accuracy analysis of explicit Runge–Kutta methods applied to the incompressible Navier–Stokes equations. J. Comput. Phys. 231, 3041–3063 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sanderse, B., Verstappen, R.W.C.P., Koren, B.: Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier–Stokes equations. J. Comput. Phys. 257, 1472–1505 (2014)

    Article  MathSciNet  Google Scholar 

  47. Shankar, P.N., Deshpande, M.D.: Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech. 32, 93–136 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  48. 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(23), 8619–8646 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  49. Shishkina, O., Wagner, C.: A fourth order finite volume scheme for turbulent flow simulations in cylindrical domains. Comput. Fluids 36(2), 484–497 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  50. Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51(1), 82–126 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. Trebotich, D.P., Colella, P.: A projection method for incompressible viscous flow on moving quadrilateral grids. J. Comput. Phys. 166, 191–217 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  52. Trottenberg, U., Oosterlee, C., Schuller, A.: Multigrid. Elsevier Academic Press (2001). ISBN:0-12-701070-X

  53. Twizell, E.H., Gumel, A.B., Arigu, M.A.: Second-order, L0-stable methods for the heat equation with time-dependent boundary conditions. Adv. Comput. Math. 6, 333–352 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  54. Verstappen, R., Veldman, A.: Direct numerical simulation of turbulence at lower costs. J. Eng. Math. 32(2–3), 143–159 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  55. Zerroukat, M., Wood, N., Staniforth, A.: The parabolic spline method (PSM) for conservative transport problems. Int. J. Numer. Meth. Fluids 51, 1297–1318 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  56. Zhang, Q.: Highly accurate Lagrangian flux calculation via algebraic quadratures on spline-approximated donating regions. Comput. Methods Appl. Mech. Engrg. 264, 191–204 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  57. Zhang, Q.: A fourth-order approximate projection method for the incompressible Navier–Stokes equations on locally-refined periodic domains. Appl. Numer. Math 77(C), 16–30 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  58. Zhang, Q., Fogelson, A.: Fourth-order interface tracking in two dimensions via an improved polygonal area mapping method. SIAM J. Sci. Comput. 36, A2369–A2400 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhang, Q., Fogelson, A.: MARS: An analytic framework of interface tracking via mapping and adjusting regular semi-algebraic sets. SIAM J. Numer. Anal. (2016, to appear)

  60. Zhang, Q., Johansen, H., Colella, P.: A fourth-order accurate finite-volume method with structured adaptive mesh refinement for solving the advection–diffusion equation. SIAM J. Sci. Comput. 34(2), B179–B201 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhang, Q., Liu, P.L.-F.: A new interface tracking method: the polygonal area mapping method. J. Comput. Phys. 227(8), 4063–4088 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  62. Zhang, Q., Liu, P.L.-F.: Handling solid–fluid interfaces for viscous flows: explicit jump approximation vs. ghost cell approaches. J. Comput. Phys. 229(11), 4225–4246 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to two anonymous referees, whose comments lead to an improvement of the quality of this paper. The author also thanks Prof. Chi-Wang Shu for the extra time and effort he spent in handling the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Utah, 155 S. 1400 E. Rm233, Salt Lake City, UT, 84112, USA

    Qinghai Zhang

Authors
  1. Qinghai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qinghai Zhang.

Additional information

In memory of my mentor, Madam Yin, Ping.

Appendix

Appendix

Figure 10 shows the Matlab code that generates all the formulas for filling ghost cells and extrapolating cell averages to face averages on the domain boundary.

Fig. 10
figure 10

A Matlab code for generating the formulas in Sect. 3.3 for filling ghost cells and extrapolating cell averages to face averages on the domain boundary

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Q. GePUP: Generic Projection and Unconstrained PPE for Fourth-Order Solutions of the Incompressible Navier–Stokes Equations with No-Slip Boundary Conditions. J Sci Comput 67, 1134–1180 (2016). https://doi.org/10.1007/s10915-015-0122-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0122-4

Keywords

Mathematics Subject Classification

Search

Navigation