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

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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.

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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.

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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.

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

    Qinghai Zhang

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Correspondence to Qinghai Zhang.

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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

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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

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