Abstract
This work concerns with the development of fast and high order algorithms for solving a single variable Poisson’s equation with rectangular domains and uniform meshes, but involving staggered boundaries. Here the staggered boundary means that the boundary is located midway between two adjacent grid nodes. Due to the popularity of staggered grids in scientific computing for solving multiple variables partial differential equations (PDEs), the planned development deserves further studies, but is rarely reported in the literature, because grand challenges exist for spectral methods, compact finite differences, and fast Fourier transform (FFT) algorithms in handling staggered boundaries. A systematic approach is introduced in this paper to attack various open problems in this regard, which is a natural generalization of a recently developed Augmented Matched Interface and Boundary (AMIB) method for non-staggered boundaries. Formulated through immersed boundary problems with zero-padding solutions, the AMIB method combines arbitrarily high order central differences with the FFT inversion. Over staggered boundaries, the proposed AMIB method can handle Dirichlet, Neumann, Robin or any combination of boundary conditions. Convergence orders in four, six and eight are numerically validated for the AMIB method in both two and three dimensions. Moreover, the proposed AMIB method performs well for some challenging problems, such as low regularity solution near boundary, PDE solution not satisfying the boundary condition, and involving both staggered and non-staggered boundaries on two ends.
Similar content being viewed by others
References
Abide, S., Zeghmati, B.: Multigrid defect correction and fourth-order compact scheme for Poisson’s equation. Comput. Math. Appl. 73, 1433–1444 (2017)
Averbuch, A., Israeli, M., Vozovoi, L.: A fast Poisson solver of arbitrary order accuracy in rectangular regions. SIAM J. Sci. Comput. 19, 933–952 (1998)
Boisvert, R.F.: A fourth order accurate Fourier method for the Helmholtz equation in three dimensions. ACM Trans. Math. Softw. (TOMS) 13, 221–234 (1987)
Braverman, E., Israeli, M., Averbuch, A., Vozovoi, L.: A fast 3D Poisson solver of arbitrary order accuracy. J. Comput. Phys. 144, 109–136 (1998)
Braverman, E., Israeli, M., Averbuch, A.: A fast spectral solver for a 3D Helmholtz equation. SIAM J. Sci. Comput. 20, 2237–2260 (1999)
Bruger, A., Nilsson, J., Kress, W.: A compact higher order finite difference method for the incompressible Navier–Stokes equations. J. Sci. Comput. 17, 551–560 (2002)
Feng, H., Long, G., Zhao, S.: An augmented matched interface and boundary (MIB) method for solving elliptic interface problem. J. Comput. Appl. Math. 361, 426–433 (2019)
Feng, H., Zhao, S.: FFT-based high order central difference schemes for the three-dimensional Poisson equation with various types of boundary conditions. J. Comput. Phys. 410, 109391 (2020)
Feng, H., Zhao, S.: A fourth order finite difference method for solving elliptic interface problems with the FFT acceleration. J. Comput. Phys. 419, 109677 (2020)
Fornberg, B.: Calculation of weights in finite difference formulas. SIAM Rev. 40, 685–691 (1998)
Golub, G.H., Huang, L.C., Simon, H., Tang, W.: A fast Poisson solver for the finite difference solution of the incompressible Navier–Stokes equations. SIAM J. Comput. 19, 1606–1624 (1998)
Gupta, M.M., Kouatchou, J., Zhang, J.: Comparison of second and fourth order discretization multigrid Poisson solvers. J. Comput. Phys. 132, 226–232 (1997)
Ge, Y.: Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D Poisson equation. J. Comput. Phys. 229, 6381–6391 (2010)
Haidvoge, D., Zang, T.: The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials. J. Comput. Phys. 30, 167–180 (1979)
Kampanis, N.A., Ekaternaris, J.A.: A staggered grid, high-order accurate method for the incompressible Navier–Stokes equations. J. Comput. Phys. 215, 589–613 (2006)
Lai, M.-C.: A simple compact fourth-order Poisson solver on polar geometry. J. Comput. Phys. 182, 337–345 (2002)
Ma, Z.H., Qian, L., Causon, D.M., Gu, H.B., Mingham, C.G.: A cartesian ghost-cell multigrid poisson solver for incompressible flows. Int. J. Numer. Meth. Eng. 85, 230–246 (2011)
Nagel, J.R.: Solving the Generalized Poisson’s Equation Using the Finite-Difference Method (FDM). University of Utah, Salt Lake City, Department of Electrical and Computer Engineering (2011)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)
Schumann, U., Sweet, R.: A direct method for the solution of Poisson’s equation with neumann boundary conditions on a staggered grid of arbitrary size. J. Comput. Phys. 20, 171–182 (1976)
Schumann, U., Sweet, R.: Fast Fourier transforms for direct solution of Poisson’s equation with staggered boundary conditions. J. Comput. Phys. 75, 123–137 (1988)
Shen, J., Tang, T., Wang, L.L.: Spectral methods: Algorithm, Analysis and Application. Springer Series in Computational Mathematics. Springer, Berlin (2011)
Sun, X.H., Zhuang, Y.: A high-order direct solver for helmholtz equations with neumann boundary conditions. Technical Report. Institute for Computer Applications in Science and Engineering (ICASE) (1997)
Swarztrauber, P.N.: Symmetric FFTs. Math. Comput. 47, 323–346 (1986)
Swarztrauber, P., Sweet, R.: Algorithm 541: efficient Fortran subprograms for the solution of separable elliptic partial differential equations. ACM Trans. Math. Softw. (TOMS) 5, 352–364 (1979)
Trottenberg, U., Oosterlee, C.W.: Multigrid. Academic Press, Cambridge (2001)
Wang, H., Zhang, Y., Ma, X., Qiu, J., Liang, Y.: An efficient implementation of fourth-order compact finite difference scheme for Poisson’s equation with Dirichlet boundary conditions. Comput. Math. Appl. 71, 1843–1860 (2016)
Wang, Y., Zhang, J.: Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D Poisson equation. J. Comput. Phys. 228, 137–146 (2009)
Wesseling, P.: An Introduction to Multigrid Methods. Pure and Applied Mathematics. Wiley, Hoboken (1992)
Zhang, K.K.O., Shotorban, B., Minkowycz, W.J., Mashayek, A.: A compact finite difference method on staggered grid for Navier–Stokes flow. Int. J. Numer. Methods Fluids 52, 867–881 (2006)
Zhao, S., Wei, G.W.: High-order FDTD methods via derivative matching for Maxwell’s equations with material interfaces. J. Comput. Phys. 200, 60–103 (2004)
Zhao, S., Wei, G.W., Xiang, Y.: DSC analysis of free-edged beams by an iteratively matched boundary method. J. Sound Vib. 284, 487–493 (2005)
Zhao, S.: On the spurious solutions in the high-order finite difference methods. Comput. Methods Appl. Mech. Eng. 196, 5031–5046 (2007)
Zhao, S.: A fourth order finite difference method for waveguides with curved perfectly conducting boundaries. Comput. Methods Appl. Mech. Eng. 199, 2655–2662 (2010)
Zhao, S., Wei, G.W.: Matched interface and boundary (MIB) for the implementation of boundary conditions in high order central finite differences. Int. J. Numer. Methods Eng. 77, 1690–1730 (2009)
Zhou, Y.C., Zhao, S., Feig, M., Wei, G.W.: High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular source. J. Comput. Phys. 213, 1–30 (2006)
Zhuang, Y., Sun, X.: A high-order fast direct solver for singular Poisson equations. J. Comput. Phys. 20, 79–94 (2001)
Acknowledgements
This research is partially supported by the National Science Foundation (NSF) Grant DMS-1812930, the Natural Science Foundation of China under Grant 11461011, and the key project of Guangxi Provincial Natural Science Foundation of China under Grants AD20238065, 2017GXNSFDA198014 and 2018GXNSFDA050014.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Feng, H., Long, G. & Zhao, S. FFT-Based High Order Central Difference Schemes for Poisson’s Equation with Staggered Boundaries. J Sci Comput 86, 7 (2021). https://doi.org/10.1007/s10915-020-01379-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-020-01379-7
Keywords
- Fast Poisson solver
- Fast Fourier transform (FFT)
- High order central difference schemes
- Matched interface and boundary (MIB) method
- Staggered grid