Higher-Order Accurate Finite Volume Discretization of the Three-Dimensional Poisson Equation Based on An Equation Error Method
DOI:
https://doi.org/10.31686/ijier.vol6.iss6.1076Keywords:
Finite volume, local control volume, equation error expansion, uniform higherorder accurate convergence, local residual error, three-dimensional Poisson’s equation, local compact fluxes, flux integral formulationAbstract
Efficient higher-order accurate finite volume schemes are developed for the threedimensional Poisson’s equation based on optimizations of an equation error expansion on local control volumes. A weighted quadrature of local compact fluxes and the flux integral form of the equation are utilized to formulate the local equation error expansions. Efficient quadrature weights for the schemes are then determined through a minimization of the error expansion for higher-order accurate discretizations of the equation. Consequently, the leading numerical viscosity coefficients are more accurately and completely determined to optimize the weight parameters for uniform higher-order convergence suitable for effective numerical modeling of physical phenomena. Effectiveness of the schemes are evaluated through the solution of the associated eigenvalue problem. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology.
Downloads
Download data is not yet available.
References
[1] T. Barth and M. Ohlberger, Encyclopedia of Computational Mechanics: Finite Volume Methods: Foundation and Analysis, vol. I, John Wiley & Sons Ltd, Hoboken, USA, 2004.
[2] B. Diskin and J. L. Thomas, Accuracy analysis for mixed-element finite-volume discretization schemes, NIA Report NO. 2007-08 (2007).
[3] Q. Du, J. R. Kamm, R. B. Lehoucq, and M. L. Parks, A new approach for a nonlocal, nonlinear conservation law, SIAM J. APPL. MATH. 72 (2012), no. 1, 464–487.
[4] S. Gabersek and D. R. Durran, Gap Flows through Idealized Topography. Part II: Effects of Rotation and Surface Friction, J. Atmos. Sci. 63 (2006), 2720–2315.
[5] Lixin Ge and Jun Zhang, Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coeffcients, J. comput. Appl. Math 143 (2002), 9–27.
[6] K. Ito, Y. Kyei, and Z. Li, Higher-Order, Cartesian Grid Based Finite Difference Schemes for Elliptic Equations on Irregular Domains , SIAM J. Sci. Comput. 27 (2005), 346–367.
[7] C. Katz and A. Jameson, A Comparison of Various Meshless Schemes Within a Unified Algorithm, 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition 2009-897 (2009).
[8] R. V. Keer and H. D. Schepper, Finite Element Approximation for 2nd Order Elliptic Eigenvalue Problems with Nonlocal Boundary or Transition Conditions, Appl. Math. Comput. 82 (1997), 1–16.
[9] P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer-Verlag, New York, Inc, 2003.
[10] E. P. C. Koh, H. M. Tsai, and F. Liu, Euler Solution Using Cartesian Grid with a Gridless Least-Squares Boundary Treatment, AIAA Journal 43 (2005), no. 2.
[11] Y. Kyei, Space-time finite volume differencing framework for effective higher-order accurate discretizations of parabolic equations, SIAM J. Sci. Comput. 34 (2012), no. 3, A1432– A1459.
[12] Y. Kyei and K. Edoh, Higher-order accurate finite volume discretization of helmholtz equations with pollution effects reductions, Int. J. for Innovation Education and Research 6 (2018), no. 4.
[13] Y. Kyei, J. P. Roop, and G. Tang, A family of sixth-order compact finite difference schemes for poisson equation, Adv. Numer. Anal. Article ID 352174 (2010), 1–17.
[14] S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992), 1–42.
[15] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems , SIAM, 2007.
[16] J. Levesley and D. L. Ragozin, Local Approximation on Manifolds Using Radial Functions and Polynomials, International Conference on Curves and Surfaces [4th], Saint-Malo,France, 1-7 July 1999 Proceedings, Volume 2. Curve and Surface Fitting, 291–300.
[17] M. Li and T. Tang, A Compact Fourth-Order Finite Difference Scheme for Unsteady Viscous Incompressible Flows, J. Sci. Comput. 16 (2001), 29–45.
[18] S. Liang, X. Ma, and A. Zhou, Finite volume mehtods for eigenvalue problems, BIT 41 (2001), no. 2, 345–363.
[19] Y. Liu, M. Vinokur, and Z.J. Wang, Spectral (finite) volume method for conservation laws on unstructured grids V: Extension to three-dimensional systems, J. Comput. Phys. 212 (2006), 454–472.
[20] F. Lorcher, G. Gassner, and C. D. Munz, An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations, J. Comput. Phys. 227 (2008), 5649–5670.
[21] W. H. Mason, Applied computational aerodynamics, Text/notes, http://www.dept.aoe.vt.edu/ maxon/Mason f/CAtextTop.html, 1997.
[22] P. Ming and X. Yue, Numerical methods for multiscale elliptic problems, J. Comput. Phys. 214 (2006), 421–445.
[23] M. Piller and E. Stalio, Finite–volume compact schemes on staggered grids, J. Comput. Phys. 197 (2004), 1064–1094.
[24] J. Santos and P. de Oliveira, A converging finite volume scheme for hyperbolic conservation laws with source terms, J. Comput. App. Math. 111 (1999), 239–251.
[25] A. Shiferaw and R. C. Mittal, An efficient direct method to solve the three dimensional poissons equation, American J. Comput. Math. 1 (2011), 285 – 293.
[26] W. F. Spotz and G. F. Carey, A high-order compact formulation for the 3D Poisson equation, Numer. Meth. Partial Differential Equations 12 (1996), 235–243.
[27] J. C. Strikwerda, Finite Difference and Partial Differential Equations, Wadsworth & BrooksCole Advanced Books & Software, 1989.
[28] G. Sutmann and B. Steffen, High-order compact solvers for the three-dimensional poisson equation, J. Comput. and Appl. Math. 187 (2006), no. 2, 142 – 170.
[29] A. K. Verma, S. M. Bhallamudi, and V. Eswaran, Overlapping control volume method for solute transport, J. Hydr. Engrg. 5 (2000), 308–316.
[30] E. Weinan and B. Engquist, Multiscale Modeling and Computation, Notices Of The AMS 50 (2003), no. 9, 1062–1070.
[31] Y. S. Wong and G. Li, Exact finite difference schemes for solving helmholtz equation at any wavenumber, Int. J. OF NUMER ANAL AND MODELING, SERIES B 2 (2011), no. 2,
91–108.
[2] B. Diskin and J. L. Thomas, Accuracy analysis for mixed-element finite-volume discretization schemes, NIA Report NO. 2007-08 (2007).
[3] Q. Du, J. R. Kamm, R. B. Lehoucq, and M. L. Parks, A new approach for a nonlocal, nonlinear conservation law, SIAM J. APPL. MATH. 72 (2012), no. 1, 464–487.
[4] S. Gabersek and D. R. Durran, Gap Flows through Idealized Topography. Part II: Effects of Rotation and Surface Friction, J. Atmos. Sci. 63 (2006), 2720–2315.
[5] Lixin Ge and Jun Zhang, Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coeffcients, J. comput. Appl. Math 143 (2002), 9–27.
[6] K. Ito, Y. Kyei, and Z. Li, Higher-Order, Cartesian Grid Based Finite Difference Schemes for Elliptic Equations on Irregular Domains , SIAM J. Sci. Comput. 27 (2005), 346–367.
[7] C. Katz and A. Jameson, A Comparison of Various Meshless Schemes Within a Unified Algorithm, 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition 2009-897 (2009).
[8] R. V. Keer and H. D. Schepper, Finite Element Approximation for 2nd Order Elliptic Eigenvalue Problems with Nonlocal Boundary or Transition Conditions, Appl. Math. Comput. 82 (1997), 1–16.
[9] P. Knabner and L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer-Verlag, New York, Inc, 2003.
[10] E. P. C. Koh, H. M. Tsai, and F. Liu, Euler Solution Using Cartesian Grid with a Gridless Least-Squares Boundary Treatment, AIAA Journal 43 (2005), no. 2.
[11] Y. Kyei, Space-time finite volume differencing framework for effective higher-order accurate discretizations of parabolic equations, SIAM J. Sci. Comput. 34 (2012), no. 3, A1432– A1459.
[12] Y. Kyei and K. Edoh, Higher-order accurate finite volume discretization of helmholtz equations with pollution effects reductions, Int. J. for Innovation Education and Research 6 (2018), no. 4.
[13] Y. Kyei, J. P. Roop, and G. Tang, A family of sixth-order compact finite difference schemes for poisson equation, Adv. Numer. Anal. Article ID 352174 (2010), 1–17.
[14] S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103 (1992), 1–42.
[15] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems , SIAM, 2007.
[16] J. Levesley and D. L. Ragozin, Local Approximation on Manifolds Using Radial Functions and Polynomials, International Conference on Curves and Surfaces [4th], Saint-Malo,France, 1-7 July 1999 Proceedings, Volume 2. Curve and Surface Fitting, 291–300.
[17] M. Li and T. Tang, A Compact Fourth-Order Finite Difference Scheme for Unsteady Viscous Incompressible Flows, J. Sci. Comput. 16 (2001), 29–45.
[18] S. Liang, X. Ma, and A. Zhou, Finite volume mehtods for eigenvalue problems, BIT 41 (2001), no. 2, 345–363.
[19] Y. Liu, M. Vinokur, and Z.J. Wang, Spectral (finite) volume method for conservation laws on unstructured grids V: Extension to three-dimensional systems, J. Comput. Phys. 212 (2006), 454–472.
[20] F. Lorcher, G. Gassner, and C. D. Munz, An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations, J. Comput. Phys. 227 (2008), 5649–5670.
[21] W. H. Mason, Applied computational aerodynamics, Text/notes, http://www.dept.aoe.vt.edu/ maxon/Mason f/CAtextTop.html, 1997.
[22] P. Ming and X. Yue, Numerical methods for multiscale elliptic problems, J. Comput. Phys. 214 (2006), 421–445.
[23] M. Piller and E. Stalio, Finite–volume compact schemes on staggered grids, J. Comput. Phys. 197 (2004), 1064–1094.
[24] J. Santos and P. de Oliveira, A converging finite volume scheme for hyperbolic conservation laws with source terms, J. Comput. App. Math. 111 (1999), 239–251.
[25] A. Shiferaw and R. C. Mittal, An efficient direct method to solve the three dimensional poissons equation, American J. Comput. Math. 1 (2011), 285 – 293.
[26] W. F. Spotz and G. F. Carey, A high-order compact formulation for the 3D Poisson equation, Numer. Meth. Partial Differential Equations 12 (1996), 235–243.
[27] J. C. Strikwerda, Finite Difference and Partial Differential Equations, Wadsworth & BrooksCole Advanced Books & Software, 1989.
[28] G. Sutmann and B. Steffen, High-order compact solvers for the three-dimensional poisson equation, J. Comput. and Appl. Math. 187 (2006), no. 2, 142 – 170.
[29] A. K. Verma, S. M. Bhallamudi, and V. Eswaran, Overlapping control volume method for solute transport, J. Hydr. Engrg. 5 (2000), 308–316.
[30] E. Weinan and B. Engquist, Multiscale Modeling and Computation, Notices Of The AMS 50 (2003), no. 9, 1062–1070.
[31] Y. S. Wong and G. Li, Exact finite difference schemes for solving helmholtz equation at any wavenumber, Int. J. OF NUMER ANAL AND MODELING, SERIES B 2 (2011), no. 2,
91–108.
Downloads
Published
2018-06-01
Issue
Section
Journal Articles
License
Copyright (c) 2018 Yaw Kyei
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Copyrights for articles published in IJIER journals are retained by the authors, with first publication rights granted to the journal. The journal/publisher is not responsible for subsequent uses of the work. It is the author's responsibility to bring an infringement action if so desired by the author for more visit Copyright & License.
How to Cite
Kyei, Y. (2018). Higher-Order Accurate Finite Volume Discretization of the Three-Dimensional Poisson Equation Based on An Equation Error Method. International Journal for Innovation Education and Research, 6(6), 107-123. https://doi.org/10.31686/ijier.vol6.iss6.1076
