Effective Source Term Discretizations for Higher Accuracy Finite Volume Discretization of Parabolic Equations 135 89
DOI:
https://doi.org/10.31686/ijier.vol9.iss8.3305Keywords:
Finite volume method, space-time control volume, local equation error expansion, her-order convergence, Heat equation, one-step scheme, numerical domain of dependence, general weighted quadratures, conservative flux integralAbstract
A finite volume method is applied to develop space-time discretizations for parabolic equations based on an equation error method.
A space-time expansion of the local equation error based on flux integral formulation of the equation is first designed using a desired
framework of neighboring quadrature points for the solution and local source terms. The quadrature weights are then determined through a
minimization process for the error which constitutes all local compact fluxes about each centroid within the computational domain.
In utilizing a local source term distribution to account for diffusive fluxes, the right minimizing quadrature weights and collocation
points including subgrid points for the source terms may be determined and optimized for higher accuracies as well as robust higher-order
computational convergence. The resulting local residuals form a more complete description of the truncation errors which are then utilized
to assess the computational performances of the resulting schemes. The effectiveness of the discretization method is demonstrated by the
results and analysis of the schemes.
References
A. Adcroft, C. Hill, and J. Marshall. Representation of Topography by Shaved Cells in a Height Coordinate Ocean Model. Monthly Weather Rev., 125:2293–2315, 1997. DOI: https://doi.org/10.1175/1520-0493(1997)125<2293:ROTBSC>2.0.CO;2
C. W. Hirt A. A. Amsden and J. L. Cook. An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds. J. Comput. Phys., 14:227–253, 1974. DOI: https://doi.org/10.1016/0021-9991(74)90051-5
A. Berman and R. J. Plemmons. Nonegative Matrices in the Mathematical sciences. Computer Science and Applied Mathematics; A series of Monographs and textbooks. Academic Press, Inc, New York San Francisco London, 1979. DOI: https://doi.org/10.1016/B978-0-12-092250-5.50011-4
C. L. Bottasso. On the computation of the boundary integral of space-time deforming finite elements. Comm. Numer. Meth. Engrg., 13:53–59, 1997. DOI: https://doi.org/10.1002/(SICI)1099-0887(199702)13:2<53::AID-CNM29>3.0.CO;2-I
D. Britz, O. Østerby, and J. Strutwolf. Damping of crank-nicolson error oscillations. Computational biology and chemistry, 27:253–63, 8 2003. DOI: https://doi.org/10.1016/S0097-8485(02)00075-X
Z. Cai, J. Mandel, and S. McCormick. The finite volume element method for diffusion equation on general triangulations. Siam J. Numer. Anal., 28:392–402, 1991. DOI: https://doi.org/10.1137/0728022
M. O. Devile, P. F. Fischer, and E. H. Mund. Higher-Order Methods for Incompressible Fluid Flow. Cambridge University Press, Cambridge · New York · Port Melbourne · Madrid · Cape Town, 2002. DOI: https://doi.org/10.1017/CBO9780511546792
C. A. J. Fletcher. Computational Galerkin Methods. Springer Series in Computational Physics. Springer-Verlag, New York Berlin Heidelberg Tokyo, 1984.
D. A. French. A space-time finite element method for the wave equation. Comput. Methods. Appl. Mech. Engrg., 107:145–157, 1993. DOI: https://doi.org/10.1016/0045-7825(93)90172-T
P. Frolkovic. Maximum Principle and Local Mass Balance for Numerical Solutions of Transport Equation Coupled with Variable Density Flow. Acta Math. Univ. Comenianae, 1:137–157, 1998.
S. Gabersek and D. R. Durran. Gap Flows through Idealized Topography. Part II: Effects of Rotation and Surface Friction. J. Atmos. Sci., 63:2720–2315, 2006. DOI: https://doi.org/10.1175/JAS3786.1
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, 143:9–27, 2002. DOI: https://doi.org/10.1007/BF03012338
L. Gorelick, M. Blank, E. Shechtman, M. Irani, and R. Basri. Actions as Space-Time Shapes. IEEE Trans. Pattern Analysis And Machine Intelligence, 29:1–7, 2007. DOI: https://doi.org/10.1109/TPAMI.2007.70711
M. N. Guddati and B. Yue. Modified integration rules for reducing dispersion error in finite element methods. Comput. Methods Appl. Mech. Engrg. 193 (2004), 193:275–287, 2004. DOI: https://doi.org/10.1016/j.cma.2003.09.010
T.J.R. Hughes and G.M. Huibert. Space-time finite element methods for elastodynamics: Formulations and error estimates. Comput. Methods. Appl. Mech. Engrg., 66:145–157, 1988. DOI: https://doi.org/10.1016/0045-7825(88)90006-0
T.J.R. Hughes and J. Stewart. A space time formulation for multiscale phenomena. J. Comput. Appl. Math., 74:217–229, 1996. DOI: https://doi.org/10.1016/0377-0427(96)00025-8
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., 27:346–367, 2005. DOI: https://doi.org/10.1137/03060120X
C. Kim and A. Jameson. A Robust and Accurate LED-BGK Solver on Unstructured Adaptive Meshes. J. Comput. Phys., 143:598–627, 1998. DOI: https://doi.org/10.1006/jcph.1998.5973
Y. Kyei. Space-time finite volume differencing framework for effective higher-order accurate discretizations of parabolic equations. SIAM J. Sci. Comput., 34(3):A1432–A1459, 2012. DOI: https://doi.org/10.1137/10080498X
Y. Kyei. Higher-order accurate finite volume discretization of the three-dimensional poisson equation based on an equation error method. Int. J. for Innovation Education and Research, 6(6):107 – 123, 2018. DOI: https://doi.org/10.31686/ijier.vol6.iss6.1076
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 352174:1–17, 2010. DOI: https://doi.org/10.1155/2010/352174
M. N. Levya, R. D. Nairb, and H. M. Tufo. High-order Galerkin methods for scalable global atmospheric models. Computers & Geosciences, 233:1022–1035, 2007. DOI: https://doi.org/10.1016/j.cageo.2006.12.004
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., 227:5649–5670, 2008. DOI: https://doi.org/10.1016/j.jcp.2008.02.015
W. H. Mason. Applied Computational Aerodynamics. Text/notes, http://www.dept.aoe.vt.edu/ maxon/Mason f/CAtextTop.html, 1997.
A. Masud and T.J.R. Hughes. A Space-time Galerkin/least-squares finite element formulation of the Navier-Staokes equations for moving domain problems. Comput. Methods Appl. Mech Engr, 146:91–126, 1997. DOI: https://doi.org/10.1016/S0045-7825(96)01222-4
C. Mattiussi. The finite volume, finite element, and finite difference methods as numerical methods for physical field problems. Advances in Imaging and Electron Physics, 113:1– 146, 2000. DOI: https://doi.org/10.1016/S1076-5670(00)80012-9
S. Mittal and T.E. Tezduyar. Notes on the stabilized space-time finite-element formulation of unsteady incompressible flows. Comput. Phys. Commun., 73:93–112, 1992. DOI: https://doi.org/10.1016/0010-4655(92)90031-S
E. Onate and M. Manzan. A general procedure for deriving stabilized space-time finite element methods for advective-diffusive problems. Int. J. Numer. Meth. Fluids, 31:203–221, 1999. DOI: https://doi.org/10.1002/(SICI)1097-0363(19990915)31:1<203::AID-FLD964>3.0.CO;2-Z
W. M. Rohsenow, J. R. Hartnett, and Y. I. Cho. Handbook of Heat Transfer. McGraw-Hill, New York, third edition, 1998.
J. Santos and P. de Oliveira. A converging finite volume scheme for hyperbolic conservation laws with source terms. J. Comput. App. Math., 111:239–251, 1999. DOI: https://doi.org/10.1016/S0377-0427(99)00146-6
M. A. Tolstykh. Vorticity-Divergence Semi-Lagrangian Shallow-Water Model of the Sphere Based on Compact Finite Differences. J. Comput. Phys., 179:180–200, 2002. DOI: https://doi.org/10.1006/jcph.2002.7050
E F. Toro and A. Hidalgo. ADER finite volume schemes for nonlinear reaction-diffusion equations. App. Numer. Math., 59:73–100, 2009. DOI: https://doi.org/10.1016/j.apnum.2007.12.001
H. van der Ven and J. J. W. van der Vegt. Space-time discontinuous Galerkin finite element method wih dynamic grid motiton for inviscid compressible flows II. Efficient flux quadrature. Comput. Methods Appl. Mech. Engr., 191:4747–4780, 2002. DOI: https://doi.org/10.1016/S0045-7825(02)00403-6
A. K. Verma, S. M. Bhallamudi, and V. Eswaran. Overlapping control volume method for solute transport. J. Hydr. Engrg., 5:308–316, 2000. DOI: https://doi.org/10.1061/(ASCE)1084-0699(2000)5:3(308)
Downloads
Published
Issue
Section
License
Copyright (c) 2021 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.
