Abstract
In this paper, a new nonlinear finite volume scheme preserving positivity for 3D anisotropic diffusion equation is proposed. A distinct feature of our new scheme is that the auxiliary vertex unknowns appearing in the nonlinear two-point flux approximation are interpolated by the least-square method combined with the so-called correction function. The correction function is local, and only solved when the coefficient discontinuity occurs. Compared with other existing 3D interpolation formulas, our interpolation formula is designed for the general polyhedron mesh and coefficient discontinuity with lower and adaptive computational complexity. The scheme is proven to be positivity-preserving. Numerical examples are presented to demonstrate the second-order accuracy and positivity-preserving property for various anisotropic diffusion problems on the distorted meshes.
Similar content being viewed by others
References
Aavatsmark, I., Eigestad, G.T.: Numerical convergence of the MPFA O-method and U-method for general quadrilateral grids. Int. J. Numer. Meth. Fluids 51, 939–961 (2006)
Agelas, L., Eymard, R., Herbin, R.: A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. C. R. Acad. Sci. Paris Ser. I(347), 673–676 (2009)
Burdakov, O., Kapyrin, I., Vassilevski, Y.: Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions. J. Comput. Phys. 231, 3126–3142 (2012)
Chang, J., Nakshatrala, K.B.: Variational inequality approach to enforcing the non-negative constraint for advection-diffusion equations. Comput. Methods Appl. Mech. Eng. 320, 287–334 (2017)
Cances, C., Cathala, M., Potier, C.L.: Monotone corrections for generic cell-centered finite volume approximations of anisotropic diffusion equations. Numer. Math. 125(1), 387–417 (2013)
Chen, L.: iFEM: an integrated finite element method package in MATLAB, Technical Report, University of California at Irvine, (2009)
Chen, G., Li, D., Su, Z.: Difference scheme by integral interpolation method for three dimensional diffusion equations. Chin. J. Comput. Phys. 20, 205–209 (2003)
Ciarlet, P.G., Raviart, P.A.: Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2, 17–31 (1973)
Danilov, A.A., Vassilevski, Y.V.: A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes. Russ. J. Numer. Anal. Math. Modell. 24, 207–227 (2009)
Draganescu, A., Dupont, T.F., Scott, L.R.: Failure of the discrete maximum pricinple for an elliptic finite element problems. Math. Comput. 74, 1–23 (2004)
Droniou, J.: Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. (M3AS) 24, 1575–1619 (2014)
Edwards, M.G., Zheng, H.: Double-families of quasi-positive Darcy-flux approximations with highly anisotropic tensors on structured and unstructured grids. J. Comput. Phys. 229, 594–625 (2010)
Edwards, M.G., Zheng, H.: Quasi M-matrix multifamily continuous Darcy-flux approximations with full pressure support on structured and unstructured grids in three dimensions. SIAM J. Sci. Comput. 33, 455–487 (2011)
Eymard, R., Henry, G., Herbin, R.: et al., 3D bencnmark on discretization schemes for anisotropic diffusion problems on general grids. In: Finite Volumes for Complex Applications VI Problems and Perspectives, (2011), 895–930
Gao, Z., Wu, J.: A second-order positivity-preserving finite volume scheme for diffusion equations on general meshes. SIAM J. Sci. Comput. 37, 420–438 (2015)
Gao, Z., Wu, J.: A linearity-preserving cell-centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes. Int. J. Numer. Meth. Fluids 67, 2157–2183 (2011)
Gao, Z., Wu, J.: A small stencil and extremum-preserving scheme for anisotropic diffusion problems on arbitrary 2D and 3D meshes. J. Comput. Phys. 250, 308–331 (2013)
Hajibeygi, H., Bonfigli, G., Hesse, M.A., Jenny, P.: Iterative multiscale finite-volume method. J. Comput. Phys. 227, 8604–8621 (2008)
Karimi, S., Nakshatrala, K.B.: Do current lattice Boltzmann methods for diffusion and advection-diffusion equations respect maximum principle and the non-negative constraints? Commun. Comput. Phys. 20, 374–404 (2016)
Lai, X., Sheng, Z., Yuan, G.: Monotone finite volume scheme for three dimensional diffusion equation on tetrahedral meshes. Commun. Comput. Phys. 21(1), 162–181 (2017)
Lan, K., Liu, J., Li, Z., et al.: Progress in octahedral spherical hohlraum study. Matter Radiat. Extrem. 1, 8–27 (2016)
Lu, C., Huang, W., Qiu, J.: Maximum principle in linear finite element approximations of anisotropic diffusion-convection-reaction problems. Numer. Math. 127, 515–537 (2014)
Lu, C., Huang, W., Van Vleck, E.S.: The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations. J. Comput. Phys. 242, 24–36 (2013)
Lindl, J.: Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain. Phys. Plasmas 2, 3933–4023 (1995)
Lipnikov, K., Shashkov, M., Svyatskiy, D., Vassilevski, Y.: Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. J. Comput. Phys. 227, 492–512 (2007)
Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: Anderson acceleration for nonlinear finite volume scheme for advection-diffusion problems. SIAM J. Sci. Comput. 35, 1120–1136 (2013)
Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: Minimal stencil finite volume scheme with the discrete maximum principle. Russ. J. Numer. Anal. Math. Model. 27, 369–385 (2012)
Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes. J. Comput. Phys. 228, 703–716 (2009)
Lipnikov, K., Svyatskiy, D., Vassilevski, Y.: A monotone finite volume method for advection-diffusion equations on unstructured polygonal meshes. J. Comput. Phys. 229, 4017–4032 (2010)
Lipnikov, K., Manzini, G., Svyatskiy, D.: Analysis of monotonicity conditions in the mimetic finite difference method for elliptic problems. J. Comput. Phys. 230, 2620–2642 (2011)
Liska, R., Shashkov, M.: Enforcing the discrete maximum pricinple for linear finite element solutions of second-order elliptic problems. Commun. Comput. Phys. 3, 852–877 (2008)
Le Potier, C.: Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes, C. R. Acad. Sci. Paris, Ser. I, 341 (2005), 787–792
Mudunuru, M.K., Nakshatrala, K.B.: On enforcing maximum pricinples and achieving element-wise species for advection-diffusion-reaction equations under the finite element method. J. Comput. Phys. 305, 448–493 (2016)
Mudunuru, M.K., Nakshatrala, K.B.: On mesh restrictions to satisfy comparison principles, maximum principles, and the non-negative constraint: Recent developments and new results. Mech. Adv. Mater. Struc. 24, 556–590 (2017)
Nordbotten, J.M., Aavatsmark, I., Eigestad, G.T.: Monotonicity of control volume methods. Numer. Math. 106, 255–288 (2007)
Nagarajan, H., Nakshatrala, K.B.: Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids. Int. J. Numer. Meth. Fluids 67, 820–847 (2011)
Nakshatrala, K.B., Nagarajan, H., Shabouei, M.: A numerical methodology for enforcing maximum principles and the non-negative constraint for transient diffusion equations. Commun. Comput. Phys. 19, 53–93 (2016)
Schneider, M., Flemisch, B., Helmig, R., Terekhov, K., Tchelepi, H.: Monotone nonlinear finite-volume method for challenging grids. Comput. Geosci. 22(2), 565–586 (2018)
Schneider, M., Agélas, L., Enchéry, G., Flemisch, B.: Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes. J. Comput. Phys. 351, 80–107 (2017)
Sheng, Z., Yuan, G.: The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes. J. Comput. Phys. 230, 2588–2604 (2011)
Sheng, Z., Yue, J., Yuan, G.: Monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes. SIAM J. Sci. Comput. 31, 2915–2934 (2009)
Su, S., Dong, Q., Wu, J.: A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes. J. Comput. Phys. 372, 773–798 (2018)
Stüben, K.: A review of algebraic multigrid. J. Comput. Appl. Math. 128, 281–309 (2001)
Terekhov, K.M., Mallison, B.T., Tchelepi, H.A.: Cell-centered nonlinear finite volume methods for the heterogeneous anisotropic diffusion problem. J. Comput. Phys. 330, 245–267 (2017)
Wang, S., Hang, X., Yuan, G.: A pyramid scheme for three-dimensional diffusion equations on polyhedral meshes. J. Comput. Phys. 350, 590–606 (2017)
Wang, S., Hang, X., Yuan, G.: A positivity-preserving pyramid scheme for anisotropic diffusion problems on general hexahedral meshes with nonplanar cell faces. J. Comput. Phys. 371, 152–167 (2018)
Wang, S., Yuan, G., Li, Y., Sheng, Z.: Discrete maximum principle based on repair technique for diamond type scheme of diffusion problems. Int. J. Numer. Meth. Fluids 70, 1188–1205 (2012)
Xie, H., Xu, X., Zhai, C.L., Yong, H.: A positivity-preserving finite volume scheme for heat conduction equation on generalized polyhedral meshes. Commun. Comput. Phys. 24, 1375–1408 (2018)
Xie, H., Zhai, C.L., Xu, X., et al.: A monotone finite volume scheme with fixed stencils for 3D heat conduction equation. Commun. Comput. Phys. 26, 1118–1142 (2019)
Yong, H., Song, P., Zhai, C.L., et al.: Numerical simulation of 2-D radiation-drive ignition implosion process. Commun. Theor. Phys. 59, 737–744 (2013)
Yuan, G., Sheng, Z.: Monotone finite volume schemes for diffusion equations on polygonal meshes. J. Comput. Phys. 227, 6288–6312 (2008)
Zhang, W., Kobaisi, M.A.: Cell-centered nonlinear finite-volume methods with improved robustness. SPE J. (2020). https://doi.org/10.2118/195694-PA
Zhang, X., Su, S., Wu, J.: A vertex-centered and positivity-preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids. J. Comput. Phys. 344, 419–436 (2017)
Zhao, F., Lai, X., Yuan, G., Sheng, Z.: A new interpolation for auxiliary unknowns of the monotone finite volume scheme for 3D diffusion equations. Commun. Comput. Phys. 27, 1201–1233 (2020)
Acknowledgements
This work was partially supported by the National Natural Science Foundation of China (Nos. 11901043,11671048,11671302,11771055).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interests
The authors have no conflicts of interest to declare that are relevant to the content of this article.
Data availability statements
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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
Xie, H., Xu, X., Zhai, C. et al. A Positivity-Preserving Finite Volume Scheme with Least Square Interpolation for 3D Anisotropic Diffusion Equation. J Sci Comput 89, 53 (2021). https://doi.org/10.1007/s10915-021-01629-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-021-01629-2