Nonlinear finite volume scheme preserving positivity for 2D convection-diffusion equations on polygonal meshes. (English) Zbl 1459.65216
Summary: In this paper, a nonlinear finite volume scheme preserving positivity for solving 2D steady convection-diffusion equation on arbitrary convex polygonal meshes is proposed. First, the nonlinear positivity-preserving finite volume scheme is developed. Then, in order to avoid the computed solution beyond the upper bound, the cell-centered unknowns and auxiliary unknowns on the cell-edge are corrected. We prove that the present scheme can avoid the numerical solution beyond the upper bound. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results show that our scheme preserves the above conclusion and has second-order accuracy for solution.
MSC:
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
References:
| [1] | Edwards, M. G.; Zheng, H., A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support, Journal of Computational Physics, 227, 22, 9333-9364 (2008) · Zbl 1231.76178 · doi:10.1016/j.jcp.2008.05.028 |
| [2] | Sheng, Z.; Yuan, G., The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes, Journal of Computational Physics, 230, 7, 2588-2604 (2011) · Zbl 1218.65120 · doi:10.1016/j.jcp.2010.12.037 |
| [3] | Morton, K., Numerical Solution of Convection-Diffusion Problems (1996), London, UK: Chapman & Hall, London, UK · Zbl 0861.65070 |
| [4] | Pert, G. J., Physical constraints in numerical calculations of diffusion, Journal of Computational Physics, 42, 1, 20-52 (1981) · Zbl 0498.76076 · doi:10.1016/0021-9991(81)90231-x |
| [5] | Nordbotten, J. M.; Aavatsmark, I., Monotonicity conditions for control volume methods on uniform parallelogram grids in homogeneous media, Computational Geosciences, 9, 1, 61-72 (2005) · doi:10.1007/s10596-005-5665-2 |
| [6] | Lapin, A., Mixed hybrid finite element method for a variational inequality with a quasi-linear operator, Computational Methods in Applied Mathematics, 9, 4, 354-367 (2009) · Zbl 1186.65085 · doi:10.2478/cmam-2009-0023 |
| [7] | Friis, H. A.; Edwards, M. G., A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids, Journal of Computational Physics, 230, 1, 205-231 (2011) · Zbl 1427.76161 · doi:10.1016/j.jcp.2010.09.012 |
| [8] | Lipnikov, K.; Manzini, G.; Svyatskiy, D., Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems, Journal of Computational Physics, 230, 7, 2620-2642 (2011) · Zbl 1218.65117 · doi:10.1016/j.jcp.2010.12.039 |
| [9] | Le Potier, C., Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés, Comptes Rendus Mathematique, 341, 12, 787-792 (2005) · Zbl 1081.65086 · doi:10.1016/j.crma.2005.10.010 |
| [10] | Lipnikov, K.; Shashkov, M.; Svyatskiy, D.; Vassilevski, Y., Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes, Journal of Computational Physics, 227, 1, 492-512 (2007) · Zbl 1130.65113 · doi:10.1016/j.jcp.2007.08.008 |
| [11] | Yuan, G.; Sheng, Z., Monotone finite volume schemes for diffusion equations on polygonal meshes, Journal of Computational Physics, 227, 12, 6288-6312 (2008) · Zbl 1147.65069 · doi:10.1016/j.jcp.2008.03.007 |
| [12] | Danilov, A.; Vassilevski, Y., A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes, Russian Journal of Numerical Analysis and Mathematical Modelling, 24, 207-227 (2009) · Zbl 1166.65394 · doi:10.1515/rjnamm.2009.014 |
| [13] | Lipnikov, K.; Svyatskiy, D.; Vassilevski, Y., Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes, Journal of Computational Physics, 228, 3, 703-716 (2009) · Zbl 1158.65083 · doi:10.1016/j.jcp.2008.09.031 |
| [14] | Sheng, Z.; Yue, J.; Yuan, G., Monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes, SIAM Journal on Scientific Computing, 31, 4, 2915-2934 (2009) · Zbl 1195.65115 · doi:10.1137/080721558 |
| [15] | Sheng, Z.; Yuan, G., An improved monotone finite volume scheme for diffusion equation on polygonal meshes, Journal of Computational Physics, 231, 9, 3739-3754 (2012) · Zbl 1242.65219 · doi:10.1016/j.jcp.2012.01.015 |
| [16] | Wang, S.; Yuan, G.; Li, Y.; Sheng, Z., A monotone finite volume scheme for advection-diffusion equations on distorted meshes, International Journal for Numerical Methods in Fluids, 69, 7, 1283-1298 (2012) · Zbl 1253.76069 · doi:10.1002/fld.2640 |
| [17] | Sheng, Z.; Yuan, G., A cell-centered nonlinear finite volume scheme preserving fully positivity for diffusion equation, Journal of Scientific Computing, 68, 2, 521-545 (2016) · Zbl 1371.65114 · doi:10.1007/s10915-015-0148-7 |
| [18] | Sheng, Z.; Yuan, G., A new nonlinear finite volume scheme preserving positivity for diffusion equations, Journal of Computational Physics, 315, 182-193 (2016) · Zbl 1349.65582 · doi:10.1016/j.jcp.2016.03.053 |
| [19] | Lan, B.; Sheng, Z.; Yuan, G., A monotone finite volume scheme with second order accuracy for convection-diffusion equations on deformed meshes, Computer Physics Communication, 24, 1455-1476 (2018) · Zbl 1475.65170 · doi:10.4208/cicp.oa-2017-0127 |
| [20] | Lan, B.; Sheng, Z.; Yuan, G., A new finite volume scheme preserving positivity for radionuclide transport calculations in radioactive waste repository, International Journal of Heat and Mass Transfer, 121, 736-746 (2018) · doi:10.1016/j.ijheatmasstransfer.2018.01.023 |
| [21] | Lan, B.; Sheng, Z.; Yuan, G., A new positive scheme for 2D convection-diffusion equation, ZAMM-Journal of Applied Mathematics and Mechanics, 99, 1-13 (2019) · Zbl 07805127 |
| [22] | Bertolazzi, E.; Manzini, G., A cell-centered second-order accurate finite volume method for convection-diffusion problems on unstructured meshes, Mathematical Models and Methods in Applied Sciences, 14, 8, 1235-1260 (2004) · Zbl 1079.65113 · doi:10.1142/s0218202504003611 |
| [23] | Lipnikov, K.; Svyatskiy, D.; Vassilevski, Y., A monotone finite volume method for advection-diffusion equations on unstructured polygonal meshes, Journal of Computational Physics, 229, 11, 4017-4032 (2010) · Zbl 1192.65140 · doi:10.1016/j.jcp.2010.01.035 |
| [24] | Zhang, Q.; Sheng, Z.; Yuan, G., A finite volume scheme preserving extremum principle for convection-diffusion equations on polygonal meshes, International Journal for Numerical Methods in Fluids, 84, 10, 616-632 (2017) · doi:10.1002/fld.4366 |
| [25] | Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (2001), Berlin, Germany: Springer-Verlag, Berlin, Germany · Zbl 1042.35002 |
| [26] | Sheng, Z.; Yuan, G., A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes, SIAM Journal on Scientific Computing, 30, 3, 1341-1361 (2008) · Zbl 1167.65049 · doi:10.1137/060665853 |
| [27] | van Lee, B., Towards the ultimate conservative difference scheme. V.A second-order sequel to Godunov’s method, Journal of Computational Physics, 32, 101-136 (1979) · Zbl 1364.65223 · doi:10.1016/0021-9991(79)90145-1 |
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