A truly two-dimensional discretization of drift-diffusion equations on Cartesian grids. (English) Zbl 1403.65037
Summary: A genuinely two-dimensional discretization of general drift-diffusion (including incompressible Navier-Stokes) equations is proposed. Its numerical fluxes are derived by computing the radial derivatives of “bubbles” which are deduced from available discrete data by exploiting the stationary Dirichlet-Green function of the convection-diffusion operator. These fluxes are reminiscent of Scharfetter and Gummel’s in the sense that they contain modified Bessel functions which allow one to pass smoothly from diffusive to drift-dominating regimes. For certain flows, monotonicity properties are established in the vanishing viscosity limit (“asymptotic monotony”) along with second-order accuracy when the grid is refined. Practical benchmarks are displayed to assess the feasibility of the scheme, including the “western currents” with a Navier-Stokes-Coriolis model of ocean circulation.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N80 | Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76U05 | General theory of rotating fluids |
| 86A05 | Hydrology, hydrography, oceanography |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
References:
| [1] | M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, John Wiley, New York, 1972. · Zbl 0543.33001 |
| [2] | M. Ainsworth and W. Dorfler, Fundamental systems of numerical schemes for linear convection-diffusion equations and their relationship to accuracy, Computing, 66 (2001), pp. 199–229. · Zbl 0988.65096 |
| [3] | M. Al-Jaboori and D. Wirosoetisno, Navier-Stokes equations on the \(β\)-plane, Discrete Contin. Dynam. Syst. Ser. B, 16 (2011), pp. 687–701, . · Zbl 1222.35029 |
| [4] | O. Axelson, E. Glishkov, and N. Glishkova, The local Green’s function method in singularly perturbed convection-diffusion problem, Math. Comp., 78 (2009), pp. 153–170. · Zbl 1198.65058 |
| [5] | I. Babuška and J.E. Osborn, Generalized finite element methods: Their performance and their relation to mixed methods, SIAM J. Numer. Anal., 20 (1983), pp. 510–536, . · Zbl 0528.65046 |
| [6] | F. Bouchut, Y. Jobic, R. Natalini, R. Occelli, and V. Pavan, Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations, J. Comput. Math., 4 (2018), pp. 1–56, . · Zbl 1416.76243 |
| [7] | F. Brezzi and A. Russo, Choosing bubbles for advection-diffusion problems, Math. Models Methods Appl. Sci., 4 (1994), pp. 571–587. · Zbl 0819.65128 |
| [8] | F. Brezzi, L.D. Marini, and P. Pietra, Two-dimensional exponential fitting and applications to drift-diffusion models, SIAM J. Numer. Anal., 26 (1989), pp. 1342–1355, . · Zbl 0686.65088 |
| [9] | K. Bryan, A numerical investigation of a nonlinear model of a wind-driven ocean, J. Atmospheric Sci., 20 (1963), pp. 594–606. |
| [10] | A.-L. Dalibard and L. Saint-Raymond, Mathematical study of degenerate boundary layers: A large scale ocean circulation problem, Mem. Amer. Math. Soc., 253 (2018), no. 1206. · Zbl 1423.35099 |
| [11] | D.R. Duffy, Green Functions and Applications, Chapman & Hall/CRC Press, Boca Raton, FL, 2001. · Zbl 0983.35003 |
| [12] | M. Endoh, A numerical experiment on the variations of western boundary currents. Part I, J. Oceanog. Soc. Japan, 29 (1973), pp. 16–27. |
| [13] | G.J. Fix, Finite element models for ocean circulation problems, SIAM J. Appl. Math., 29 (1975), pp. 371–387, . · Zbl 0329.76092 |
| [14] | S. Franz and N. Kopteva, Green’s function estimates for a singularly perturbed convection-diffusion problem, J. Differential Equations, 252 (2012), pp. 1521–1545. · Zbl 1235.35087 |
| [15] | E.C. Gartland, Jr., Discrete weighted mean approximation of a model convection-diffusion equation, SIAM J. Sci. Statist. Comput., 3 (1982), pp. 460–472, . · Zbl 0496.76092 |
| [16] | U. Ghia, K.N. Ghia, and C.T. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48 (1982), pp. 387–411. · Zbl 0511.76031 |
| [17] | L. Gosse, A two-dimensional version of the Godunov scheme for scalar balance laws, SIAM J. Numer. Anal., 52 (2014), pp. 626–652, . · Zbl 1295.65084 |
| [18] | L. Gosse, Viscous equations treated with \(\mathcal{L}\)-splines and Steklov-Poincaré operator in two dimensions, in Innovative Algorithms Analysis, Springer INdAM Series 16, Springer, Cham, 2017, pp. 167–195 . · Zbl 1367.65128 |
| [19] | H. Han, Z. Huang, and R.B. Kellogg, A tailored finite point method for a singular perturbation problem on an unbounded domain, J. Sci. Comput., 36 (2008), pp. 243–261. · Zbl 1203.65221 |
| [20] | T.Y. Hou and B.T.R. Wetton, Second-order convergence of a projection scheme for the incompressible Navier–Stokes with boundaries, SIAM J. Numer. Anal., 30 (1993), pp. 609–629, . · Zbl 0776.76055 |
| [21] | A.M. Il’in, A difference scheme for a differential equation with a small parameter affecting the highest derivative, Mat. Zametki, 6 (1969), pp. 237–248 (in Russian). · Zbl 0185.42203 |
| [22] | L.R.M. Maas, A closed form Green function describing diffusion in a strained flow field, SIAM J. Appl. Math., 49 (1989), pp. 1359–1373, . · Zbl 0679.76098 |
| [23] | M. Stynes and E. O’Riordan, A uniformly accurate finite element method for a singular perturbation problem in conservative form, SIAM J. Numer. Anal., 23 (1986), pp. 369–375, . · Zbl 0595.65091 |
| [24] | E. O’Riordan and M. Stynes, A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions, Math. Comp., 57 (1991), pp. 47–62. · Zbl 0733.65063 |
| [25] | H.-G. Roos, Ten ways to generate the Il’in and related schemes, J. Comput. Appl. Math., 53 (1993), pp. 43–59. · Zbl 0817.65063 |
| [26] | P.L. Roe and D. Sidilkover, Optimum positive linear schemes for advection in two and three dimensions, SIAM J. Numer. Anal., 29 (1992), pp. 1542–1568, . · Zbl 0765.65093 |
| [27] | H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems, Springer Ser. Comput. Math. 24, Springer-Verlag, Berlin, 2008. · Zbl 1155.65087 |
| [28] | L. St.-Raymond, The role of boundary layers in large-scale ocean circulation, in Mathematical Models and Methods for Planet Earth, Springer INdAM Series 6, Springer, Cham, 2014, pp. 11–24. · Zbl 1425.86008 |
| [29] | R. Sacco and M. Stynes, Finite element methods for convection-diffusion problems using exponential splines on triangles, Comput. Math. Appl., 35 (1998), pp. 35–45. · Zbl 0907.65110 |
| [30] | A.A. Samarskii, The Theory of Difference Schemes, Chapman & Hall/CRC Press, Boca Raton, FL, 2001. · Zbl 0971.65076 |
| [31] | H.L. Scharfetter and H.K. Gummel, Large signal analysis of a silicon Read diode oscillator, IEEE Trans. Electron Devices, 16 (1969), pp. 64–77. |
| [32] | L.L. Schumaker, Spline Functions: Basic Theory, 3rd ed., Cambridge University Press, Cambridge, UK, 2007. · Zbl 1123.41008 |
| [33] | Y. Shih, J.-Y. Cheng, and K.-T. Chen, An exponential-fitting finite element method for convection-diffusion problems, Appl. Math. Comput., 217 (2011), pp. 5798–5809. · Zbl 1211.65157 |
| [34] | J.B.R. do Val and M.G. Andrade, On mean value solutions for the Helmholtz equation on square grids, Appl. Numer. Math., 41 (2002), pp. 459–479. · Zbl 1005.65117 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
