Aliasing and two-dimensional well-balanced for drift-diffusion equations on square grids

Gosse, Laurent

doi:10.1090/mcom/3451

Aliasing and two-dimensional well-balanced for drift-diffusion equations on square grids
HTML articles powered by AMS MathViewer

by Laurent Gosse;

Math. Comp. 89 (2020), 139-168

DOI: https://doi.org/10.1090/mcom/3451

Published electronically: July 16, 2019

HTML | PDF | Request permission

Abstract:

A notion of “2D well-balanced” for drift-diffusion is proposed. Exactness at steady-state, typical in 1D, is weakened by aliasing errors when deriving “truly 2D” numerical fluxes from local Green’s function. A main ingredient for proving that such a property holds is the optimality of the trapezoidal rule for periodic functions. In accordance with practical evidence, a “Bessel scheme” previously introduced in [SIAM J. Numer. Anal. 56 (2018), pp. 2845–2870] is shown to be “2D well-balanced” (along with former algorithms known as “discrete weighted means” or “tailored schemes”. Some $L^2$ stability estimates are established, too.

References

M. Ainsworth and W. Dörfler, Fundamental systems of numerical schemes for linear convection-diffusion equations and their relationship to accuracy, Computing 66 (2001), no. 2, 199–229. Archives for scientific computing. Numerical methods for transport-dominated and related problems (Magdeburg, 1999). MR 1825804, DOI 10.1007/s006070170035
Debora Amadori and Laurent Gosse, Error estimates for well-balanced schemes on simple balance laws, SpringerBriefs in Mathematics, Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2015. One-dimensional position-dependent models; With a foreword by François Bouchut; BCAM SpringerBriefs. MR 3442979, DOI 10.1007/978-3-319-24785-4
M. G. Andrade and J. B. R. do Val, A numerical scheme based on mean value solutions for the Helmholtz equation on triangular grids, Math. Comp. 66 (1997), no. 218, 477–493. MR 1401937, DOI 10.1090/S0025-5718-97-00825-9
V. B. Andreev, On the accuracy of grid approximations of nonsmooth solutions of a singularly perturbed reaction-diffusion equation in the square, Differ. Uravn. 42 (2006), no. 7, 895–906, 1005 (Russian, with Russian summary); English transl., Differ. Equ. 42 (2006), no. 7, 954–966. MR 2294140, DOI 10.1134/S0012266106070044
Akio Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. I [J. Comput. Phys. 1 (1966), no. 1, 119–143], J. Comput. Phys. 135 (1997), no. 2, 101–114. With an introduction by Douglas K. Lilly; Commemoration of the 30th anniversary {of J. Comput. Phys.}. MR 1486265, DOI 10.1006/jcph.1997.5697
Giles Auchmuty, Sharp boundary trace inequalities, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 1, 1–12. MR 3164533, DOI 10.1017/S0308210512000601
Owe Axelsson, Evgeny Glushkov, and Natalya Glushkova, The local Green’s function method in singularly perturbed convection-diffusion problems, Math. Comp. 78 (2009), no. 265, 153–170. MR 2448701, DOI 10.1090/S0025-5718-08-02161-3
O. Axelsson and W. Layton, Defect correction methods for convection-dominated convection-diffusion problems, RAIRO Modél. Math. Anal. Numér. 24 (1990), no. 4, 423–455 (English, with French summary). MR 1070965, DOI 10.1051/m2an/1990240404231
Alan E. Berger, Jay M. Solomon, and Melvyn Ciment, An analysis of a uniformly accurate difference method for a singular perturbation problem, Math. Comp. 37 (1981), no. 155, 79–94. MR 616361, DOI 10.1090/S0025-5718-1981-0616361-0
Roberta Bianchini and Laurent Gosse, A truly two-dimensional discretization of drift-diffusion equations on Cartesian grids, SIAM J. Numer. Anal. 56 (2018), no. 5, 2845–2870. MR 3855393, DOI 10.1137/17M1151353
Garrett Birkhoff and Surender Gulati, Optimal few-point discretizations of linear source problems, SIAM J. Numer. Anal. 11 (1974), 700–728. MR 362933, DOI 10.1137/0711057
Charles L. Epstein, How well does the finite Fourier transform approximate the Fourier transform?, Comm. Pure Appl. Math. 58 (2005), no. 10, 1421–1435. MR 2162785, DOI 10.1002/cpa.20064
E. C. Gartland Jr., Discrete weighted mean approximation of a model convection-diffusion equation, SIAM J. Sci. Statist. Comput. 3 (1982), no. 4, 460–472. MR 677099, DOI 10.1137/0903030
Eugene C. Gartland Jr., Strong stability of compact discrete boundary value problems via exact discretizations, SIAM J. Numer. Anal. 25 (1988), no. 1, 111–123. MR 923929, DOI 10.1137/0725009
Eugene C. Gartland Jr., On the uniform convergence of the Scharfetter-Gummel discretization in one dimension, SIAM J. Numer. Anal. 30 (1993), no. 3, 749–758. MR 1220650, DOI 10.1137/0730037
U. Ghia, K.N. Ghia, C.T. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982) 387–411.
Laurent Gosse, Computing qualitatively correct approximations of balance laws, SIMAI Springer Series, vol. 2, Springer, Milan, 2013. Exponential-fit, well-balanced and asymptotic-preserving. MR 3053000, DOI 10.1007/978-88-470-2892-0
Laurent Gosse, Dirichlet-to-Neumann mappings and finite-differences for anisotropic diffusion, Comput. & Fluids 156 (2017), 58–65. MR 3693203, DOI 10.1016/j.compfluid.2017.06.026
L. Gosse, Viscous equations treated with ${\mathcal L}$-splines and Steklov-Poincaré operator in two dimensions, in Innovative Algorithms & Analysis, DOI: 10.1007/978-3-319-49262-9_6.
Laurent Gosse, ${\mathcal L}$-splines and viscosity limits for well-balanced schemes acting on linear parabolic equations, Acta Appl. Math. 153 (2018), 101–124. MR 3745732, DOI 10.1007/s10440-017-0122-5
J. M. Greenberg and A. Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33 (1996), no. 1, 1–16. MR 1377240, DOI 10.1137/0733001
Houde Han, Zhongyi Huang, and R. Bruce Kellogg, A tailored finite point method for a singular perturbation problem on an unbounded domain, J. Sci. Comput. 36 (2008), no. 2, 243–261. MR 2434846, DOI 10.1007/s10915-008-9187-7
H. Han and R. B. Kellogg, Differentiability properties of solutions of the equation $-\epsilon ^2\Delta u+ru=f(x,y)$ in a square, SIAM J. Math. Anal. 21 (1990), no. 2, 394–408. MR 1038899, DOI 10.1137/0521022
J. P. Hennart, On the numerical analysis of analytical nodal methods, Numer. Methods Partial Differential Equations 4 (1988), no. 3, 233–254. MR 1012482, DOI 10.1002/num.1690040306
William Layton, Optimal difference schemes for $2$-D transport problems, J. Comput. Appl. Math. 45 (1993), no. 3, 337–341. MR 1216076, DOI 10.1016/0377-0427(93)90051-C
Konstantin Lipnikov, Gianmarco Manzini, and Mikhail Shashkov, Mimetic finite difference method. part B, J. Comput. Phys. 257 (2014), no. part B, 1163–1227. MR 3133437, DOI 10.1016/j.jcp.2013.07.031
P. L. Butzer, G. Schmeisser, and R. L. Stens, An introduction to sampling analysis, Nonuniform sampling, Inf. Technol. Transm. Process. Storage, Kluwer/Plenum, New York, 2001, pp. 17–121. MR 1875676
Anita Mayo, Fast high order accurate solution of Laplace’s equation on irregular regions, SIAM J. Sci. Statist. Comput. 6 (1985), no. 1, 144–157. MR 773287, DOI 10.1137/0906012
Yoichi Miyazaki, Sobolev trace theorem and the Dirichlet problem in the unit disk, Milan J. Math. 82 (2014), no. 2, 297–312. MR 3277700, DOI 10.1007/s00032-014-0222-x
Eugene O’Riordan and Martin Stynes, A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions, Math. Comp. 57 (1991), no. 195, 47–62. MR 1079029, DOI 10.1090/S0025-5718-1991-1079029-1
Qazi I. Rahman and Gerhard Schmeisser, Characterization of the speed of convergence of the trapezoidal rule, Numer. Math. 57 (1990), no. 2, 123–138. MR 1048307, DOI 10.1007/BF01386402
Hans-Görg Roos, Martin Stynes, and Lutz Tobiska, Robust numerical methods for singularly perturbed differential equations, 2nd ed., Springer Series in Computational Mathematics, vol. 24, Springer-Verlag, Berlin, 2008. Convection-diffusion-reaction and flow problems. MR 2454024
R. Sacco and M. Stynes, Finite element methods for convection-diffusion problems using exponential splines on triangles, Comput. Math. Appl. 35 (1998), no. 3, 35–45. MR 1605547, DOI 10.1016/S0898-1221(97)00277-0
H. L. Scharfetter, H. K. Gummel, Large signal analysis of a silicon Read diode oscillator, IEEE Trans. Electron Devices 16 (1969) 64–77.
Larry L. Schumaker, Spline functions: basic theory, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2007. MR 2348176, DOI 10.1017/CBO9780511618994
Eitan Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM J. Numer. Anal. 23 (1986), no. 1, 1–10. MR 821902, DOI 10.1137/0723001
João B. R. do Val and Marinho G. Andrade, On mean value solutions for the Helmholtz equation on square grids, Appl. Numer. Math. 41 (2002), no. 4, 459–479. MR 1905218, DOI 10.1016/S0168-9274(01)00127-1
Lloyd N. Trefethen and J. A. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Rev. 56 (2014), no. 3, 385–458. MR 3245858, DOI 10.1137/130932132
Yau Shu Wong and Guangrui Li, Exact finite difference schemes for solving Helmholtz equation at any wavenumber, Int. J. Numer. Anal. Model. Ser. B 2 (2011), no. 1, 91–108. MR 2866996
A. A. Žensykbaev, Best quadrature formula for the class $W^{r}L_{2}$, Anal. Math. 3 (1977), no. 1, 83–93 (English, with Russian summary). MR 447924, DOI 10.1007/BF02333255