Top

Journal of Scientific Computing

Published in:

25-07-2016

Uniform Convergent Tailored Finite Point Method for Advection–Diffusion Equation with Discontinuous, Anisotropic and Vanishing Diffusivity

Authors: Min Tang, Yihong Wang

Published in: Journal of Scientific Computing | Issue 1/2017

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We propose two tailored finite point methods for the advection–diffusion equation with anisotropic tensor diffusivity. The diffusion coefficient can be very small in one direction in some part of the domain and be discontinuous at the interfaces. When flows advect from the vanishing-diffusivity region towards the non-vanishing diffusivity region, standard numerical schemes tend to cause spurious oscillations or negative values. Our proposed schemes have uniform convergence in the vanishing diffusivity limit, even when the solution exhibits interface and boundary layers. When the diffusivity is along the coordinates, the positivity and maximum principle can be proved. We use the value as well as their derivatives at the grid points to construct the scheme for nonaligned case, which makes it can achieve good accuracy and convergence for the derivatives as well, even when there exhibit boundary or interface layers. Numerical experiments are presented to show the performance of the proposed scheme.

previous article Large-Scale Optimization-Based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies

next article On Second Order Semi-implicit Fourier Spectral Methods for 2D Cahn–Hilliard Equations

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)MathSciNetCrossRefMATH

Ashby, S.F., Bosl, W.J., Falgout, R.D., Smith, S.G., Tompson, A.F., Williams, T.J.: A numerical simulation of groundwater flow and contaminant transport on the CRAY T3D and C90 supercomputers. Int. J. High Perform. Comput. Appl. 13(1), 80–93 (1999)CrossRef

Ern, A., Stephansen, A.F., Zunino, P.: A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity. IMA J. Numer. Anal. 29, 235–256 (2009)MathSciNetCrossRefMATH

Han, H., Huang, Z., Kellogg, B.: A tailored finite point method for a singular perturbation problem on an unbounded domain. J. Sci. Comp. 36, 243–261 (2008)MathSciNetCrossRefMATH

Han, H., Huang, Z.Y.: Tailored finite point method for a singular perturbation problem with variable coefficients in two dimensions. J. Sci. Comput. 49, 200–220 (2009)MathSciNetCrossRefMATH

Han, H., Huang, Z.Y.: Tailored finite point method for steady-state reaction–diffusion equation. Commun. Math. Sci. 8, 887–899 (2010)MathSciNetCrossRefMATH

Han, H., Huang, Z.Y.: Tailored finite point method based on exponential bases for convection–diffusion–reaction equation. Math. Comput. 82, 213–226 (2013)MathSciNetCrossRefMATH

Han, H., Huang, Z.Y., Ying, W.J.: A semi-discrete tailored finite point method for a class of anisotropic diffusion problems. Comput. Math. Appl. 65, 1760–1774 (2013)MathSciNetCrossRefMATH

Hyman, J., Morel, J., Shashkov, M., Steinberg, S.: Mimetic finite difference methods for diffusion equations. Comput. Geosci. 6, 333–352 (2002)MathSciNetCrossRefMATH

10.

Herbin,R.,Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Eymard, R., Herard, J.-M. (eds.) Finite Volumes for Complex Applications V, pp. 659–692. John Wiley & Sons (2008)

11.

Kellogg, R.B., Houde, Han: Numerical analysis of singular perturbation problems. Hemisphere Publishing Corp, USA. Distributed outside North America by Springer-Verlag, pp. 323–335 (1983)

12.

Lipnikov, K., Shashkov, M., Svyatskiy, D., Vassilevski, Yu.: Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes regimes. J. Comput. Phys. 227, 492–512 (2007)MathSciNetCrossRefMATH

13.

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)MathSciNetCrossRefMATH

14.

Motzkin, T.S., Wasow, W.: On the approximation of linear elliptic differential equations by difference equations with positive coefficients. J. Math. Phys. 31, 253–259 (1953)MathSciNetCrossRefMATH

15.

Leveque, R.J., Li, Z.L.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044 (1994)MathSciNetCrossRefMATH

16.

Oberman, A.M.: Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44(2), 879–895 (2006)MathSciNetCrossRefMATH

17.

Pasdunkorale, J., Turner, I.W.: A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media. J. Comput. Math. 23, 1–16 (2005)MathSciNetMATH

18.

Sheng, Z.Q., Yue, J.Y., Yuan, G.W.: Monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes. SIAM J. Sci. Comput. 31(4), 2915–2934 (2009)MathSciNetCrossRefMATH

19.

Treguier, A.M.: Modelisation numerique pour loceanographie physique. Ann. Math. Blaise Pascal 9(2), 345–361 (2002)MathSciNetCrossRef

20.

van Esa, B., Koren, B., de Blank, H.J.: Finite-difference schemes for anisotropic diffusion. J. Comput. Phys. 272, 526–549 (2014)MathSciNetCrossRefMATH

21.

Wu, J.M., Gao, Z.M.: Interpolation-based second-order monotone finite volume schemes for anisotropic diffusion equations on general grids. J. Comput. Phys. 274, 569–588 (2014)MathSciNetCrossRefMATH

22.

Younes, A., Fontaine, V.: Efficiency of mixed hybrid finite element and multipoint flux approximation methods on quadrangular grids and highly anisotropic media. Int. J. Numer. Methods Eng. 76, 314–336 (2008)MathSciNetCrossRefMATH

23.

Yuan, G.W., Sheng, Z.Q.: Monotone finite volume schemes for diffusion equations on polygonal meshes. J. Comput. Phys. 227(12), 6288–6312 (2008)MathSciNetCrossRefMATH

Title: Uniform Convergent Tailored Finite Point Method for Advection–Diffusion Equation with Discontinuous, Anisotropic and Vanishing Diffusivity
Authors: Min Tang
Yihong Wang
Publication date: 25-07-2016
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2017
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-016-0254-1

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1/2017

Suboptimal Feedback Control of PDEs by Solving HJB Equations on Adaptive Sparse Grids

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

On Second Order Semi-implicit Fourier Spectral Methods for 2D Cahn–Hilliard Equations

High-Order Accurate Local Schemes for Fractional Differential Equations

Accuracy of Finite Element Methods for Boundary-Value Problems of Steady-State Fractional Diffusion Equations

A Galerkin Finite Element Method for a Class of Time–Space Fractional Differential Equation with Nonsmooth Data

Premium Partner