On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations

Journal of Computational Physics

Volumn 231, Issue 4, 2012, Pages 1570-1586

  John, Volker ^a,b   Novo, Julia ^c  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

^b FREIE UNIVERSITÄT BERLIN   (Germany)

^c UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

Author keywords

ENO schemes; FEM FCT schemes; Time dependent convection diffusion reaction equations; Under and overshoots; WENO schemes

Indexed keywords

DIFFUSION IN LIQUIDS; HEAT CONVECTION; RUNGE KUTTA METHODS; TENSORS;

CONVECTION-DIFFUSION EQUATIONS; CONVECTION-DIFFUSION-REACTION EQUATIONS; DISCRETIZATIONS; ENO SCHEME; ESSENTIALLY NON-OSCILLATORY; FEM-FCT SCHEME; FINITE-DIFFERENCE DISCRETIZATION; TIME DEPENDENT CONVECTION-DIFFUSION-REACTION EQUATIONS; UNDER- AND OVERSHOOT; WENO SCHEMES;

FINITE ELEMENT METHOD;

EID: 84855192222     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2011.10.025     Document Type: Article

References (35)

1. Anderson D.G. Iterative procedures for nonlinear integral equations. J. Assoc. Comput. Mach. 1965, 12:547-560.
2. Bauer R.B. A hybrid adaptive ENO scheme. J. Comput. Phys. 1997, 136:180-196.
3. R. Bordás, V. John, E. Schmeyer, D. Thévenin, Measurement and simulation of a droplet population in a turbulent flow field. Preprint 1590, WIAS, 2011.
4. Bueno-Orovio A., Pérez-García V.M., Fenton F.H. Spectral methods for partial differential equations in irregular domains: the spectral smoothed boundary method. SIAM J. Sci. Comput. 2006, 28:886-900.
5. Bürger R., Kozakevicius A. Adaptive multiresolution WENO schemes for multi-species kinematic flow models. J. Comput. Phys. 2007, 224:1190-1222.
6. Burman E., Fernández M.A. Finite element methods with symmetric stabilization for the transient convection-diffusion-reaction equation. Comput. Methods Appl. Mech. Eng. 2009, 198:2508-2519.
7. Davis T.A. Algorithm 832: UMFPACK V4.3 - an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 2004, 30:196-199.
8. Fletcher C.A.J. The group finite element formulation. Int. J. Numer. Methods Fluids 1983, 37:225-243.
9. Gresho P.M., Griffiths D.F., Silvester D.J. Adaptive time-stepping for incompressible flow part I: Scalar advection-diffusion. SIAM J. Sci. Comput. 2008, 30:2018-2054.
10. Gustafsson K., Lundh M., Söderlind G. A PI stepsize control for the numerical solution of ordinary differential equations. BIT 1988, 28:270-287.
11. Harten A., Enquist B., Osher S., Chakravarthy S. Uniformly high order essentially non-oscillatory schemes III. J. Comput. Phys. 1987, 71:231-303.
12. Jiang G.-S., Shu C.-W. Efficient implementation of weighted ENO schemes. J. Comput. Phys. 1996, 126:202-228.
13. John V., Matthies G. MooNMD - a program package based on mapped finite element methods. Comput. Visual. Sci. 2004, 6:163-170.
14. John V., Novo J. Error analysis of the supg finite element discretization of evolutionary convection-diffusion-reaction equations. SIAM J. Numer. Anal. 2011, 49:1149-1176.
15. John V., Roland M., Mitkova T., Sundmacher K., Tobiska L., Voigt A. Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Eng. Sci. 2009, 64:733-741.
16. John V., Schmeyer E. Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion. Comput. Methods Appl. Mech. Eng. 2008, 198:475-494.
17. John V., Schmeyer E. On finite element methods for 3d time-dependent convection-diffusion-reaction equations with small diffusion. Lecture Notes in Computational Science and Engineering 2009, vol. 69:173-181. Springer. A. Hegarty (Ed.).
18. Kozakevicius A.J., Santos L.C.C. ENO adaptive method for solving one-dimensional conservation laws. Appl. Numer. Math. 2009, 59:2337-2355.
19. Kuzmin D. Explicit and implicit FEM-FCT algorithms with flux linearization. J. Comput. Phys. 2009, 228:2517-2534.
20. Kuzmin D., Möller M. Algebraic flux correction I. Scalar conservation laws. Flux-Corrected Transport: Principles, Algorithms and Applications 2005, 155-206. Springer. R. Löhner, D. Kuzmin, S. Turek (Eds.).
21. Kuzmin D., Turek S. Flux correction tools for finite elements. J. Comput. Phys. 2002, 175:525-558.
22. LeVeque R.J. High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 1996, 33:627-665.
23. Liu X.-D., Osher S., Chan T. Weighted essentially non-oscillatory schemes. J. Comput. Phys. 1994, 115:200-212.
24. Löhner R., Morgan K., Peraire J., Vahdati M. Finite element flux-correct transport (FEM-FCT) for the Euler and the Navier-Stokes equations. Int. J. Numer. Methods Fluids 1987, 7:1093-1109.
25. Luo H., Baum J.D., Löhner R. A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys. 2007, 225. 686-713.
26. M. Möller. Adaptive high-resolution finite element schemes, Ph.D thesis, Technical University, Dortmund, 2008.
27. Roos H.-G., Stynes M., Tobiska L. Robust numerical methods for singularly perturbed differential equations. Springer Series in Computational Mathematics 2008, vol. 24. Springer. 2nd ed.
28. Saad Y. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 1993, 14(2):461-469.
29. Shu C.-W. High order weighted essentially nonoscillatoy schemes for convection dominated problems. SIAM Rev. 2009, 51:82-126.
30. Shu C.-W., Osher S. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 1988, 77:439-471.
31. Turek S. Efficient solvers for incompressible flow problems: an algorithmic and computational approach. Lecture Notes in Computational Science and Engineering 1999, vol. 6. Springer.
32. Valli A.M.P., Carey G.F., Coutinho A.L.G.A. Control strategies for timestep selection in finite element simulation of incompressible flows and coupled reaction-convection-diffusion processes. Int. J. Numer. Methods Fluids 2005, 47:201-231.
33. Walker H.F., Ni P. Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 2011, 49:1715-1735.
34. Zalesak S.T. Fully multi-dimensional flux corrected transport algorithms for fluid flow. J. Comput. Phys. 1979, 31:335-362.
35. Zienkiewicz O.C., Zhu J.Z. A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 1987, 24(2):337-357.