A time adaptive scheme for the solution of the advection equation in the presence of a transient flow velocity

CMES - Computer Modeling in Engineering and Sciences

Volumn 12, Issue 1, 2006, Pages 41-53

  Selvadurai, A P S ^a   Dong, Wenjun ^a  

^a MCGILL UNIVERSITY   (Canada)

Author keywords

Advection equation; Courant number; Fourier analysis; Modified Least Squares method; Time adaptive scheme

Indexed keywords

ADAPTIVE SYSTEMS; COMPUTATIONAL METHODS; FOURIER TRANSFORMS; MATHEMATICAL TECHNIQUES; NUMERICAL ANALYSIS; VELOCITY MEASUREMENT;

COURANT NUMBERS; MODIFIED LEAST SQUARES (MLS); TIME MARCHING SCHEME; TRANSIENT HYDRAULIC BOUNDARY CONDITIONS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 33646883025     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (38)

1. Atluri, S.N. (2004): The Meshless Local-Petrov-Galerkin Method for Domain & BIE Discretizations, Tech Science Press, Encino, CA.
2. Atluri S.N.; Han Z.D.; Rajendran A.M. (2004): A new implementation of the meshless finite volume method, through the MLPG "Mixed" approach. CMES: Computer Modeling in Engineering & Sciences, vol.6, pp. 491-513.
3. Atluri, S.N.; Zhu, T. (1998): A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech, vol.22, pp. 117-127.
4. Brezzi, F.; Bristeau, M.O.; Franca, L.P.; Mallet, M.; Rogé, G. (1992): A relationship between stabilized finite element methods and the Galerkin method with bubbles for advection-diffusion problems. Comput Methods Appl Mech Engrg, vol.96, pp. 117-129.
5. Brezzi F.; Franca L. P.; Hughes T. J. R.; Russo A. (1997): 'b =fg'. Comput Methods Appl Mech Engrg, vol. 145, pp. 329-339.
6. Cardle J.A. (1995): A modification of the PetrovGalerkin method for the transient convection-diffusion equation. Int J Numer Meth Engng, vol.38, pp. 171-181.
7. Carey, G.F.; Jiang, B.N. (1988): Least squares finite elements for first-order hyperbolic systems. Int J Numer Meth Engng, vol. 26, pp. 81-93.
8. Christie, I.; Griffiths, D.; Mitchell, A.; Zienkiewicz, O.C. (1976): Finite element methods for second-order differential equations with significant first derivatives. Int J Numer Meth Engng. vol. 10. pp. 1389-1396.
9. Codina, R. (1993): Stability analysis of the forward Euler scheme for the convection-diffusion equation using the SUPG formulation in space. Int J Numer Meth Engng. vol.36, pp. 1445-1464.
10. Codina, R. (1998): Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comput Methods Appl Mech Engrg. vol.156, pp. 185-210.
11. Donea, J.; Giulian, S.; Laval, H.; Quartapelle, L. (1984): Time-accurate solution of advection-diffusion problem by finite elements. Comput Methods Appl Mech Engrg, vol.45, pp. 123-145.
12. Franca L.P.; Farhat C. (1995): Bubbles functions prompt unusual stabilized finite element methods. Coinput Methods Appl Mech Engrg, vol. 123, vol. 299-308.
13. Franca, L. P.; Hwang, F. (2002): Refining the submesh strategy in the two-level finite element method: application to the advection-diffusion equation. Int J Numer Meth Fluids, vol. 39, pp. 161-87.
14. Freeze, R.A.; Cherry, J. A. (1979): Groundwater, Prentice-Hall, Englewood Cliffs, N.J.
15. Ganzha, V.G.; Vorozhtsov, E.V. (1998): Computer Aided Analysis of Difference Schemes for Partial Differential Equations. CRC Press, Boca Raton.
16. Hauke, G.; Doweidar, M.H. (2005): Fourier analysis of semi-discrete and space-time stabilized methods for the advective-diffusive-reactive equation: II SGS. Comput Methods Appl Mech Engrg, vol. 194, pp. 691 -725.
17. Hughes, T.J.R. (1995): Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles, and the origins of stabilized methods. Comput Methods Appl Mech Engrg, vol. 127. pp. 387-401.
18. Hughes, T.J.R.; Brooks, A. (1982): A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: Application to the StreamlineUpwind Procedure. In: R.H. Gallagher et al (ed.) Finite Elements in Fluids, Wiley, Chichester, vol. 4, pp. 47-65.
19. Jiang, B.N. (1998): The Least-Squares Finite Element Method: Theory and Application in Computational Fluid Dynamics and Electromagnetics, Springer-Verlag, Berlin Heidelberg.
20. LeVeque, R.J. (1992): Numerical Method for Conservation Laws, Birkhäuser Verlag Basel-Boston Berlin.
21. Lin H.; Atluri S.N. (2000): Meshless Local PetrovGalerkin (MLPG) method for convection-diffusion problems. CMES: Computer Modeling in Engineering & Sciences, vol.1, pp. 45-60.
22. Morton, K.W. (1980): Stability of finite difference approximations to a diffusion-convection equation. Int J Numer Meth Engng, vol. 15, pp. 677-683.
23. Morton, K.W. (1996): Numerical Solution of the Convection-Diffusion Problem, Chapman & Hall, London.
24. Onate, E.; Garcia, J.; Idelsohn, S. (1997): Computation of the stabilization parameter for the finite element solution of advective-diffusive problems. Int J Numer Meth Fluids, vol. 25, pp. 1385-1407.
25. Pereira, J.M.C. ; Pereira, J.C.F. (2001): Fourier analysis of several finite difference schemes for the one-dimensional unsteady convection-diffusion equation. Int J Numer Meth Fluids, vol. 36, pp. 417-439.
26. Quarteroni, A.; Valli, A. (1997): Numerical Approximation of Partial Differential Equations. 2nd ed. Springer Series in Computational Mathematics, Springer-Verlag, Berlin.
27. Raymond, W.H.; Garder, A. (1976): Selective damping in a Galerkin method for solving wave problems with variable grids. Monthly Weather Rev, vol. 104, pp. 1583-1590.
28. Richtmeyer, R.D.; Morton, K.W. (1967): Difference Methods for Initial Value Problem, Wiley, New York.
29. Selvadurai, A.P.S. (2000): Partial Differential Equations in Mechanics, Vol. 1, Fundamentals, Laplace's Equation, Diffusion Equation, Wave Equation, Springer-Verlag, Berlin.
30. Selvadurai, A.P.S. (2002a): The advective transport of a chemical from a cavity in a porous medium. Computers and Geotechnics, vol. 29, pp. 525-546.
31. Selvadurai, A.P.S. (2002b): Some remarks on the elastic drive equation, in: L.Vulliet, L. Laloui and B.A. Schrefler (eds). Environmental Geomechanics, Proceedings of the International Workshop on Environmental Geomechanics. Monte Verita, Ascona. EPFL Press, Lausanne, pp. 253-258.
32. Selvadurai, A.P.S. (2003): Contaminant migration from an axisymmetric source in a porous medium. Water Resources Res, vol. 39:1203. DOI: 10.1029/2002 WR001442.
33. Shakib, F.; Hughes, T.J.R. (1991): A new finite element formulation for computational fluid dynamics: IX. Fourier Analysis of space-time Galerkin/least-squares algorithms. Comput Methods Appl Mech Engrg, vol. 87, pp. 35-58.
34. Tezduyar, T.E.; Ganjoo, D.K. (1986): Petrov-Galerkin formulation with weighting functions dependent upon spatial and temporal discretization: applications to transient convection-diffusion problems. Comput Methods Appl Mech Engrg, vol. 59, pp. 49-71.
35. Vichnevetsky, R.; Bowles, J.B. (1982): Fourier Analysis of Numerical Approximations of Hyperbolic Equations, SIAM Philadelphia.
36. Wang, Y; Hutter, K. (2001): Comparisons of numerical methods with respect to convectively dominated problems. Int J Numer Meth Fluids, vol. 37, pp.721-745.
37. Wendland, E.; Schmid, G.A. (2000): Symmetrical Streamline Stabilization scheme for high advective transport, Int J Numer Anal Meth Geomech, vol. 24, pp. 29-45.
38. Yu, C.C.; Heinrich, J.C. (1986): Petrov-Galerkin methods for the time-dependent convective transport equation. Int J Numer Meth Engng, vol. 23, pp. 883-901.