A numerical solution of 2D Buckley-Leverett equation via gradient reproducing kernel particle method

CMES - Computer Modeling in Engineering and Sciences

Volumn 32, Issue 1, 2008, Pages 17-33

  Shodja, Hossein M ^a   Hashemian, Alireza ^a  

^a SHARIF UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Buckley Leverett equation; Gradient reproducing kernel particle method; Meshless; Nonlinear evolutionary partial differential equation; Steep gradient; Two phase flow

Indexed keywords

BOUNDARY VALUE PROBLEMS; DIFFERENTIAL EQUATIONS; FLUID DYNAMICS; FLUID STRUCTURE INTERACTION; FUNCTION EVALUATION; HYDRODYNAMICS; MULTIPHASE FLOW; ORDINARY DIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROBABILITY DENSITY FUNCTION; TWO PHASE FLOW; VISCOSITY;

BUCKLEY-LEVERETT EQUATION; GRADIENT REPRODUCING KERNEL PARTICLE METHOD; MESHLESS; NONLINEAR EVOLUTIONARY PARTIAL DIFFERENTIAL EQUATION; STEEP GRADIENT;

NONLINEAR EQUATIONS;

EID: 56849117588     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (40)

1. Atluri, S.N.; Cho, J.Y.; Kim, H.G. (1999): Analysis of thin beams, using the meshless local Petrov-Galerkin method, with generalized moving least squares interpolations. Comput Mech, vol. 24, pp. 334-347.
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.; Liu, H.T.; Han, Z.D. (2006a): Meshless local Petrov-Galerkin (MLPG) mixed collocation method for elasticity problems. CMES: Computer Modeling in Engineering & Sciences, vol. 14, pp. 141-152.
4. Atluri, S.N.; Liu, H.T.; Han, Z.D. (2006b): Meshless local Petrov-Galerkin (MLPG) mixed finite difference method for solid mechanics. CMES: Computer Modeling in Engineering & Sciences, vol. 15, pp. 1-16.
5. Atluri, S.N.; Shen, S. (2002a): The Meshless Local Petrov-Galerkin (MLPG) Method, Tech Science Press, New York.
6. Atluri, S.N.; Shen, S. (2002b): The meshless local Petrov-Galerkin (MLPG) method: A simple & less costly alternative to the finite element and boundary element method. CMES: Computer Modeling in Engineering & Sciences, vol. 3, pp. 11-52.
7. Atluri, S.N.; Shen, S. (2005): Simulation of a 4th order ODE: illustration of various primal & mixed MLPG methods. CMES: Computer Modeling in Engineering & Sciences, vol. 7, pp. 241-268.
8. Atluri, S.N.; Zhu, T. (1998): A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech, vol. 22, pp. 117-127.
9. Aziz, k.; Settari, A. (1979): Petroleum Reservoir Simulation, Applied Science Publishers, London.
10. Babuška, I.; Melenk, J. (1997): The partition of unity method. Int J Numer Methods Eng, vol. 40, pp. 727-758.
11. Belytschko, T.; Lu, Y. Y.; Gu, L. (1994a): Element-free Galerkin methods. Int J Numer Methods Eng, vol. 37, pp. 229-256.
12. Belytschko, T.; Lu, Y. Y.; Gu, L. (1994b): Fracture and crack growth by element-free Galerkin method. Model Simul Mater Sci Eng, vol. 2, pp. 519-534.
13. Berger, M.; Oliger, J. (1984): Adaptive mesh refinement for hyperbolic partial differential equations. J Comput Physics, vol. 53, pp. 484-512.
14. Buckley, J. M.; Leverett, M. C. (1942): Mechanism of fluid displacement in sands. Trans AIME, vol. 146, pp. 107-116.
15. Chen, J. S.; Pan, C.; Wu, C. T.; Liu, W. K. (1996): Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng, vol. 139, pp. 195-227.
16. Cho, J. Y.; Atluri, S. N. (2001): Analysis of shear flexible beams, using the meshless local Petrov-Galerkin method, based on a locking-free formulation. Eng Comput, vol. 18, pp. 215-240.
17. Durate, C. A.; Oden, J. T. (1996): An h-p adaptive method using clouds. Comput Methods Appl Mech Eng, vol. 139, pp. 237-262.
18. Gear, C. W. (1971): Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs.
19. Gelinas, R. J.; Doss, S. K.; Miller, K. (1981): The moving finite element method: applications to general partial differential equations with multiple large gradients. J Comput Physics, Vol. 40, pp. 202-249.
20. Hashemian, A. (2000): A Study of Columns' Buckling Loads and Modes by Meshfree Methods, Master's Thesis, Sharif University of Technology, Tehran.
21. Hashemian, A.; Shodja, H. M. (2008a): Gradient reproducing kernel particle method. J Mech Mater Sturct, vol. 3, pp. 127-152.
22. Hashemian, A.; Shodja, H. M. (2008b): A meshless approach for solution of Burgers' equation. J Comput Appl Math, vol. 220, pp. 226-239.
23. Johnson, I. W.; Wathen, A. J.; Baines, M. J. (1988): Moving finite element methods for evolutionary problems, II, applications. J Comput Physics, vol. 79, pp. 270-297.
24. Lambert, J. D. (1972): Computational Methods in Ordinary Differential Equations, John Wiley & Sons, New York.
25. Li, S.; Liu, W. K. (2002): Meshfree and particle methods and their applications. Appl Mech Rev, vol. 55, pp. 1-34.
26. Li, H.; NG, T. Y.; Cheng, J. Q.; Lam, K. Y. (2003): Hermite-Cloud: a novel true meshless method. Comput Mech, vol. 33, pp. 30-41.
27. Liu, W. K.; Jun, S.; Zhang, Y. F. (1995): Reproducing kernel particle methods. Int J Numer Methods Fluids, vol. 20, pp. 1081-1106.
28. Liu, W. K.; Chen, Y. J.; Uras, R. A.; Chang C. T. (1996): Generalized multiple scale reproducing kernel particle methods. Comput Methods Appl Mech Eng, vol. 139, pp. 91-158.
29. Long, S.; Atluri, S. N. (2002): A meshless local Petrov-Galerkin method for solving the bending problem of a thin plate. CMES: Computer Modeling in Engineering & Sciences, vol. 3, pp. 53-63.
30. Lucy, L. B. (1977): A numerical approach to the testing of the fission hypothesis. Astron J, vol. 82, pp. 1013-1024.
31. Nayroles, B.; Touzot, G.; Villon, P. (1992): Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech, vol. 10, pp. 307-318.
32. Parker, J. C. (1989): Multiphase flow and transport in porous media. Rev Geophysics, vol. 27, pp. 311-328.
33. Raju, I. S.; Phillips , D. R. (2003): Further developments in the MLPG method for beam problems. CMES: Computer Modeling in Engineering & Sciences, vol. 4, pp. 141-159.
34. Rivière, B.; Bastian, P. (2004): Discontinuous Galerkin Methods for Two-Phase Flow in Porous Media, Technical Report 2004-28, University of Heidelberg.
35. Shodja, H. M.; Hashemian, A. (2007a): A remedy to gradient type constraint dilemma encountered in RKPM. Adv Eng Softw, vol. 38, pp. 229-243.
36. Shodja, H. M.; Hashemian, A. (2007b): 2-D GRKPM: analysis of the Kirchhoff plate. In: A. Ferreira, G. Fasshauer, E. Kansa, V. M. A. Leitão (eds) Second ECCOMAS thematic conference on meshless methods. Faculty of Engineering of the University of Porto, Porto, pp. 286-293.
37. Wagner, G. J.; Liu, W. K. (2000): Application of essential boundary conditions in mesh-free methods: a corrected collocation method. Int J Numer Methods Eng, vol. 47, pp. 1367-1379.
38. Zhu, T.; Atluri, S. N. (1998): A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method. Comput Mech, vol. 21, pp. 211-222.
39. Zhu, T.; Zhang, J. D.; Atluri, S. N. (1998a): A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Comput Mech, vol. 21, pp. 223-235.
40. Zhu, T.; Zhang, J. D.; Atluri, S. N. (1998b): A meshless local boundary integral equation (LBIE) method for solving nonlinear problems. Comput Mech, vol. 22, pp. 174-186.