An Element-Driven Boundary Integral Treatment for Nonlinearity: A Coupled Nonlinear Two-Dimensional Burger’s Equation Test Case
To further improve on the competitiveness of the boundary element method (BEM), a hybrid version of it is used for a numerical solution of two dimensional nonlinear coupled viscous Burger’s equation. Adopting this approach to a discretized 2D spatial domain, the resulting integral equations arising from the singular integral theory are applied locally to each of the elements. The resulting nonlinear discrete equations are finally solved by the Picard iteration algorithm. The simulation results obtained, not only concur with analytical solutions, but also display high accuracy and are in agreement with those available in literature.
Okey Oseloka Onyejekwe,
An Element-Driven Boundary Integral Treatment for Nonlinearity: A Coupled Nonlinear Two-Dimensional Burger’s Equation Test Case, International Journal of Fluid Mechanics & Thermal Sciences.
Vol. 3, No. 1,
2017, pp. 1-15.
Cole, J. D., “On the quasilinear parabolic equations occurring in aerodynamics” Quart. Appld. Math. vol. 9 pp. 225-236, 1951.
Nee, J., Duan J., “Limit set of trajectories of the coupled viscous Burger’s equations” Appld. Math. Letters vol. 11 (1) pp. 57-61, 1998.
Esipov, S. E., “Coupled Burgers’ equations: a model for poly-dispersive sedimentation” Phs. Rev. vol. 52 p 3711, 1995.
Fletcher, C. A. J., “Generating exact solutions of the two-dimensional Burgers’ equation” Int. Jnl. Numer. Mthds Fluids vol. 3 pp. 213-216, 1983.
Abazari R., and Borhanifar A., “A numerical study of solution of the Burgers’ equations by a differential transformation method”, Computers and mathematics with Applications vol. 59 pp. 2711-2722, 2010.
Ablowitz M. J. and Clarkson P. A., Solitons, nonlinear evolution equation and inverse scattering Cambridge: Cambridge University Press, 1991.
Hirota R., “Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons”, Phys. Rev. Letters, vol. 27 pp. 1192-1194, 1971.
Bender C. M., Pinsky K. S. and Simmons L. M., “A new perturbative approach to nonlinear problems’’, Journal of Math. Physics, vol. 30(7) pp. 1447-1455, 1989.
Kaya D., “An explicit solution of coupled viscous Burgers’ equations by decomposition method”, Int. Jnl. Math. Sci., vol. 27 pp. 675-680, 2001.
Zhu H., Shu H. and Ding M., “Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method” Computers and Mathematics with Applications, vol. 60 (3) pp. 840-848, 2010.
Fletcher, C. A. J., “A comparison of finite element and finite difference solution of the one and two-dimensional Burgers’ equations” Jnl. Comput. Physics, vol. 3 pp. 159-188, 1983.
Abdou M. A., Soliman A. A., (2004) “ Variational iteration method for solving Burger’s and coupled Burger’s equations”, Jnl. Comput. Appld. Math., vol. 181 (2) pp. 245-251, 2004.
Kelleci A, and Yildrim A., “An efficient numerical method for solving coupled Burgers’ equation by combining homotopy perturbation and Pade techniques” Num. Mthds. For Partial Differential Equations, vol. 27 (4), pp. 982-995, 2011.
Kutluay, S. and Ucar Y., “Numerical solutions of the coupled Burgers’ equation by the Galerkin quadratic B-spline finite element method “Mathematical Methods in the Applied Sceinces (wileyonlinelibrary.com), DOI:10.1002/mma.2767, 2013.
Soliman, A. A., “On the solution of two-dimensional coupled burgers’ equations by variational iteration method”, Chaos Solitons and Fractals, vol. 40 (3), pp. 1146-1155, 2009.
Odibat, Z. M. and Momani S., “Application of variational iteration method to nonlinear differential equations of fractional order”, Int. Jnl. Nonlinear Sci. Num. Simulation, vol. 7 pp. 27-34, 2006.
Bahadir, A. R., “A fully implicit finite difference scheme for two-dimensional Burgers’ equations”, Appld. Math. and Comput., vol. 137 pp. 131-137, 2003.
Srivastava, V. K. Tamsir, M. and Bhardwaj, Y. and Sanyasiraju, Y., “Crank-Nicolson scheme for numerical solutions of two dimensional coupled Burgers’ equations”, Jnl. Scientific and Engnr. Research, vol. 2 (5) pp. 1-7, 2011.
Srivastava, V. K., Singh S. and Awasthi M. K., “Numerical solutions of coupled Burgers’ equations by an implicit finite difference scheme”, AIP Advances, vol. 3 p. 082131, 2013.
Wubs, F. W. and de Goede, E. D., “An explicit-implicit method for a class of time- dependent partial differential equations”, Appld. Numer. Math., vol. 9 pp. 157-181, 1992.
Goyon, O., “Multilevel schemes for solving unsteady equations”, Int. Jnl. Num. Mthds. Fluids, vol. 22 pp. 937-959, 1996.
Onyejekwe, O. O., A modified boundary integral solution of coupled linear and nonlinear one dimensional transport equations, Proceed. Boundary elements XXII, eds. Brebbia C. A. and Power H, 2000.
Grigoriev, M. M., “A boundary element method for the solution of convective diffuaion and Burger’s equation”, Int. Jnl. Num. Mthds Heat and Fluid Flow, vol. 4 (6) pp. 527-552, 1994.
Siraj-ul-Islam, B., Sarler, B. R., Vertnik S., and Kosec G., (2012) “Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations”, Appld. Math. Modelling, vol. 36 pp. 1148-1160, 2012.
Toutip, W., The dual reciprocity boundary element method for linear and nonlinear problems Ph. D thesis, University of Hertfordshire, England, 2001.
Percher, R., Harris, S. D., Knipe, R. J. Elliot, L., Ingham, D. B., “New formulation of the Green element method to maintain its second-order accuracy in 2D/FD”, Engnr. Analy. Bdry Elem., vol. 25, pp 211-219, 2001.
Lorinczi, P., Applications of the Green element method to flow in heterogeneous porous media, Ph. D. Thesis University of Leeds, 2006.
Lorinczi. P., Harris S. D. and Elliot, L., (2010) “Unsteady flux-vector-based Green element method”, Transport Porous Media, vol. 87 (1) pp 207-228, 2010.
Archer, R., Horne, R. N. and Onyejekwe O. O., Petroleum reservoir engineering applications of the dual reciprocity boundary element method and the Green element method, 21st world conference on the boundary element method, 25-27 August, 1999, Oxford University England, 1999.
Archer, R., Computing flow and pressure transients in heterogeneous media using boundary element methods, Ph. D. dissertation, Stanford University, 2000.
Onyejekwe, O. O., “A comparison of time discretization schemes for the Green element solution of transient heat conduction equation”, Num. Heat Transfer, Part B: Fundamentals: An International Journal of Computation and Methodology, vol. 38 (4) pp. 405-422, 2000.
Onyejekwe, O. O., “Experiences in solutions of non-linear transport equations with the Green element method”, Int. Jnl. Num. Mthds. Heat and Fluid Flow, vol. 10 (7) 675-686, 2000.
Onyejekwe, O. O. and Onyejekwe, O. N., “Numerical solutions of the one-phase classical Stefan problem using an enthalpy Green element formulation”, Adv. Engnr. Software, vol. 42 (10) pp. 743-749, 2011.
Archer, R. and Horne R. N., (1998) Flow simulations in heterogeneous reservoirs using the dual reciprocity boundary element method and the Green element method, European Conference on the Mathematics of Oil Recovery VI, September 8-11, Peebles, Scotland, 1998.
Taigbenu, A. E., “The Green element method ”, Int. Jnl. Num. Mthds in Engnr., vol. 38 pp. 2241-2263, 1995.
Onyejekwe, O. O., A Green element application to the diffusion equation, Proceedings 35th Heat transfer and Mechanics Institute, California State University, Sacramento California, pp. 75-90 1995.
Onyejekwe, O. O. “Boundary integral procedures for unsaturated flow problems”, Transport in Porous Media, vol. 3, pp. 313-330, 1998.
Taigbenu, A. E. and Onyejekwe, O. O., “Green element simulations of the transient nonlinear unsaturated flow equation”, Appld. Math. Mod., vol. 19 pp. 675-684, 1995.
Hibersek, M. and Skerget L, “Domain decomposition methods for fluid flow problems by boundary integral method” Zeitschrift fur Angewandle Mathematik und Mechanic Vol. 76 pp. 115-139, 1996.
Onyejekwe, O. O., “A Boundary element-Finite element equation solutions to flow in heterogeneous porous media” Transport in Porous Media, vol. 31 (3) pp. 293-312, 1998.
Grigoriev, M. M. and Dargush, G. F., “Boundary element methods for transient convective diffusion Part 1: General formulation and 1D implementation” Computer Methods in Appld. Mech. And Engnr., vol. 192 pp. 4281-4298, 2003.
Perata, A. and Popov, V., “Numerical stability of the BEM for advection-diffusion problems” http://dx.doi.org/10.1002/num.2009.
Portapilla, M. and Power H., “iterative solution schemes for quadratic DRM-MD” Num. Mthds. Partial Differential Equations, vol. 24 pp. 1430-1459, 2008.
Sladeck, J., Sladeck V., and Zhang C., “A local BIEM for analysis of transient heat conduction with nonlinear source terms in FMG’s” Engnr. Analy. Bdry. Elelm., vol. 28 pp. 1-11, 2004.
Abashar, M. E., “Application of the Green element method to chemical engineering problems” Journal of King Saud University, vol 17(1) pp. 47-59, 2004.
Mohammadi, M., Hematiyan, M. R., Marin L. “Boundary element analysis of nonlinear transient heat conduction problems involving nonhomogeneous and nonlinear heat sources using time dependent fundamental solutions”, Engnr. Analysis Bdry. Elements, vol. 34 pp. 655-665, 2010.
Onyejekwe, O. O., “Green element method for 2D Helmholtz and convection diffusion problems with variable velocity coefficients”, Num. Mthds. Partial Diff. Equations, vol. 21 (2) pp. 229-241, 2005.
Onyejekwe, O. O., “A note on Green element method discretization for Poisson equation in polar coordinates”, Applied Math Letters, vol. 19 pp 785-788, 2006.
Nyirenda, E., A time dependent Green element Method for potential flow study in 3D Publ. AV Akademi, Kervalag GMBH, 2012.
Bagherinezhad, A. and Pishvaie M. R., “A new approach to counter-current spontaneous imbibition simulation using Green element method”, Journ. Petrol. Science and Engnr., vol. 119 pp. 163-168, 2014.
Archer, R. A. and Horne, R. N., (2002) “Green element method and singularity programming for numerical well test analysis”, Engnr. Analy. Bdry. Elem, vol. 26 pp 537-546, 2002.
Archer, R. A., “C1 continuous solutions from the Green element method using Overhauser elements”, Appld. Numerical Math., vol 56 (2) pp 222-229, 2006.
Onyejekwe, O. O., “An effective boundary integral approach for the solution of nonlinear transient thermal diffusion problems”, Italian Jnl. Pure and Appld. Math., vol. 34 pp. 397-412, 2015.
Onyejekwe, O. O., “The effect of time-stepping on the accuracy of the Green element formulation of unsteady convective transport” Jnl. Applied Math and Physics, vol. 2 pp 621-633, 2015
Onyejekwe, O. O., “An Hermitian Boundary integral hybrid formulation for nonlinear Fisher-type equations” Applied and Computaional Mathematics, vol. 4 (3) pp 83-99, 2015.
Onyejekwe, O. O., “Green element procedures accompanied by nonlinear reaction”, Int. Jnl. Thermal Sci., vol. 42 pp. 813-820 2003.
Tamsir, M. and Srivastava V. K., “A semi-implicit finite-difference approach for two-dimensional coupled Burgers’ equations” Int. Jnl. Of Scientific and Engnr. Res., vol. 2 (6) pp 1-6, 2011.
G lkac, V., “Numerical solutions of two-dimensional Burgers’ equations” International Journal of Scientific and Engineering Research, vol. 6 (4) pp. 215-218, 2015.
Jain, P. C. and Holla, D. N., “Numerical solution of coupled Burgers’ equations” Int. Jnl. Num. Mthds. Engngr., vol. 12 pp. 415-428, 1978.
Shukla, H. S., Tamsir, M., Srivastava V. K. and Kumar, J., “Numerical solution of two dimensional coupled viscous Burger equation using modified cubic B-Spline differential quadrature method”, AIP advances, vol. 4 p. 117134-1, 2014.
Zhao, G., Yu X., and Zhang R., “The new numerical method for solving the system of two-dimensional Burgers’ equations” Computers and Mathematics with Applications, vol. 62 pp. 3279-3291, 2011.