Global thin plate spline differential quadrature as a meshless numerical solution for two-dimensional viscous Burgers' equation

Document Type : Article

Authors

Department of Civil Engineering, Persian Gulf University, Shahid Mahini St., Bushehr, P.O. Box 75169, Iran

Abstract

This paper is aimed to present the Global Thin Plate Spline Differential Quadrature method for the numerical solution of viscous Burgers’ equation. This mesh-less and high-order model is introduced with the motive of diminishing computational effort and dealing with irregular geometries. Thin Plate Spline Radial basis function is used as a test function to determine coefficients of derivatives in differential quadrature. The present algorithm is applied to discretize and solve two-dimensional Burgers’ equation in both rectangular and irregular non-rectangular computational domains with randomly distributed computation nodes. To evaluate the capability of the present model, several problems with different boundary and initial conditions and Reynolds Numbers are solved and the obtained results are compared with the analytical solutions and other previous numerical models. The obtained results show the higher accuracy of the present model for solving Berger's equation with fewer computational nodes compared to the previous models even in irregular domains.

Keywords

References

References:
1. Bateman, H. "Some recent researches on the motion of  fluids", Mon. Weather Rev., 43(4), pp. 163-170 (1915).
2. Burgers, J.M. "A mathematical model illustrating the theory of turbulence", In Advances in Applied Mechanics, Elsevier, 1(C), pp. 171-199 (1948).
3. Wazwaz, A.-M., Partial Differential Equations: Methods and Applications, AA Balkema (2003).
4. Wei, G.W., Zhang, D.S., Kouri, D.J., et al. "Distributed approximating functional approach to Burgers' equation in one and two space dimensions", Comput. Phys. Commun., 111(1-3), pp. 93-109 (1998).
5. Hopf, E. "The partial differential equation ut+uux = xx", Commun. Pure Appl. Math., 3(3), pp. 201-230 (1950).
6. Cole, J.D. "On a quasi-linear parabolic equation occurring in aerodynamics", Q. Appl. Math., 9(3), pp. 225-236 (1951).
7. Fletcher, C.A.J. "Generating exact solutions of the two-dimensional Burgers' equations", Int. J. Numer. Methods Fluids, 3(3), pp. 213-216 (1983).
8. Varoglu, E. and Liam Finn, W.D. "Space-time finite elements incorporating characteristics for the burgers' equation", Int. J. Numer. Methods Eng., 16(1), pp. 171-184 (1980).
9. Caldwell, J., Wanless, P., Cook, A.E., et al. "A finite element approach Burgers' equation to", Appl. Math. Model., 5(June), pp. 189-193 (1981).
10. Kutluay, S., Bahadir, A.R., and Ozdes, A. "Numerical solution of one-dimensional Burgers equation: explicit and exact-explicit finite difference methods", J. Comput. Appl. Math., 103(2), pp. 251-261 (1999).
11. Evans, D.J. and Abdullah, A.R. "The group explicit method for the solution of Burger's equationDie gruppenexplizite methode fur die Losung der Burgerschen Gleichung", Computing, 32(3), pp. 239-253 (1984).
12. Kakuda, K. and Tosaka, N. "The generalized boundary element approach to Burgers' equation", Int. J. Numer. Methods Eng., 29(2), pp. 245-261 (1990).
13. Ozis, T., Ozdes, A., Of, J., et al. "Guidelines for the safe use of Doppler Ultrasound for clinical applications", Eur. J. Ultrasound, 2(2), pp. 167-168 (1995).
14. Bar-Yoseph, P., Moses, E., Zrahia, U., et al. "Spacetime spectral element methods for one-dimensional nonlinear advection-diffusion problems", J. Comput. Phys., 119(1), pp. 62-74 (1995).
15. Zhang, D.S., Wei, G.W., Kouri, D.J., et al. "Burgers' equation with high Reynolds number", Phys. Fluids, 9(6), pp. 1853-1855 (1997).
16. Mittal, R.C. and Jiwari, R. "Differential quadrature method for two-dimensional Burgers' equations", Int. J. Comput. Methods Eng. Sci. Mech., 10(6), pp. 450- 459 (2009).
17. Vaghefi, M., Rahideh, H., Haghighi, M.R.G., et al. "Distributed approximating functional approach to Burgers' equation using element differential quadrature method", J. Appl. Sci. Environ. Manag., 16(1), pp. 143-149 (2012).
18. Esen, A. and Tasbozan, O. "Numerical solution of time fractional burgers equation by cubic B-spline finite elements", Mediterr. J. Math., 13(3), pp. 1325-1337 (2016).
19. Aswin, V.S., Awasthi, A., and Rashidi, M.M. "A differential quadrature based numerical method for highly accurate solutions of Burgers' equation", Numer. Methods Partial Differ. Equ., 33(6), pp. 2023- 2042 (2017).
20. Erdo, U., Ozis, T., Ozis, T., et al. "Numerical solution of Burgers' equation with high order splitting methods", J. Comput. Appl. Math., 291(January 2016), pp. 410-421 (2016).
21. Sakar, M.G., Saldr, O., and Erdogan, F. "Numerical solution of time-fractional Burgers' equation in reproducing kernel space", arXiv Prepr. arXiv1805.06953 (2018).
22. Hussain, M., Haq, S., Ghafoor, A., et al. "Numerical solutions of time-fractional coupled viscous Burgers' equations using meshfree spectral method", Comput. Appl. Math., 39(1), p. 6 (2019).
23. Wu, Y.L. and Shu, C. "Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli", Comput. Mech., 29(6), pp. 477-485 (2002).
24. Sugihara, M. and Fujino, S. "Numerical solutions of Burgers' equation with a large Reynolds number", Reliab. Comput., 2(2), pp. 173-179 (1996).
25. Bellman, R. and Casti, J. "Differential quadrature and long-term integration", J. Math. Anal. Appl., 34(2), pp. 235-238 (1971).
26. Motaman, F., Rakhshandehroo, G.R., Hashemi, M.R., et al. "Application of RBF-DQ method to timedependent analysis of unsaturated seepage", Transp. Porous Media, 125(3), pp. 543-564 (2018).
27. Parand, K. and Hashemi, S. "RBF-DQ method for solving non-linear differential equations of Lane- Emden type", Ain Shams Eng. J., 9(4), pp. 615-629 (2018).
28. Sun, D., Ai, Y., Zhang, W., et al. "Direct solution of Navier-Stokes equations by using an upwind local RBF-DQ method", J. Vibroengineering, 16(1), pp. 78- 89 (2014).
29. Homayoon, L., Abedini, M.J., and Hashemi, S.M.R. "RBF-DQ solution for shallow water equations", J. Waterw. Port, Coastal, Ocean Eng., 139(1), pp. 45-60 (2012).
30. Behroozi, A.M. and Vaghefi, M. "Multi-block DQM/RBF-DQ as a meshless model for numerical investigation of laminar  flow and forced convection in a channel with two circular fins", Eng. Anal. Bound. Elem., 125, pp. 33-45 (2021).
31. Ding, H., Shu, C., and Tang, D.B. "Error estimates of local multiquadric-based differential quadrature (LMQDQ) method through numerical experiments", Int. J. Numer. Methods Eng., 63(11), pp. 1513-1529 (2005).
32. Bookstein, F.L. "Principal warps: thin-plate splines and the decomposition of deformations", IEEE Trans. Pattern Anal. Mach. Intell., 11(6), pp. 567-585 (1989).
33. Behroozi, A.M. and Vaghefi, M. "Radial basis function differential quadrature for hydrodynamic pressure on dams with arbitrary reservoir and face shapes affected by earthquake", J. Appl. Fluid Mech., 13(06), pp. 1759-1768 (2020).
34. Behroozi, A.M. and Vaghefi, M. "Numerical simulation of water hammer using implicit Crank-Nicolson local multiquadric based differential quadrature", Int. J. Press. Vessel. Pip., 181, p. 104078 (2020).
35. Abbaszadeh, M. and Dehghan, M. "An upwind local radial basis functions-differential quadrature (RBFs- DQ) technique to simulate some models arising in water sciences", Ocean Eng., 197, p. 106844 (2020).
36. Watson, D.W., Karageorghis, A., and Chen, C.S. "The radial basis function-differential quadrature method for elliptic problems in annular domains", J. Comput. Appl. Math., 363, pp. 53-76 (2020).
37. Liu, J., Li, X., and Hu, X. "A RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation", J. Comput. Phys., 384, pp. 222-238 (2019).
38. Ghalandari, M., Shamshirband, S., Mosavi, A., et al. "Flutter speed estimation using presented differential quadrature method formulation", Eng. Appl. Comput. Fluid Mech., 13(1), pp. 804-810 (2019).
39. Jafarabadi, A. and Shivanian, E. "Numerical simulation of nonlinear coupled Burgers' equation through meshless radial point interpolation method", Eng. Anal. Bound. Elem., 95, pp. 187-199 (2018).
40. Mohammadi, M., Mokhtari, R., and Panahipour, H. "A Galerkin-reproducing kernel method: Application to the 2D nonlinear coupled Burgers' equations", Eng. Anal. Bound. Elem., 37(12), pp. 1642-1652 (2013).
41. Ali, A., Siraj-ul-Islam, I., and Haq, S. "A computational meshfree technique for the numerical solution of the two-dimensional coupled Burgers' equations", Int. J. Comput. Methods Eng. Sci. Mech., 10(5), pp. 406- 422 (2009).
42. Bahadir, A.R. "A fully implicit finite-difference scheme for two-dimensional Burgers' equations", Appl. Math. Comput., 137(1), pp. 131-137 (2003).
43. Siraj-ul-Islam, I., Sarler, B., Vertnik, R., et al. "Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers' equations", Appl. Math. Model., 36(3), pp. 1148-1160 (2012).
44. Young, D.L., Fan, C.M., Hu, S.P., et al. "The Eulerian-Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers' equations", Eng. Anal. Bound. Elem., 32(5), pp. 395-412 (2008).
45. Arminjon, P. and Beauchamp, C. "Numerical solution of Burgers' equations in two space dimensions", Comput. Methods Appl. Mech. Eng., 19(3), pp. 351-365 (1979).
Scientia Iranica
Volume 30, Issue 6
Transactions on Civil Engineering (A)
November and December 2023
Pages 1942-1954

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History

  • Receive Date: 10 April 2022
  • Revise Date: 31 August 2022
  • Accept Date: 07 November 2022

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