Abstract
In this work, the numerical solutions of advection-diffusion equation are investigated through the finite element method. The quartic-trigonometric tension (QTT) B-spline which presents advantages over the well-known existing B-splines is adapted as the base of the numerical algorithm. Space integration of the model partial differential equation is achieved through QTT B-spline Galerkin method. The resultant system of time-dependent differential equations is integrated using the Crank-Nicolson technique. The stability of the current scheme is accomplished and proved to be unconditionally stable. Simulation of several sample problems are carried out for verification of the proposed numerical scheme. Solutions obtained by numerically computed scheme are compared to the existing literature.
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References
Abbas M, Majid AA, Md. Ismail AI, Rashid A (2014) Numerical method using cubic B-spline for a strongly coupled reaction-diffusion system. PLoS One 9(1):e83265. https://doi.org/10.1371/journal.pone.0083265
Ahmadian A, Salahshour S, Chan CS, Baleanu D (2018) Numerical solutions of fuzzy differential equations by an efficient Runge-Kutta method with generalized differentiability. Fuzzy Sets Syst 331:47–67. https://doi.org/10.1016/j.fss.2016.11.013
Alinia N, Zarebnia M (2019) A numerical algorithm based on a new kind of tension B-spline function for solving Burgers-Huxley equation. Numer Algorithms 82:1121–1142. https://doi.org/10.1007/s11075-018-0646-4
Allahviranloo T, Khezerloo M, Sedaghatfar O, Salahshour S (2013) Toward the existence and uniqueness of solutions of second-order fuzzy volterra integro-differential equations with fuzzy kernel. Neural Comput Appl 22(1):133–141. https://doi.org/10.1007/s00521-012-0849-x
Chakraborty A, Mondal SP, Alam S, Ahmadian A, Senu N, De D, Salahshour S (2019) The pentagonal fuzzy number: its different representations, properties, ranking, defuzzification and application in game problems. Symmetry 11(2):248. https://doi.org/10.3390/sym11020248
Chatwin PC, Allen CM (1985) Mathematical models of dispersion in rivers and estuaries. Annu Rev Fluid Mech 17(1):119–149. https://doi.org/10.1146/annurev.fl.17.010185.001003
Crank J, Nicolson P (1947) A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc Camb Philos Soc 43:50–67. https://doi.org/10.1017/S0305004100023197
Dağ Canivar A, Şahin A (2011) Taylor-Galerkin method for advection-diffusion equation. Kybernetes 40(5/6):762–777. https://doi.org/10.1108/03684921111142304
Dag I, Irk D, Tombul M (2006) Least-squares finite element method for the advection diffusion equation. Appl Math Comput 173:554–565. https://doi.org/10.1016/j.amc.2005.04.054
Dehghan M (2004) Weighted finite difference techniques for the one-dimensional advection-diffusion equation. Appl Math Comput 147(2):307–319. https://doi.org/10.1016/S0096-3003(02)00667-7
Dhawan S, Kapoor S, Kumar S (2012) Numerical method for advection diffusion equation using FEM and B-splines. J Comput Sci 3(5):429–437. https://doi.org/10.1016/j.jocs.2012.06.006
Evans DJ, Abdullah AR (1985) A new explicit method for the diffusion convection Equation. Comput Math Appl 11:145–154. https://doi.org/10.1016/0898-1221(85)90143-9
Fattah QN, Hoopes JA (1985) Dispersion in anisotropic, homogeneous, porous media. J Hydraul Eng 111(5):810–827. https://doi.org/10.1061/(ASCE)0733-9429(1985)111:5(810)
Gane CR, Stephenson PL (1979) An explicit numerical method for solving transient combined heat conduction and convection problems. Int J Numer Methods Eng 14(8):1141–1163. https://doi.org/10.1002/nme.1620140804
Gorgulu MZ, Dursun I (2019) The Galerkin Finite Element Method for advection diffusion equation. Sigma J Eng Nat Sci 37(1):119–128
Gurarslan G, Karahan H, Alkaya D, Sari M, Yasar M (2013) Numerical solution of advection-diffusion equation using a sixth-order compact finite difference method. Math Probl Eng. https://doi.org/10.1155/2013/672936
Hepson OE (2020) A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system. Eng Comput. https://doi.org/10.1108/EC-05-2020-0289
Irk D, Dag I, Tombul M (2015) Extended cubic B-spline solution of advection diffusion equation. KSCE J Civ Eng 19(4):929–934. https://doi.org/10.1007/s12205-013-0737-7
Isenberg J, Gutfinger C (1973) Heat transfer to a draining film. Int J Heat Mass Transf 16(2):505–512. https://doi.org/10.1016/0017-9310(73)90075-6
Kadalbajoo MK, Arora P (2010) Taylor-Galerkin B-spline finite element method for the one-dimensional advection-diffusion equation. Numer Methods Partial Differ Equ 26(5):1206–1223. https://doi.org/10.1002/num.20488
Korkmaz A, Dağ I (2012) Cubic B-spline differential quadrature methods for the advection-diffusion equation. Int J Numer Methods Heat Fluid Flow 22(8):1021–1036. https://doi.org/10.1108/09615531211271844
Korkmaz A, Dağ I (2016) Quartic and quintic B-spline methods for advection-diffusion equation. Appl Math Comput 274:208–219. https://doi.org/10.1016/j.amc.2015.11.004
Kumar N (1983) Unsteady flow against dispersion in finite porous media. J Hydrol 63(3–4):345–358. https://doi.org/10.1016/0022-1694(83)90050-1
Lees M (1967) An extrapolated Crank-Nicolson difference scheme for quasilinear parabolic equations. In: Nonlinear partial differential equations. Academic Press. https://doi.org/10.1016/B978-1-4831-9647-3.50017-0
Mat Zin S, Abbas M, Abd Majid A, Md Ismail AI (2014) A new trigonometric spline approach to numerical solution of generalized nonlinear Klien-Gordon equation. PLoS One 9(5):e95774. https://doi.org/10.1371/journal.pone.0095774
Mittal RC, Jain RK (2012) Numerical solution of convection-diffusion equation using cubic B-splines collocation methods with Neumann\(^{\prime }\)s boundary conditions. Int J Appl Math Comput 4(2):115–127. https://doi.org/10.0000/IJAMC.2012.4.2.470
Mittal RC, Rohila R (2020) The numerical study of advection-diffusion equations by the fourth-order cubic B-spline collocation method. Math Sci 14(4):409–423. https://doi.org/10.1007/s40096-020-00352-7
Nazir T, Abbas M, Ismail AIM, Majid AA, Rashid A (2016) The numerical solution of advection-diffusion problems using new cubic trigonometric B-splines approach. Appl Math Model 40(7–8):4586–4611. https://doi.org/10.1016/j.apm.2015.11.041
Nazir T, Abbas M, Yaseen M (2017) Numerical solution of second-order hyperbolic telegraph equation via new cubic trigonometric B-splines approach. Cogent Math Stat 4(1):1382061. https://doi.org/10.1080/23311835.2017.1382061
Parbange JY (1980) Water transport in soils. Annu Rev Fluid Mech 2:77–102
Salahshour S, Allahviranloo T, Abbasbandy S (2012) Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun Nonlinear Sci Numer Simul 17(3):1372–1381. https://doi.org/10.1016/j.cnsns.2011.07.005
Salahshour S, Ahmadian A, Ismail F, Baleanu D (2017) A fractional derivative with non-singular kernel for interval-valued functions under uncertainty. Optik 130:273–286. https://doi.org/10.1016/j.ijleo.2016.10.044
Salama AA, Zidan HZ (2006) Fourth-order schemes of exponential type for singularly perturbed parabolic partial differential equations. Rocky Mt J Math 36(3):1049–1068. https://doi.org/10.1216/rmjm/1181069445
Salmon JR, Liggett JA, Gallagher RH (1980) Dispersion analysis in homogeneous lakes. Int J Numer Methods Eng 15(11):1627–1642. https://doi.org/10.1002/nme.1620151106
Sankaranarayanan S, Shankar NJ, Cheong HF (1998) Three-dimensional finite difference model for transport of conservative polluants. Ocean Eng 25(6):425–442. https://doi.org/10.1016/S0029-8018(97)00008-5
Sharma D, Jiwari R, Kumar S (2011) Galerkin-finite element method for the numerical solution of advection-diffusion equation. Int J Pure Appl Math 70(3):389–399
Sunarsih S, Purwantoro Sasongko D, Sutrisno S, Agnes Putri G, Hadiyanto H (2020) Analysis of wastewater facultative pond using advection-diffusion model based on explicit finite difference method. Environ Eng Res 26(3):190496. https://doi.org/10.4491/eer.2019.496
Wang G, Fang M (2008) Unified and extended form of three types of splines. J Comput Appl Math 216:498–508. https://doi.org/10.1016/j.cam.2007.05.031
Xu G, Wang G (2007) AHT Bezier curves and NUAHT B-spline curves. J Comput Sci Technol 22:597–607. https://doi.org/10.1007/s11390-007-9073-z
Ya-Juan L, Guo-Zhao W (2005) Two kinds of B-basis of the algebraic hyperbolic space. J Zhejiang Univ-Sci A 6(7):750–759. https://doi.org/10.1631/jzus.2005.A0750
Zlatev Z, Berkowicz R, Prahm LP (1984) Implementation of a variable stepsize variable formula method in the time-integration part of a code for treatment of long-range transport of air pollutants. J Comput Phys 55(2):278–301. https://doi.org/10.1016/0021-9991(84)90007-X
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Ersoy Hepson, O., Yigit, G. Quartic-trigonometric tension B-spline Galerkin method for the solution of the advection-diffusion equation. Comp. Appl. Math. 40, 141 (2021). https://doi.org/10.1007/s40314-021-01526-2
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DOI: https://doi.org/10.1007/s40314-021-01526-2