Abstract
We construct in this paper two efficient spectral algorithms in the frequency space for solving unsteady advection–reaction–diffusion equations with constant and variable coefficients. We first consider a Jacobi–Galerkin method for solving linear equations with constant coefficients. We then develop the direct solution algorithm for the linear advection–reaction–diffusion equations with variable coefficients using the Jacobi–Galerkin method with numerical integration. The proposed Jacobi–Galerkin methods, both in temporal and spatial discretizations, are successfully developed to handle the two-dimensional unsteady advection–reaction–diffusion equations with constant and variable coefficients and with fractional orders. In these methods, the model solution is expanded in both space and time in terms of polynomials bases built upon a linear combination of Jacobi polynomials. The homogeneous initial and Dirichlet boundary conditions are satisfied exactly by expanding the model solution in terms of these polynomials. The proposed Jacobi–Galerkin methods yield an exponential rate of convergence when the solution is smooth and allow a great flexibility to handle multi-dimensional time fractional advection–reaction–diffusion equations. Finally, a series of numerical examples are presented to demonstrate the efficiency and flexibility of the methods.
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References
Leonard BP, MacVean MK, Lock AP (1995) The flux integral method for multidimensional convection and diffusion. Appl Math Modell 19:333–342
Patel MK, Markatos NC, Cross M (1984) A critical evaluation of seven discretization schemes for convection-diffusion equations. Int J Numer Methods Fluids 5:225–244
Cifani S, Jakobsen ER, Karlsen KH (2011) The discontinuous Galerkin method for fractional degenerate convection-diffusion equations. BIT Numer Math 51:809–844
Castillo P, Cockburn B, Schötzau D, Schwab C (2002) Optimal a priori error estimates for the \(hp\)-version of the local discontinuous Galerkin method for convection-diffusion problems. Math Comput 71(238):455–478
Cheng Y, Zhang Q (2017) Local analysis of the local discontinuous Galerkin method with generalized alternating numerical flux for one-dimensional singularly perturbed problem. J Sci Comput 72:792–819
Cheng Y, Meng X, Zhang Q (2017) Application of generalized Gauss-Radau projections for the local discontinuous Galerkin method for linear convection-diffusion equations. Math Comput 86:1233–1267
Wang H, Zhang Q, Shu C (2018) Third order implicit-explicit Runge-Kutta local discontinuous Galerkin methods with suitable boundary treatment for convection-diffusion problems with Dirichlet boundary conditions. J Comput Appl Math 342:164–179
Zhou L, Xu Y (2018) Stability analysis and error estimates of semi-implicit spectral deferred correction coupled with local discontinuous Galerkin method for linear convection-diffusion equations. J Sci Comput 77:1001–1029
Burman E, Ern A (2007) Continuous interior penalty \(hp\)-finite element methods for advection and advection-diffusion equations. Math Comput 76:1119–1140
Matthies G, Skrzypacz P, Tobiska L (2008) Stabilization of local projection type applied to convection-diffusion problems with mixed boundary conditions. Electron Trans Numer Anal 32:90–105
Braack M, Lube G (2009) Finite elements with local projection stabilization for incompressible flow problems. J Comput Math 27:116–147
Ern A, Guermond J (2004) Theory and Practice of Finite Elements. Applied mathematical sciences. Springer, New York
Codina R (2011) Finite element approximation of the convection-diffusion equation: subgrid-scale spaces, local instabilities and anisotropic space-time discretizations. In: Lecture Notes in computational science and engineering, vol 81, pp 85–97
Saadatmandi A, Dehghan M, Azizi M (2012) The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients. Commun Nonlinear Sci Numer Simul 17:4125–4136
Dehghan M, Abbaszadeh M, Mohebbi A (2015) Error estimate for the numerical solution of fractional reaction-subdiffusion process based on a meshless method. J Comput Appl Math 280:14–36
Kalita JC, Dalal DC, Dass AK (2002) A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients Int. J Numer Methods Fluids 38:1111–1131
Dehghan M, Safarpoor M, Abbaszadeh M (2015) Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations. J Comput Appl Math 290:174–195
Noye BJ, Tan HH (1988) Finite difference methods for solving the two-dimensional advection-diffusion equation Int. J Numer Methods Fluids 26:1615–1629
Tian ZF, Ge YB (2007) A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems. J Comput Appl Math 198:268–286
Dehghan M, Abbaszadeh M (2017) A finite element method for the numerical solution of Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Eng Comput 33(3):587–605
Dehghan M, Mohebbi A (2008) High-order compact boundary value method for the solution of unsteady convection-diffusion problems. Math Comput Simul 79:683–699
Zaky MA, Tenreiro Machado JA (2017) On the formulation and numerical simulation of distributed-order fractional optimal control problems. Commun Nonlinear Sci Numer Simul 52:177–189
Alsuyuti MM, Doha EH, Ezz-Eldien SS, Bayoumi BI, Baleanu D (2019) Modified Galerkin algorithm for solving multitype fractional differential equations. Math Meth Appl Sci 42(5):1389–1412
Doha EH, Bhrawy AH, Abd-Elhameed WM (2009) Jacobi spectral Galerkin method for elliptic Neumann problems. Numer Algorithms 50:67–91
Doha EH, Bhrawy AH, Hafez RM (2011) A Jacobi-Jacobi dual-Petrov-Galerkin method for third-and fifth-order differential equations. Math Comput Model 53:1820–1832
Zaky MA, Doha EH, Tenreiro Machado JA (2018) A spectral numerical method for solving distributed-order fractional initial value problems. J Comput Nonlinear Dynam 13(10):1–8
Doha EH, Hafez RM, Youssri YH (2019) Shifted Jacobi spectral-Galerkin method for solving hyperbolic partial differential equations. Comput Math Appl 78(3):889–904
Zaky MA, Doha EH, Tenreiro JA (2018) A spectral framework for fractional variational problems based on fractional Jacobi functions. Appl Numer Math 132:51–72
Khosravian-Arab H, Dehghan M, Eslahchi MR (2017) Fractional spectral and pseudo-spectral methods in unbounded domains: Theory and applications. J Comput Phys 338:527–566
Doha EH (2004) On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. J Phys A: Math Gen 37:657–675
Luke Y (1969) The special functions and their approximations, vol 2. Academic Press, New York
Doha EH, Zaky MA, Abdelkawy MA (2019) Spectral methods within fractional calculus. Appl Eng Life Soc Sci Part B. https://doi.org/10.1515/9783110571929-008
Zaky MA (2019) Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems. Appl Numer Math. https://doi.org/10.1016/j.apnum.2019.05.008
Zaky MA (2019) Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions. J Comput Appl Math 357:103–122
Zaky MA, Ameen IG (2019) A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions. Numer Algor. https://doi.org/10.1007/s11075-019-00743-5
Zaky MA (2018) An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade fluid. Comput Math Appl 75:2243–2258
Abdelkawy MA, Lopes MA, Zaky MA (2019) Shifted fractional Jacobi spectral algorithm for solving distributed order time-fractional reaction-diffusion equations. Comput Appl Math 38(81):1–21
Teodoro GS, Machado JA, Oliveira EC (2019) A review of definitions of fractional derivatives and other operators. J Comput Phys 388:195–208
Bhrawy AH, Zaky MA (2015) A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J Comput Phys 281:876–895
Nazir T, Abbas M, Ismail AI, Majid A, Rashid A (2016) The numerical solution of advection-diffusion problems using new cubic trigonometric B-splines approach. Appl Math Model 40:4586–4611
Mittal RC, Jain RK (2012) Redefined cubic B-spline collocation method for solving convection diffusion equations. Appl Math Model 36:5555–5573
Goh J, Majid AA, Ismail AIBMd (2012) Cubic B-spline collocation method for one-dimensional heat and advection-diffusion equations. J Appl Math 2012(458701):1–8
Mohammadi R (2013) Exponential B-spline solution of convection-diffusion equations. Appl Math 4:933–944
Mohebbi A, Dehghan M (2010) High-order compact solution of the one dimensional heat and advection-diffusion equations. Appl Math Model 34:3071–3084
Golbabai A, Arabshahi MM (2010) A numerical method for diffusion-convection equation using high-order difference schemes. Comput Phys Commun 181:1224–1230
Cao H, Liu L, Zhang Y, Fu S (2011) A fourth-order method of the convection-diffusion equations with Neumann boundary conditions. Appl Math Comput 217:9133–9141
Zhou F, Xu X (2014) Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets. Appl Math Comput 247:353–367
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We would like to thank anonymous referees for their careful reading of this paper and their many valuable comments and suggestions for improving the presentation of this work.
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Hafez, R.M., Zaky, M.A. High-order continuous Galerkin methods for multi-dimensional advection–reaction–diffusion problems. Engineering with Computers 36, 1813–1829 (2020). https://doi.org/10.1007/s00366-019-00797-y
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DOI: https://doi.org/10.1007/s00366-019-00797-y
Keywords
- Advection–reaction–diffusion equations
- Fractional diffusion equations, Galerkin methods
- Jacobi polynomials