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Doostaki, Reza; Hosseini, Mohammad Mehdi; Salemi, Abbas

A new simultaneously compact finite difference scheme for high-dimensional time-dependent PDEs. (English) Zbl 1540.65286

Math. Comput. Simul. 212, 504-523 (2023).
Summary: This paper presents a new compact finite difference scheme for solving linear high-dimensional time-dependent partial differential equations (PDEs). Despite the different spatial and time conditions in time-dependent problems, we propose a new compact finite difference scheme simultaneously both in time and space with arbitrary order accuracy. The merit of the proposed method is that the approximation of partial derivatives are derived simultaneously at all grid points. Also, by substituting the partial derivatives in the linear time-dependent PDEs a linear system of equations is derived. To show the efficiency and applicability of the proposed method, the fourth, sixth, and eighth-order simultaneously compact finite difference schemes are used for solving parabolic and convection-diffusion equations which have both time and spatial partial derivatives. The numerical results show the accuracy of the proposed method.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Keywords:

compact finite difference scheme; parabolic equations; convection-diffusion equations; high-order accuracy

Software:

Algorithm 986
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Full Text: DOI

References:

[1] Chen, C.; Zhang, X.; Liu, Z.; Zhang, Y., A new high-order compact finite difference scheme based on precise integration method for the numerical simulation of parabolic equations, Adv. Difference Equ., 2020, 1, 1-28 (2020) · Zbl 1487.65114
[2] Dehghan, M.; Mohebbi, A., High-order compact boundary value method for the solution of unsteady convection-diffusion problems, Math. Comput. Simulation, 79, 3, 683-699 (2008) · Zbl 1155.65075
[3] Doostaki, R., Convergence rate of GMRES on tridiagonal block Toeplitz linear systems, Linear Multilinear Algebra, 64, 12, 2533-2544 (2016) · Zbl 1368.65049
[4] Doostaki, R.; Sadeghi Goughery, H., The upper and lower bounds for generalized minimal residual method on a tridiagonal Toeplitz linear system, Int. J. Comput. Math., 93, 3, 567-577 (2016) · Zbl 1339.65046
[5] Gatiso, A. H.; Belachew, M. T.; Wolle, G. A., Sixth-order compact finite difference scheme with discrete sine transform for solving Poisson equations with Dirichlet boundary conditions, Results Appl. Math., 10, Article 100148 pp. (2021) · Zbl 1478.65101
[6] Gidey, H.; Reddy, B., Operator-splitting methods for the 2D convective Cahn-Hilliard equation, Comput. Math. Appl., 77, 12, 3128-3153 (2019) · Zbl 1442.65194
[7] Gürarslan, G., Numerical modelling of linear and nonlinear diffusion equations by compact finite difference method, Appl. Math. Comput., 216, 8, 2472-2478 (2010) · Zbl 1193.65156
[8] Kalita, J. C.; Dalal, D.; Dass, A. K., A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients, Internat. J. Numer. Methods Fluids, 38, 12, 1111-1131 (2002) · Zbl 1094.76546
[9] Karaa, S.; Zhang, J., High order ADI method for solving unsteady convection-diffusion problems, J. Comput. Phys., 198, 1, 1-9 (2004) · Zbl 1053.65067
[10] Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 1, 16-42 (1992) · Zbl 0759.65006
[11] Li, J.; Chen, Y.; Liu, G., High-order compact ADI methods for parabolic equations, Comput. Math. Appl., 52, 8-9, 1343-1356 (2006) · Zbl 1121.65092
[12] Li, L.; Jiang, Z.; Yin, Z., Fourth-order compact finite difference method for solving two-dimensional convection-diffusion equation, Adv. Difference Equ., 2018, 1, 1-24 (2018) · Zbl 1446.65142
[13] Mehra, M.; Patel, K. S., Algorithm 986: A suite of compact finite difference schemes, ACM Trans. Math. Software, 44, 2, 1-31 (2017) · Zbl 1484.65359
[14] Mohebbi, A.; Dehghan, M., High-order compact solution of the one-dimensional heat and advection-diffusion equations, Appl. Math. Model., 34, 10, 3071-3084 (2010) · Zbl 1201.65183
[15] Noye, B.; Tan, H., Finite difference methods for solving the two-dimensional advection-diffusion equation, Internat. J. Numer. Methods Fluids, 9, 1, 75-98 (1989) · Zbl 0658.76079
[16] Peaceman, D. W.; Rachford, H. H., The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3, 1, 28-41 (1955) · Zbl 0067.35801
[17] Roache, P., Computational Fluid Dynamics (1972), Hermosa Press: Hermosa Press Albuquerque, NM · Zbl 0251.76002
[18] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM · Zbl 1002.65042
[19] Seydaoğlu, M., An accurate approximation algorithm for Burgers’ equation in the presence of small viscosity, J. Comput. Appl. Math., 344, 473-481 (2018) · Zbl 1464.65092
[20] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 3, 506-517 (1968) · Zbl 0184.38503
[21] Sun, J.; Eichholz, J. A., Splitting methods for differential approximations of the radiative transfer equation, Appl. Math. Comput., 322, 140-150 (2018) · Zbl 1426.82059
[22] Sun, H.; Zhang, J., A high-order compact boundary value method for solving one-dimensional heat equations, Numer. Methods Partial Differential Equations, 19, 6, 846-857 (2003) · Zbl 1038.65084
[23] Tian, Z.; Ge, Y., A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems, J. Comput. Appl. Math., 198, 1, 268-286 (2007) · Zbl 1104.65086
[24] Wang, H.; Zhang, Y.; Ma, X.; Qiu, J.; Liang, Y., An efficient implementation of fourth-order compact finite difference scheme for Poisson equation with Dirichlet boundary conditions, Comput. Math. Appl., 71, 9, 1843-1860 (2016) · Zbl 1443.65288
[25] Xu, B.; Zhang, X., An efficient high-order compact finite difference scheme based on proper orthogonal decomposition for the multi-dimensional parabolic equation, Adv. Difference Equ., 2019, 1, 1-22 (2019) · Zbl 1485.65095
[26] Zhang, Q.; Zhang, C.; Wang, L., The compact and Crank-Nicolson ADI schemes for twodimensional semilinear multidelay parabolic equations, J. Comput. Appl. Math., 306, 217-230 (2016) · Zbl 1382.65261
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