A fourth-order two-level factored implicit scheme for solving two-dimensional unsteady transport equation with time-dependent dispersion coefficients. (English) Zbl 07881089
Summary: This paper analyzes a two-level factored implicit scheme in a numerical solution of two-dimensional unsteady advection-diffusion equation with time dependent dispersion coefficients subjects to initial and boundary conditions. The proposed approach is fast and efficient: unconditionally stable, second order accurate in time, spatial fourth order convergent and it requires less computing time. In fact, the two-level factored technique reduces to solve a tridiagonal system of linear equations at each calculation step. This reduces the computational cost of the algorithm. The analysis of the stability of the numerical scheme considers the \(L^\infty( t_0, T_f; L^2)\)-norm while the error estimates and convergence rate use \(L^2\)-norm. A wide set of numerical evidences are presented and discussed.
MSC:
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 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:
two-dimensional transport equation; time dependent dispersion coefficients; implicit method; fourth order two-level factored implicit scheme; stability and convergence rateReferences:
| [1] | Singh, K. M. and Tanaka, M., “On exponential variable transformation based boundary element formulation for advection-diffusion problems,” Eng. Anal. Bound. Elem., vol. 24, no. 3, pp. 225-235, 2000. DOI: . · Zbl 0942.80003 |
| [2] | Barry, D. A. and Sposito, G., “Analytic solution of a convection-dispersion model with time-dependent transport coefficients,” Water Resour. Res., vol. 25, no. 12, pp. 2407-2416, 1989. DOI: . |
| [3] | Fattah, Q. N. and Hoopes, J. A., “ Dispersion in anisotropic homogeneous porous media,” J. Hydraul. Eng., vol. 111, no. 5, pp. 810-827, 1985. DOI: . |
| [4] | Guvanasen, V. and Volker, R. E., “ Numerical solutions for solute transport in unconfined aquifers,” Int. J. Numer. Meth. Fluids, vol. 3, no. 2, pp. 103-123, 1983. DOI: . · Zbl 0513.76096 |
| [5] | Parlange, J. Y., “Water transport in soils,” Annu. Rev. Fluid Mech., vol. 12, no. 1, pp. 77-102, 1980. DOI: . · Zbl 0465.76092 |
| [6] | Salmon, J. R., Liggett, J. A., and Gallager, R. H., “ Dispersion analysis in homogeneous lakes,” Int. J. Numer. Meth. Eng., vol. 15, no. 11, pp. 1627-1642, 1980. DOI: . · Zbl 0457.76016 |
| [7] | Zlatev, Z., Berkowicz, R., and Prahm, L. P., “Implementation of a variable stepsize variable formula in the time-integration part of a code for treatment of long-range transport of air pollutants,” J. Comput. Phys., vol. 55, no. 2, pp. 278-301, 1984. DOI: . · Zbl 0558.65068 |
| [8] | Aral, M. M. and Liao, B., “Analytic solutions for two-dimensional transport equation with time-dependent dispersion coefficients,” J. Hydrol. Eng., vol. 1, no. 1, pp. 20-32, 1996. DOI: . |
| [9] | Yadav, R. R., Jaiswail, D. K., and Gulrana, “Two-dimensional solute transport for periodic flow in isotropic porous media: An analytical solution,” Hydrol. Process, vol. 26, pp. 3425-3433, 2012. DOI: . |
| [10] | Ngondiep, E., Kerdid, N., Abdulaziz Mohammed Abaoud, M., and Abdulaziz Ibrahim Aldayel, I., “A three-level time-split MacCormack method for two-dimensional nonlinear reaction-diffusion equations,” Int. J. Numer. Meth. Fluids, vol. 92, no. 12, pp. 1681-1706, 2020. DOI: . |
| [11] | Ngondiep, E., “An efficient three-level explicit time-split scheme for solving two-dimensional unsteady nonlinear coupled Burgers equations,” Int. J. Numer. Meth. Fluids, vol. 92, no. 4, pp. 266-284, 2020. DOI: . |
| [12] | Gupta, M. M., Manohar, R. P., and Stephenson, J. W., “A single cell high order scheme for the convection-diffusion equation with variable coefficients,” Int. J. Numer. Meth. Fluids, vol. 4, no. 7, pp. 641-651, 1984. DOI: . · Zbl 0545.76096 |
| [13] | Ngondiep, E., “An efficient three-level explicit time-split approach for solving two-dimensional heat conduction equation,” Appl. Math. Inf. Sci., vol. 14, no. 6, pp. 1075-1092, 2020. |
| [14] | Ngondiep, E., “Analysis of stability over long time intervals of a two-level explicit MacCormack/Crank-Nicolson approach for unsteady coupled Stokes-Darcy equations”, submitted. · Zbl 1492.76081 |
| [15] | Dennis, S. C. R. and Hudson, J. D., “Compact h^4 finite-difference approximations to operators of Navier-Stokes type,” J. Comput. Phys., vol. 85, no. 2, pp. 390-418, 1989. DOI: . · Zbl 0681.76031 |
| [16] | Namio, F. T., Ngondiep, E., Ntchantcho, R., and Ntonga, J. C., “Mathematical models of complete shallow water equations with source terms, stability analysis of Lax-Wendroff scheme,” J. Theor. Comput. Sci., vol. 2, no. 132, 2015. |
| [17] | Ngondiep, E., “Stability analysis of MacCormack rapid solver method for evolutionary Stokes-Darcy problem,” J. Comput. Appl. Math., vol. 345, pp. 269-285, 2019. DOI: . · Zbl 1397.76074 |
| [18] | Mittal, R. C. and Tripathi, A., “ Numerical solutions of generalized Burgers’-Fisher and generalized Burgers’-Huxley equations using collocation of cubic B-splines,” Int. J. Comput. Math., vol. 92, no. 5, pp. 1053-1077, 2015. DOI: . · Zbl 1314.65134 |
| [19] | Ngondiep, E., “A three-level explicit time-split MacCormack method for 2D nonlinear reaction-diffusion equations,” preprint available online fromhttp://arxiv.org/abs/1903.10877, 2019. |
| [20] | MacKinnon, R. J. and Johnson, R. W., “Differential equation based representation of truncation errors for accurate numerical simulation,” Int. J. Numer. Meth. Fluids, vol. 13, no. 6, pp. 739-757, 1991. DOI: . · Zbl 0729.76611 |
| [21] | Ngondiep, E., “A novel three-level time-split MacCormack method for solving two-dimensional viscous coupled Burgers equations,” preprint available online from http://arxiv.org/abs/1906.01544, 2019. |
| [22] | Zhang, J., “An explicit fourth-order compact finite-difference scheme for three dimensional convection-diffusion equation,” Commun. Numer. Meth. Eng., vol. 14, no. 3, pp. 209-218, 1998. DOI: . · Zbl 0912.65083 |
| [23] | Ngondiep, E., “Long time stability and convergence rate of MacCormack rapid solver method for nonstationary Stokes-Darcy problem,” Comput. Math. Appl., vol. 75, no. 10, pp. 3663-3684, 2018. DOI: . · Zbl 1419.65026 |
| [24] | Mittal, R. C. and Tripathi, A., “ Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-splines finite elements,” Eng. Comput., vol. 32, no. 5, pp. 1275-1306, 2015. DOI: . |
| [25] | Ngondiep, E., “Long time unconditional stability of a two-level hybrid method for nonstationary incompressible Navier-Stokes equations,” J. Comput. Appl. Math., vol. 345, pp. 501-514, 14, 2019. pages. DOI: . · Zbl 1397.76095 |
| [26] | Ngondiep, E., “Asymptotic growth of the spectral radii of collocation matrices approximating elliptic boundary problems,” Int. J. Appl. Math. Comput., vol. 4, pp. 199-219, 2012. |
| [27] | Zhang, J., “Multigrid method and fourth-order compact difference scheme for 2D Poisson equation unequal meshsize discretization,” J. Comput. Phys., vol. 179, no. 1, pp. 170-179, 2002. DOI: . · Zbl 1005.65137 |
| [28] | Ngondiep, E., “Error estimate of MacCormack rapid solver method for 2D incompressible Navier-Stokes problems,” preprint available online from http://arxiv.org/abs/1903.10857, 2019. · Zbl 1522.76043 |
| [29] | Ngondiep, E., Alqahtani, R., and Ntonga, J. C., “Stability analysis and convergence rate of MacCormack scheme for complete shallow water equations with source terms,” preprint available online from http://arxiv.org/abs/1903.11104, 2019. |
| [30] | Li, M., Tang, T., and Fornberg, B., “A compact fourth-order finite-difference scheme for the incompressible Navier-Stokes equations,” Int. J. Numer. Meth. Fluids, vol. 20, no. 10, pp. 1137-1151, 1995. DOI: . · Zbl 0836.76060 |
| [31] | Ngondiep, E., “An efficient explicit approach for predicting the Covid-19 spreading with undetected infectious: The case of Cameroon,” preprint available online from http://arxiv.org/abs/2005.11279, 2020. |
| [32] | Ngondiep, E., “A novel three-level time-split MacCormack scheme for two-dimensional evolutionary linear convection-diffusion-reaction equation with source term,” Int. J. Comput. Math., vol. 24, 2020. DOI: . · Zbl 1480.65219 |
| [33] | Ngondiep, E., “ A robust three-level time-split MacCormack scheme for solving two-dimensional unsteady convection-diffusion equation,” J. Appl. Comput. Mech., pp. 38,2020. |
| [34] | Mittal, R. C. and Tripathi, A., “ Numerical solutions of symmetric regularized long wave equations using B-collocation of cubic B-splines finite elements,” Int. J. Comput. Eng. Sci. Mech., vol. 16, no. 2, pp. 142-150, 2015. DOI: . · Zbl 07871408 |
| [35] | Dai, W. and Nassar, R., “Compact ADI method for solving partial differential equations,” Numer. Methods Partial Differential Eq., vol. 18, no. 2, pp. 129-142, 2002. DOI: . · Zbl 1004.65086 |
| [36] | Gartland, E. C. Jr. “Uniform high-order difference schemes for a singular value problem,” Math. Comp., vol. 48, no. 178, pp. 551-564, 1987. DOI: . · Zbl 0621.65088 |
| [37] | Gharehbaghi, A., “Third- and fifth-order finite volume schemes for advection-diffusion equation with variable coefficients in semi-infinite domain,” Water Environ. J., vol. 31, no. 2, pp. 184-193, 2017. DOI: . |
| [38] | Karaa, S. and Zhang, J., “Higher order ADI method for solving unsteady convection-diffusion problems,” J. Comput. Phys., vol. 198, no. 1, pp. 1-9, 2004. DOI: . · Zbl 1053.65067 |
| [39] | Karahan, H., “Implicit finite difference techniques for the advection-diffusion equation using spreadsheets,” Adv. Eng. Software, vol. 37, no. 9, pp. 601-608, 2006. DOI: . |
| [40] | Nazir, T., Abbas, M., Ismail, A. I. M., Majid, A. A., and Rashid, A., “The numerical solution of advection-diffusion problems using new cubic trigonometric B-splines approach,” Appl. Math. Model., vol. 40, no. 7-8, pp. 4586-4611, 2016. DOI: . · Zbl 1459.65201 |
| [41] | Ngondiep, E., “A two-level factored Crank-Nicolson method for two-dimensional nonstationary advection-diffusion equation with time dependent dispersion coefficients and source terms,” Adv. Appl. Math. Mech., pp. 20,2020. · Zbl 1488.65270 |
| [42] | Noye, B. J. and Tan, H. H., “ Finite difference methods for solving the two-dimensional advection-diffusion equation,” Int. J. Numer. Meth. Fluids, vol. 9, no. 1, pp. 75-98, 1989. DOI: . · Zbl 0658.76079 |
| [43] | Kalita, J. C., Dalal, D. C., and Dass, A. K., “A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients,” Int. J. Numer. Meth. Fluids, vol. 38, no. 12, pp. 1111-1131, 2002. DOI: . · Zbl 1094.76546 |
| [44] | Mittal, R. C. and Tripathi, A., “ Numerical solutions of two-dimensional unsteady convection-diffusion problems using modified bi-cubic B-spline finite elements,” Int. J. Comput. Math., vol. 94, no. 1, pp. 1-21, 2017. DOI: . · Zbl 1365.65229 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
