The Rothe-Newton approach to simulate the variable coefficient convection-diffusion equations

Document Type : Research Paper

Authors

1 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

2 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China

3 Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan

4 Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy

5 Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

Abstract

The current article presents a novel hybrid approach based on the Rothe time-marching algorithm and a spectral matrix collocation approach using the well-known Newton bases to deal with the spatial variable. Utilizing the Rothe approach converts the underlying convection-diffusion into initial-boundary value problems and then the Newton collocation method solves the continuous discretized time equation in each time step. The error analysis of the newly employed basis functions is established. Three numerical simulations are developed to show the accuracy and utility of the proposed hybrid strategy.
Applying the current study to other linear and nonlinear PDEs and high-order PDEs can be performed straightforwardly.

Keywords

References

[1] M. Ahmadinia, Z. Safari, S. Fouladi, Analysis of local discontinuous Galerkin method for time-space fractional convection-di usion equations, BIT Numer. Math. Vol. 58, no. 3, (2018) 533{554
[2] M. Ahmadinia, Z. Safari, Convergence analysis of a LDG method for tempered fractional convection-di usion equations, ESAIM: Math. Model. Numer. Anal. Vol. 54, no. 1, (2020) 59-78.
[3] C. Clavero, J.C. Jorge, F. Lisbona, A uniformly convergent scheme on a nonuniform mesh for convection-di usion parabolic problems, J. Comput. Appl. Math. Vol. 154, (2003) 415{429.
[4] S. Dhawan, S. Kapoor, S. Kumar, Numerical method for advection di usion equation using FEM and B-splines, J. Comput. Sci. Vol. 3, (2012) 429{437.
[5] A. Fahim, M. A. Fariborzi Araghi, Numerical solution of convection-di usion equations with memory term based on sinc method, Comput. Methods Di er. Equ. Vol. 6, no. 3,(2018) 380{395.
[6] N. Guglielmi, M. Lopez-Fernandez, G. Nino, Numerical inverse Laplace transform for convection-di usion equations, Math. Comp. Vol. 89, (2020) 1161{1191.
[7] M. Izadi, Split-step  nite di erence schemes for solving the nonlinear Fisher equation, J. Mahani Math. Res. Cent. Vol. 7, no. 1-2, (2018) 37{55.
[8] M. Izadi, Application of the Newton-Raphson method in a SDFEM for inviscid Burgers equation, Comput. Methods Di er. Equ. Vol. 8, no. 4, (2020) 708{732.
[9] M. Izadi, A second-order accurate  nite-di erence scheme for the classical Fisher-Kolmogorov-Petrovsky-Piscounov equation, J. Infor. Optim. Sci. Vol. 42, no. 2, (2021) 431{448.
[10] M. Izadi, A combined approximation method for nonlinear foam drainage equation, Sci. Iran. Vol. 29, no. 1, (2022) 70{78.
[11] M. Izadi, Two-stages explicit schemes based numerical approximations of convectiondi  usion equations, Int. J. Comput. Sci. Math. (2022) in press.
[12] M. Izadi, P. Roul, Spectral semi-discretization algorithm for a class of nonlinear parabolic PDEs with applications, Appl. Math. Comput. Vol. 429, (2022) Article ID 127226.
[13] M. Izadi, M.E. Samei, Time accurate solution to Benjamin-Bona-Mahony Burgers equation via Taylor-Boubaker series scheme, Bound. Value Probl. Vol. 2022, (2022) Article ID 17.
[14] M. Izadi, H.M. Srivastava, An optimized second order numerical scheme applied to the non-linear Fisher's reaction-di usion equation, J. Interdiscip. Math. Vol. 25, no. 2,(2022) 471{492.
[15] M. Izadi, S. Yuzbas, A hybrid approximation scheme for 1-D singularly perturbed parabolic convection-di usion problems, Math. Commun. Vol. 27, no. 1, (2022) 47{63.
[16] M. Izadi, S. Yuzbas, D. Baleanu, A Taylor-Chebyshev approximation technique to solve the 1D and 2D nonlinear Burgers equations, Math. Sci. (2021), https://doi.org/10.1007/s40096-021-00433-1,
[17] C.Y. Ku, J.E. Xiao, C.Y. Liu, Space-time radial basis function-based meshless approach for solving convection-di usion equations, Mathematics, Vol. 8 no. 10, (2020) Article ID 1735.
[18] N. Okhovati, M. Izadi, Numerical coupling of two scalar conservation laws by a RKDG method, J. Korean Soc. Ind. Appl. Math. Vol. 23, no. 3, (2019) 211{236.
[19] N. Okhovati, M. Izadi, A predictor-corrector scheme for conservation equations with discontinuous coecients, J. Math. Fund. Sci. Vol. 52, no. 3, (2020) 322-338.
[20] M. Li, Z. Zheng, An ecient multiscale-like multigrid computation for 2D convectiondi usion equations on nonuniform grids, Math. Methods Appl. Sci. Vol. 44, no. 4, (2021) 3214{3224.
[21] E. Rothe, Zweidimensionale parabolische randwertaufgaben als grenzfall eindimensionaler randwertaufgaben, Math. Ann. Vol. 102, no. 1, (1930) 650{670.
[22] H.S. Shekarabi, J. Rashidinia, Three level implicit tension spline scheme for solution of Convection-Reaction-Di usion equation, Ain Shams Eng. J. Vol. 9, (2018) 1601{1610.
[23] H. S. Shukla, M. Tamsir, An exponential cubic B-spline algorithm for multi-dimensional convection-di usion equations, Alexandria Eng. J. Vol. 57, no. 3, (2018) 1999{2006.
[24] H. M. Srivastava, H. Ahmad, I. Ahmad, P. Thounthong, M. N. Khan, Numerical simulation of 3-D fractional-order convection-di usion PDE by a local meshless method, Thermal Sci. Vol. 25 no. (1A), (2021) 347{358.
[25] H. M. Srivastava, H. I. Abdel-Gawad, K. M. Saad, Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-di usion model subjected to the Dirichlet conditions, Discrete Contin. Dyn. Syst. S Vol. 14 (2021), 3785{3801.
[26] G.W. Stewart, Afternotes on Numerical Analysis, SIAM, Vol. 49, 1996.
[27] J. Tan, An upwind  nite volume method for convection-di usion equations on rectangular mesh, Chaos Solit. Fract. Vol. 118, (2019) 159{165.
[28] V. M. Tripathi, H. M. Srivastava, H. Singh, C. Swarup, S. Aggarwal, Mathematical analysis of non-isothermal reaction-di usion models arising in spherical catalyst and spherical biocatalyst, Appl. Sci. Vol. 11 (2021), Article ID 10423.
[29] S. Yuzbas, N. Sahin, Numerical solutions of singularly perturbed one-dimensional parabolic convection-di usion problems by the Bessel collocation method, Appl. Math. Comput. Vol. 220, (2013) 305{315.
[30] F. Zhou, X. Xu. Numerical solution of the convection di usion equations by the second kind Chebyshev wavelets, Appl. Math. Comput. Vol. 247, (2014) 353{367.

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History

  • Receive Date: 14 May 2022
  • Accept Date: 21 May 2022

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