PARAMETER-UNIFORM HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION PROBLEMS WITH DISCONTINUOUS INITIAL CONDITIONS
DOI:
https://doi.org/10.61841/za1yz276Keywords:
Singularly perturbed parabolic convection-diffusion problem, finite difference scheme, Discontinuous initial condition, Interior and Boundary Layer, Piecewise-uniform S-type meshAbstract
This article presents a parameter-uniform hybrid numerical technique for singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions (DIC). It utilizes the classical backward-Euler technique for time discretization and a hybrid finite difference scheme ( which is a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions(generated by the DIC)) for spatial discretization. The scheme produces parameter-uniform numerical approximations on a piecewise- uniform Shishkin mesh. When the perturbation parameter ε(0 < ε ≪ 1) is small, it becomes difficult to solve these problems using the classical numerical methods (standard central difference or a standard upwind scheme) on uniform meshes because discontinuous initial conditions frequently appear in the solutions of this class of problems. The method is shown to converge uniformly in the discrete supremum with nearly second-order spatial accuracy. The suggested method is subjected to a stability study, and parameter-uniform error estimates are generated. In order to support the theoretical findings, numerical results are presented.
References
L. BOBISUD, Parabolic equations with a small parameter and discontinuous data, Journal of Mathematical Analysis and Applications, 26 (1969), pp. 208–220.
Z. CEN, A hybrid difference scheme for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient, Applied mathematics and computation, 169 (2005), pp. 689–699.
C. CLAVERO, J. GRACIA, AND J. JORGE, High-order numerical methods for one-dimensional parabolic singularly perturbed problems with regular layers, Numerical Methods for Partial Differential Equations: An International Journal, 21 (2005), pp. 149–169.
C. CLAVERO, J. L. GRACIA, G. I. SHISHKIN, AND L. P. SHISHKINA, An efficient numerical scheme for 1d parabolic singularly perturbed problems with an interior and boundary layers, Journal of Computational and Applied Mathematics, 318 (2017), pp. 634–645.
C. CLAVERO, J. L. GRACIA, AND M. STYNES, A simpler analysis of a hybrid numerical method for time-dependent convection–diffusion problems, Journal of computational and ap- plied mathematics, 235 (2011), pp. 5240–5248.
C. CLAVERO, J. JORGE, AND F. LISBONA, A uniformly convergent scheme on a nonuni-form mesh for convection–diffusion parabolic problems, Journal of Computational and Applied Mathematics, 154 (2003), pp. 415–429.
R. DUNNE AND E. O’RIORDAN, Interior layers arising in linear singularly perturbed differ- ential equations with discontinuous coefficients, dcu, maths. dept, Preprint series, (2007).
P. FARRELL, A. HEGARTY, J. MILLER, E. O’RIORDAN, AND G. SHISHKIN, Singularly per- turbed convection–diffusion problems with boundary and weak interior layers, Journal of Com- putational and Applied Mathematics, 166 (2004), pp. 133–151.
P. FARRELL, A. HEGARTY, J. M. MILLER, E. O’RIORDAN, AND G. I. SHISHKIN, Robust computational techniques for boundary layers, CRC Press, 2000.
J. GRACIA AND E. O’RIORDAN, Numerical approximation of solution derivatives of singularly perturbed parabolic problems of convection-diffusion type, Mathematics of Computation, 85 (2016), pp. 581–599.
J. L. GRACIA AND E. O’RIORDAN, Parameter-uniform numerical methods for singularly per- turbed parabolic problems with incompatible boundary-initial data, Applied Numerical Mathe- matics, 146 (2019), pp. 436–451.
Parameter-uniform approximations for a singularly perturbed convection-diffusion prob- lem with a discontinuous initial condition, Applied Numerical Mathematics, 162 (2021), pp. 106–123.
J. L. GRACIA AND E. O’RIORDAN, Singularly perturbed reaction–diffusion problems with
discontinuities in the initial and/or the boundary data, Journal of Computational and Applied Mathematics, 370 (2020), p. 112638.
J. L. GRACIA AND E. O’RIORDAN, Numerical approximations to a singularly perturbed convection-diffusion problem with a discontinuous initial condition, Numerical Algorithms, 88 (2021), pp. 1851–1873.
P. HEMKER AND G. SHISHKIN, Approximation of parabolic pdes with a discontinuous initial condition, East-West Journal of Numerical mathematics, 1 (1993), pp. 287–302.
Discrete approximation of singularly perturbed parabolic pdes with a discontinuous initial condition, in Bail VI Proceedings, 1994, pp. 3–4.
M. K. KADALBAJOO AND A. AWASTHI, The midpoint upwind finite difference scheme for time-dependent singularly perturbed convection-diffusion equations on non-uniform mesh, International Journal for Computational Methods in Engineering Science and Mechanics, 12 (2011), pp. 150–159.
M. K. KADALBAJOO AND V. GUPTA, A brief survey on numerical methods for solving singu- larly perturbed problems, Applied mathematics and computation, 217 (2010), pp. 3641–3716.
A. MAJUMDAR AND S. NATESAN, A higher-order hybrid numerical scheme for singularly perturbed convection-diffusion problem with boundary and weak interior layers, International Journal of Mathematical Modelling and Numerical Optimisation, 10 (2020), pp. 68–101.
J. MILLER, E. O’RIORDAN, G. SHISHKIN, AND R. B. KELLOGG, Fitted numerical methods for singular perturbation problems, SIAM Review, 39 (1997), pp. 535–537.
K. MUKHERJEE, Parameter-uniform improved hybrid numerical scheme for singularly per-turbed problems with interior layers, Mathematical Modelling and Analysis, 23 (2018), pp. 167– 189.
K. MUKHERJEE AND S. NATESAN, Parameter-uniform hybrid numerical scheme for time-dependent convection-dominated initial-boundary-value problems, Computing, 84 (2009), pp. 209–230.
ε-uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with nterior layers, Numerical Algorithms, 58 (2011), pp. 103–141.
E. O’RIORDAN AND J. QUINN, A linearised singularly perturbed convection–diffusion problem with an interior layer, Applied Numerical Mathematics, 98 (2015), pp. 1–17.
E. O’RIORDAN AND G. SHISHKIN, Singularly perturbed parabolic problems with non-smooth data, Journal of computational and applied mathematics, 166 (2004), pp. 233–245.
T. PRABHA, M. CHANDRU, AND V. SHANTHI, Hybrid difference scheme for singularly per- turbed reaction-convection-diffusion problem with boundary and interior layers, Applied Math- ematics and Computation, 314 (2017), pp. 237–256.
H.-G. ROOS, M. STYNES, AND L. TOBISKA, Robust numerical methods for singularly per-turbed differential equations: convection-diffusion-reaction and flow problems, vol. 24, Springer Science & Business Media, 2008.
S. K. SAHOO AND V. GUPTA, Higher order robust numerical computation for singularly perturbed problem involving discontinuous convective and source term, Mathematical Methods in the Applied Sciences, 45 (2022), pp. 4876–4898.
G. I. SHISHKIN, Grid approximation of singularly perturbed parabolic convection-diffusion equa- tions with a piecewise-smooth initial condition, Zhurnal Vychislitel’noi Matematiki i Matem- aticheskoi Fiziki, 46 (2006), pp. 52–76.
D. SODANO, Richardson extrapolation technique for singularly perturbed parabolic convection- diffusion problems with a discontinuous initial condition, Asian Journal of Pure and Applied Mathematics, 6(1), 1–25, (2024).
An upwind finite difference method to singularly perturbed parabolic convection-diffusion problems with discontinuous initial conditions on a piecewise-uniform mesh, Mendeley Data, V2 (2024).
F. WONDIMU GELU AND G. F. DURESSA, A novel numerical approach for singularly perturbed parabolic convection-diffusion problems on layer-adapted meshes, Research in Mathematics, 9 (2022), p. 2020400.
N. S. YADAV AND K. MUKHERJEE, Uniformly convergent new hybrid numerical method for singularly perturbed parabolic problems with interior layers, International Journal of Applied and Computational Mathematics, 6 (2020), p. 53.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Desta Sodano Sheiso (Author)
This work is licensed under a Creative Commons Attribution 4.0 International License.
You are free to:
- Share — copy and redistribute the material in any medium or format for any purpose, even commercially.
- Adapt — remix, transform, and build upon the material for any purpose, even commercially.
- The licensor cannot revoke these freedoms as long as you follow the license terms.
Under the following terms:
- Attribution — You must give appropriate credit , provide a link to the license, and indicate if changes were made . You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
Notices:
You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation .
No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material.
