Adaptive mesh based Haar wavelet approximation for a singularly perturbed integral boundary problem

Iranian Journal of Numerical Analysis and Optimization

مقاله 6، دوره 14، Issue 4 - شماره پیاپی 31، اسفند 2024، صفحه 1140-1167   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2024.87550.1421

نویسندگان

P. Shukla¹؛ S. Saini^{* 2}؛ V. Devi³

¹Department of Engineering Sciences, Indian Institute of Information Technology and Management Gwalior, Gwalior, Madhya Pradesh, 474015, India.

²Department of Mathematics, Vellore Institute of Technology, Vellore, Tamilnadu, 632014, India.

³Department of Mathematics, Bhakt Darshan Govt. P.G. College, Pauri Garhwal, Uttrakhand, 246193, India.

چکیده

This research presents a nonuniform Haar wavelet approximation of a singularly perturbed convection-diffusion problem with an integral boundary. The problem is discretized by approximating the second derivative of the solution with the help of a nonuniform Haar wavelets basis on an arbitrary nonuniform mesh. To resolve the multiscale nature of the problem, adaptive mesh is generated using the equidistribution principle. This approach allows for the dynamical adjustment of the mesh based on the solution’s behavior without requiring any information about the solution. The combination of nonuniform wavelet approximation and the use of adaptive mesh leads to improved accuracy, efficiency, and the ability to handle the multiscale behavior of the solution. On the adaptive mesh rigorous error analysis is performed showing that the proposed method is a second-order parameter uniformly convergent. Numerical stability and computational efficiency are validated in various tables and plots for numerical results obtained by the implementation of two test examples. 

کلیدواژه‌ها

Singular perturbation problems؛ Adaptive mesh؛ Nonuniform Haar wavelet؛ Parameter uniform convergence

مراجع

[1] Ahsan, M., Bohner, M., Ullah, A., Khan, A.A. and Ahmad, S. A haar wavelet multiresolution collocation method for singularly perturbed differential equations with integral boundary conditions, Math. Comput. Simul. 204 (2023), 166–180.
[2] Bakhvalov, N.S. Towards optimization of methods for solving boundary value problems in the presence of boundary layers, Zh. Vychisl. Mat. Mat. Fiz. 9, (1969), 841–859. 

[3] Beckett, G. and Mackenzie, J.A. Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem, Appl. Numer. Math. 35(2) (2000), 87–109.
[4] Beckett, G. and Mackenzie, J.A. On a uniformly accurate finite difference approximation of a singularly perturbed reaction–diffusion problem using grid equidistribution, J. Comput. Appl. Math. 131(1-2) (2001), 381–405.
[5] Boor, C. Good approximation by splines with variable knots, In: Spline functions and approximation theory: Proceedings of the Symposium Held at the University of Alberta, Edmonton May 29 to June 1, 1972, Springer, (1973) 57–72.
[6] Cakir, M. and Amiraliyev, G.M. A finite difference method for the singularly perturbed problem with nonlocal boundary condition, Appl. Math. Comput. 160(2) (2005), 539–549.
[7] Chadha, N.M. and Kopteva, N. A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction–diffusion problem, IMA J. Numer. Anal. 31(1) (2011), 188–211.
[8] Das, P. and Mehrmann, V. Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters, BIT Numer. Math. 56(1) (2016), 51–76.
[9] Das, P. and Natesan, S. Higher-order parameter uniform convergent schemes for robin type reaction-diffusion problems using adaptively generated grid, Int. J. Comput. Methods, 9(04) (2012), 1250052.
[10] Das, P., Rana, S. and Vigo-Aguiar, J. Higher order accurate approximations on equidistributed meshes for boundary layer originated mixed type reaction diffusion systems with multiple scale nature, Appl. Numer. Math. 148 (2020), 79–97.
[11] Debela, H.G. and Duressa, G.F. Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition, Int. J. Numer. Methods Fluids, 92(12) (2020), 1914– 1926. 

[12] Dubeau, F., Elmejdani, S. and Ksantini, R. Non-uniform haar wavelets, Appl. Math. Comput. 159(3) (2004), 675–693.
[13] Goswami, J.C. and Chan, A.K. Fundamentals of wavelets: Theory, algorithms, and applications, John Wiley & Sons, 2011. 
[14] Gowrisankar, S. and Natesan, S. The parameter uniform numerical method for singularly perturbed parabolic reaction–diffusion problems on equidistributed grids, Appl. Math. 26(11) (2013), 1053–1060.
[15] Gowrisankar, S. and Natesan, S. Robust numerical scheme for singularly perturbed convection–diffusion parabolic initial–boundary-value problems on equidistributed grids, Comput. Phys. Commun. 185(7) (2014), 2008–2019.
[16] Haar, A. Zur Theorie der Orthogonalen Funktionensysteme. GeorgAugust-Universitat, Gottingen, 1909.
[17] Hirsch, C. Numerical computation of internal and external flows: The fundamentals of computational fluid dynamics, Elsevier, 2007.
[18] Huang, W. and Russell, R.D. Adaptive moving mesh methods, Springer, 2010.
[19] Kopteva, N., Madden, N. and Stynes, M. Grid equidistribution for reaction–diffusion problems in one dimension, Numer. Algorithms, 40(3) (2005), 305–322.
[20] Kopteva, N., and Stynes, M.A robust adaptive method for a quasi-linear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal. 39(4) (2001), 1446–1467.
[21] Kumar, S. and Kumar, M. Parameter-robust numerical method for a system of singularly perturbed initial value problems, Numer. Algorithms, 59(2) (2012), 185–195.
[22] Kumar, S., Kumar, S. and Sumit, A posteriori error estimation for quasilinear singularly perturbed problems with integral boundary condition, Numer. Algorithms, 89(2) (2022), 791–809. 

[23] Kumar, S., Sumit and Vigo-Aguiar, J. A parameter-uniform grid equidistribution method for singularly perturbed degenerate parabolic convection–diffusion problems, J. Comput. Appl. Math. 404 (2020), 113273.
[24] Lepik, U. Numerical solution of differential equations using haar wavelets. Math. Comput. Simul. 68(2), (2005) 127–143.
[25] Lepik, Ü. and Hein, H. Haar wavelets with applications, Springer, New York, 2014.
[26] LeVeque, R.J. Finite difference methods for ordinary and partial differential equations: Steady-state and time-dependent problems, SIAM, 2007.
[27] Liu, L.-B., Long, G. and Cen, Z. A robust adaptive grid method for a nonlinear singularly perturbed differential equation with integral boundary condition, Numer. Algorithms, 83(2) (2020), 719–739.
[28] Mackenzie, J.A. Uniform convergence analysis of an upwind finitedifference approximation of a convection-diffusion boundary value problem on an adaptive grid, IMA J. Numer. Anal. 19(2) (1999), 233–249.
[29] Mallat, S. A wavelet tour of signal processing, Elsevier 1999.
[30] Melenk, J.M. Hp-finite element methods for singular perturbations, Springer, 2002.
[31] Miller, J.J.H., O’Riordan, E. and Shishkin, G.I. Fitted numerical methods for singular perturbation problems: Error estimates in the maximum norm for linear problems in one and two dimensions, World Scientific, 2012.
[32] Pandit, S. and Kumar, M. Haar wavelet approach for numerical solution of two parameters singularly perturbed boundary value problems, Appl. Math. Inf. Sci. 8(6) (2014), 2965.
[33] Podila, P.C. and Sundrani, V. A non-uniform haar wavelet method for a singularly perturbed convection–diffusion type problem with integral boundary condition on an exponentially graded mesh. Comput. Appl. Math. 42(5) (2023), 216.
[34] Qiu, Y. and Sloan, D.M. Analysis of difference approximations to a singularly perturbed two-point boundary value problem on an adaptively generated grid, J. Comput. Appl. Math. 101(1-2) (1999), 1–25.
[35] Roos, H.G., Stynes, M. and Tobiska, L. Numerical methods for singularly perturbed differential equations, Springer, 1996.
[36] Sah, K.K. and Gowrisankar, S. Richardson extrapolation technique on a modified graded mesh for singularly perturbed parabolic convectiondiffusion problems, Iran. J. Numer. Anal. Optim. 14(1) (2024), 219–264.
[37] Shishkin, G.I. A difference scheme for a singularly perturbed equation of parabolic type with discontinuous boundary conditions, USSR Comput. Math. Math. Phys. 28(6) (1988), 32–41.
[38] Wichailukkana, N., Novaprateep, B. and Boonyasiriwat, C. A convergence analysis of the numerical solution of boundary-value problems by using two-dimensional haar wavelets, Sci. Asia, 42 (2016), 346–355.
[39] Woldaregay, M.M. and Duressa, G.F. Exponentially fitted tension spline method for singularly perturbed differential difference equations, Iran. J. Numer. Anal. Optim. 11(2) (2021), 261–282.
[40] Xu, X., Huang, W., Russell, R.D. and Williams, J.F. Convergence of de Boor’s algorithm for the generation of equidistributing meshes, IMA J. Numer. Anal. 31(2) (2011), 580–596.
[41] Yapman, O. and Amiraliyev, G.M. A novel second-order fitted computational method for a singularly perturbed Volterra integro-differential equation, Int. J. Comput. Math. 97(6) (2019), 1–10.