A Comparative Study of the Numerical Approximations of the Quenching Time for a Nonlinear Reaction-Diffusion Equation

Year 2020, Volume: 3 Issue: 2, 144 - 152, 15.12.2020

Frederick Jones He Yang

https://doi.org/10.33401/fujma.755721

Abstract

In this paper, we study the numerical methods for solving a nonlinear reaction-diffusion model for the polarization phenomena in ionic conductors. In particular, we propose three types of numerical methods, including the finite difference, cubic B-spline collocation, and local discontinuous Galerkin method, to approximate the quenching time of the model. We prove the conservation properties for all three numerical methods and compare their numerical performance.

Keywords

Cubic B-spline collocation method , Finite difference method , Local discontinuous Galerkin method , Reaction-diffusion equation , Quenching time

References

• [1] B. Selcuk, N. Ozalp, Quenching behavior of semilinear heat equations with singular boundary conditions, Electron. J. Differ. Eq., 311 (2015), 1-13.
• [2] H. Kawarada, On solutions of initial-boundary problem for ut = uxx + 1 1􀀀u , Publ. RIMS, Kyoto Univ., 10 (1975), 729-736.
• [3] C. S. Chou, W. Sun, Y. Xing, H. Yang, Local discontinuous Galerkin methods for the Khokhlov-Zabolotskaya-Kuznetzov equation, J. Sci. Comput., 73 (2017), 593-616.
• [4] H. Yang, High-order energy and linear momentum conserving methods for the Klein-Gordon equation, Math., (2018), Article ID 200, 17 pages.
• [5] H. Yang, Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein-Gordon-Schrödinger equations, J. Appl. Math. Comput., 62 (2020), 377-424.
• [6] H. Yang, Optimal error estimate of a decoupled conservative local discontinuous Galerkin method for the Klein-Gordon-Schr¨odinger equations, J. Korean Soc. Ind. Appl. Math., 24 (2020), 39-78.
• [7] R. C. Mittal, R. K. Jain, Cubic B-splines collocation method for solving nonlinear parabolic partial differential equations with Neumann boundary conditions, Commun. Nonlinear. Sci., 17 (2012), 4616-4625.
• [8] S. B. G. Karakoç, Y. Uçar, N. Yagmurluğ, Numerical solutions of the MRLW equation by cubic B-spline Galerkin finite element method, Kuwait J. Sci., 42 (2015), 141-159.
• [9] H. Zeybek, S. B. G. Karakoc¸, A numerical investigation of the GRLW equation using lumped Galerkin approach with cubic B-spline, SpringerPlus, (2016), Article ID 199, 17 pages.
• [10] S. K. Bhowmik, S. B. G. Karakoc¸, Numerical approximation of the generalized regularized long wave equation using Petrov-Galerkin finite element method, Numer. Methods Partial Differ. Equ., 35 (2019), 2236-2257.
• [11] S. B. G. Karakoç, S. K. Bhowmik, Galerkin finite element solution for Benjamin-Bona-Mahony-Burgers equation with cubic B-splines, Comput. Math. Appl., 77 (2019), 1917-1932.
• [12] X. Ding, Q.-J. Meng, L.-P. Yin, Discrete-time orthogonal spline collocation method for one-dimensional sine-Gordon equation, Discrete Dyn. Nat. Soc., (2015), Article ID 206264, 8 pages.
• [13] L. Guo, Y. Yang, Positivity preserving high-order local discontinuous Galerkin method for parabolic equations with blow-up solutions, J. Comput. Phys., 289 (2015), 181-195.
• [14] L. Shao, X. Feng, Y. He, The local discontinuous Galerkin finite element method for Burger’s equation, Math. Comput. Model., 54 (2011), 2943-2954.