A Posteriori Error Estimation and Adaptive Algorithm for Singularly Perturbed Reaction-Diffusion Equations
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摘要: 研究了一类奇异摄动半线性反应扩散方程的自适应网格方法.在任意非均匀网格上建立迎风有限差分离散格式,并推导出离散格式的后验误差界,然后用该误差界设计自适应网格移动算法.数值实验结果证明了所提出的自适应网格方法的有效性.Abstract: An adaptive grid method for singularly perturbed semilinear reactiondiffusion equations was studied. The equation was discretized by means of the upwind finite difference scheme on an arbitrary nonuniform mesh. A posteriori error estimation of the presented numerical scheme was built. In turn, the posteriori error bound was derived, and the adaptive grid generation algorithm was designed. Numerical experiments prove the effectiveness of the proposed adaptive grid method.
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[1] 史娟荣, 莫嘉琪. 一类奇异摄动燃烧模型的渐近解[J]. 应用数学和力学, 2016,37(7): 691-698.(SHI Juanrong, MO Jiaqi. Asymptotic solutions to a class of singular perturbation burning models[J]. Applied Mathematics and Mechanics,2016,37(7): 691-698.(in Chinese)) [2] VULANOVIC R, TEOFANOV L. A uniform numerical method for semilinear reaction-diffusion problems with a boundary turning point[J]. Numerical Algorithms,2010,54: 431-444. [3] GENG F, QIAN S, LI S. Numerical solutions of singularly perturbed convection-diffusion problems[J]. International Journal of Numerical Methods for Heat & Fluid Flow,2014,24(6): 1268-1274. [4] RAMOS H, VIGO-AGUIAR J, NATESAN S, et al. Numerical solution of nonlinear singularly perturbed problems on nonuniform meshes by using a non-standard algorithm[J]. Journal of Mathematical Chemistry,2010,48(1): 38-54. [5] ADLER J, MACLACHLAN S, MADDEN N. A first-order system Petrov-Galerkin discretization for a reaction-diffusion problem on a fitted mesh[J]. IMA Journal of Numerical Analysis,2015,36(3): 1-29. [6] BRDAR M, ZARIN H. A singularly perturbed problem with two parameters on a Bakhvalov-type mesh[J]. Journal of Computational and Applied Mathematics,2016,292: 307-319. [7] 蔡新, 蔡丹琳, 吴瑞潜, 等. 奇异摄动反应扩散问题的高阶不等距计算方法[J]. 应用数学和力学, 2009,30(2): 171-178.(CAI Xin, CAI Danlin, WU Ruiqian, et al. High accurate non-equidistant method for singular perturbation reaction-diffusion problem[J]. Applied Mathematics and Mechanics,2009,30(2): 171-178.(in Chinese)) [8] 邵文婷, 郑烁宇. 奇异摄动内层问题的宽度估计及其数值求解[J]. 计算机工程与应用, 2020,56(4): 44-49.(SHAO Wenting, ZHENG Shuoyu. Width estimation of singular perturbed interior layer problem and its numerical solution[J]. Computer Engineering and Applications,2020,56(4): 44-49.(in Chinese)) [9] Jiang W A, Luo S K. A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems[J]. Nonlinear Dynamics, 2012, 67(1): 475-482. [10] LIU L B, CHEN Y P. A robust adaptive grid method for a system of two singularly perturbed convection-diffusion equations with weak coupling[J]. Journal of Scientific Computing,2014,61(1): 1-16. [11] 毛志, 刘利斌. 一类奇异摄动问题的自适应移动网格算法[J]. 湘潭大学学报(自然科学版), 2019,41(1): 64-71.(MAO Zhi, LIU Libin. An adaptive moving grid algorithm for a system of singularly perturbed problems[J]. Journal of Xiangtan University (Natural Science Edition),2019,41(1): 64-71.(in Chinese)) [12] CHADHA N M, KOPTEVA N. A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction-diffusion problem[J]. IMA Journal of Numerical Analysis,2011,31(1): 188-211. [13] LINB T, RADOJEV G, ZARIN H. Approximation of singularly perturbed reaction-diffusion problems by quadratic C 1-splines[J]. Numerical Algorithms,2012,61: 35-55. [14] LINB T, RADOJEV G. Robust a posteriori error bounds for spline collocation applied to singularly perturbed reaction-diffusion problems[J]. Electronic Transactions on Numerical Analysis,2016,45: 342-353. [15] LINB T. A posteriori error estimation for arbitrary order fem applied to singularly perturbed one-dimensional reaction-diffusion problems[J]. Applications of Mathematics,2014,59(3): 241-256. [16] KOPTEVA N. Maximum norm a posteriori error estimates for a 1D singularly perturbed semilinear reaction-diffusion problem[J]. IMA Journal of Numerical Analysis,2006,27(3): 1-14. [17] KOPTEVA N, MADDENB N, STYNES M. Grid equidistribution for reaction-diffusion problems in one dimension[J]. Numerical Algorithms,2005,40(3): 305-322. [18] RAO S C S, KUMAR S, KUMAR M. A parameter-uniform B-spline collocation method for singularly perturbed semilinear reaction-diffusion problems[J]. Journal of Optimization Theory and Applications,2010,146: 795-809.
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