کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4627489 1631810 2014 14 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Fully implicit time-stepping schemes and non-linear solvers for systems of reaction–diffusion equations
ترجمه فارسی عنوان
طرح های کاملا گام به گام و غیر خطی برای سیستم های واکنش معادلات دیفرانسیل
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات ریاضیات کاربردی
چکیده انگلیسی

In this article we present robust, efficient and accurate fully implicit time-stepping schemes and nonlinear solvers for systems of reaction–diffusion equations. The applications of reaction–diffusion systems is abundant in the literature, from modelling pattern formation in developmental biology to cancer research, wound healing, tissue and bone regeneration and cell motility. Therefore, it is crucial that modellers, analysts and biologists are able to solve accurately and efficiently systems of highly nonlinear parabolic partial differential equations on complex stationary and sometimes continuously evolving domains and surfaces. The main contribution of our paper is the study of fully implicit schemes by use of the Newton method and the Picard iteration applied to the backward Euler, the Crank–Nicolson (and its modifications) and the fractional-step θθ methods. Our results conclude that the fractional-step θθ method coupled with a single Newton iteration at each timestep is as accurate as the fully adaptive Newton method; and both outperform the Picard iteration. In particular, the results strongly support the observation that a single Newton iteration is sufficient to yield as accurate results as those obtained by use of an adaptive Newton method. This is particularly advantageous when solving highly complex nonlinear partial differential equations on evolving domains and surfaces. To validate our theoretical results, various appropriate numerical experiments are exhibited on stationary planary domains and in the bulk of stationary surfaces.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Applied Mathematics and Computation - Volume 244, 1 October 2014, Pages 361–374
نویسندگان
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