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Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation
Author(s) -
Liao Wenyuan,
Yan Yulian
Publication year - 2011
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20589
Subject(s) - runge–kutta methods , mathematics , diagonal , ordinary differential equation , space (punctuation) , explicit and implicit methods , alternating direction implicit method , order (exchange) , mathematical analysis , numerical methods for ordinary differential equations , scheme (mathematics) , partial differential equation , compact finite difference , finite difference method , differential equation , geometry , collocation method , computer science , finance , economics , operating system
In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011