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J. Ramos (2006)
Linearly-implicit, approximate factorization, exponential methods for multi-dimensional reaction-diffusion equationsAppl. Math. Comput., 174
A. Campo, A. Salazar, D. Celentano, M. Raydan (2014)
Accurate analytical/numerical solution of the heat conduction equationInternational Journal of Numerical Methods for Heat & Fluid Flow, 24
P. Danumjaya, A. Pani (2005)
Orthogonal cubic spline collocation method for the extended Fisher-Kolmogorov equationJournal of Computational and Applied Mathematics, 174
Idiris Dağ, O. Ersoy (2016)
The exponential cubic B-spline algorithm for Fisher equationChaos Solitons & Fractals, 86
W. Briley, H. Mcdonald (1980)
On the structure and use of linearized block implicit schemesJournal of Computational Physics, 34
J. Ramos (2017)
A conservative, spatially continuous method of lines for one-dimensional reaction-diffusion equationsInternational Journal of Numerical Methods for Heat & Fluid Flow, 27
P. Houwen, B. Sommeijer (2001)
Approximate factorization for time-dependent partial differential equationsJournal of Computational and Applied Mathematics, 128
R. Beam, R. Warming (1977)
An Implicit Factored Scheme for the Compressible Navier-Stokes EquationsAIAA Journal, 16
R. Beam, R. Warming (1976)
An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. [application to Eulerian gasdynamic equations
J. Ramos (1997)
Linearization methods for reaction-diffusion equations : 1-D problemsApplied Mathematics and Computation, 88
O. Makinde (2007)
Thermal stability of a reactive viscous flow through a porous‐saturated pipeInternational Journal of Numerical Methods for Heat & Fluid Flow, 17
G. Buscaglia, R. Enrique (1996)
Numerical solution of the thermally‐assisted diffusion of hydrogen in zirconium alloys considering hysteresis and finite‐rate kineticsInternational Journal of Numerical Methods for Heat & Fluid Flow, 6
J. Ramos (2005)
Exponential methods for one-dimensional reaction-diffusion equationsAppl. Math. Comput., 170
R. Mittal, G. Arora (2010)
Efficient numerical solution of Fisher's equation by using B-spline methodInternational Journal of Computer Mathematics, 87
Norio Kikuchi, Jozef Kačur (2003)
Convergence of Rothe's Method in Hölder SpacesApplications of Mathematics, 48
E. Steinthorsson, T. Shih (1993)
Methods for Reducing Approximate-Factorization Errors in Two- and Three-Factored SchemesSIAM J. Sci. Comput., 14
J. Ramos (2007)
Numerical methods for nonlinear second-order hyperbolic partial differential equations. II - Rothe's techniques for 1-D problemsAppl. Math. Comput., 190
Erich Rothe (1930)
Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler RandwertaufgabenMathematische Annalen, 102
Lindsey Yue, Leanne Reich, T. Simon, Roman Bader, W. Lipiński (2017)
Progress in thermal transport modeling of carbonate-based reacting systemsInternational Journal of Numerical Methods for Heat & Fluid Flow, 27
J. Ramos (1997)
Linearization methods for reaction-diffusion equations: Multidimensional problemsApplied Mathematics and Computation, 88
D. Bahuguna (1989)
Application of rothe’s method to semilinear hyperbolic equationsApplicable Analysis, 33
P. Ramachandran (2006)
Method of lines with boundary elements for 1‐D transient diffusion‐reaction problemsNumerical Methods for Partial Differential Equations, 22
W. Allee, E. Bowen (1932)
Studies in animal aggregations: Mass protection against colloidal silver among goldfishesJournal of Experimental Zoology, 61
J. Zeldowitsch, D. Frank-Kamenetzki (1988)
A Theory of Thermal Propagation of Flame
P. Verhulst, P. Verhulst
Notice sur la loi que la population pursuit dans son accroissement
J. Ramos (2006)
Iterative and non-iterative, full and approximate factorization methods for multidimensional reaction-diffusion equationsAppl. Math. Comput., 174
The purpose of this paper is to develop a new transversal method of lines for one-dimensional reaction–diffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging.Design/methodology/approachA conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers.FindingsThe transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reaction–diffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions.Originality/valueA new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone.
International Journal of Numerical Methods for Heat and Fluid Flow – Emerald Publishing
Published: Oct 17, 2019
Keywords: Conservation; Finite; Transversal method of lines; Piecewise–analytical solution; Finite–volume technique; Nonlinear reaction–diffusion equations
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