On the blow-up time convergence of semidiscretizations of reaction-diffusion equations

Abia, L. M.; López-Marcos, J.C.; Martínez, Juan Pablo Torres

doi:10.1016/s0168-9274(97)00105-0

Cited by 49 publications

(49 citation statements)

References 10 publications

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“…Nakagawa [35] In this subsection, we prove the convergence of blow-up time for Ndimensional semi-discrete problem, using our general theory, under the assumption that the L°° norm blows up. Our result contains the result of [1] as a special case (see Remark 3.3).…”

Section: So the Inequality (Ii) Holds With G(s) = ~~S~a-supporting

confidence: 53%

“…This assumption, however, does not always hold [17]. Recently, Abia, Lopez-Marcos and Martinez [1] considered a one dimensional semi-discrete problem for this equation. Here a semi-discrete problem means the system of ordinary differential equations which is obtained by discretizing the space variable (but not the time variable) via finite difference approximation.…”

Section: §1 Introduction and Main Resultsmentioning

confidence: 99%

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On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations

Ushijima¹

2000

Publ. Res. Inst. Math. Sci.

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There are many nonlinear parabolic equations whose solutions develop singularity in finite time, say T. In many cases, a certain norm of the solution tends to infinity as time t approaches T. Such a phenomenon is called blow-up, and T is called the blow-up time. This paper is concerned with approximation of blow-up phenomena in nonlinear parabolic equations. For numerical computations or for other reasons, we often have to deal with approximate equations. But it is usually not at all clear if such wild phenomena as blow-up can be well reflected in the approximate equations. In this paper we present rather simple but general sufficient conditions which guarantee that the blow-up time for the original equation is well approximated by that for approximate equations. We will then apply our result to various examples. §1. Introduction and Main ResultsThere is a large number of nonlinear partial differential equations of parabolic type whose solution for a given initial data cannot be extended globally in time and develop a singularity in finite time, say T. Such a phenomenon is called blow-up and T is called the blow-up time. In many cases, some norms of blow-up solutions tend to infinity as t approaches T.Blow-up is known to occur in various equations including those in combustion theory, chemotaxis models and equations describing crystalline formation involving curvature-driven motion (see [9], [39]). The study of blow-up phenomena is not only interesting from the mathematical point of view but also Communicated by H. Okamoto, March 9, 2000. Revised April 20, 2000. 1991 Generally speaking, it is very difficult to numerically simulate blow-up phenomena accurately. For one thing, one has to deal with numerical data that grow indefinitely as the blow-up time approaches. This is obviously not an easy task. Secondly -and more importantly -it is not at all clear if features of such a wild phenomenon as blow-up can be well reflected in the discretized equation which approximate the original equation. Most of the standard error estimates become useless as t approaches the blow-up time.There have been some attempts to establish numerical methods to capture blow-up phenomena. For example, the rescaling algorithm by Berger and Kohn [10] and the method of MMPDE [12] by Budd et al can observe the shape of blow-up solutions near the blow-up time. There are also some works concerning approximation of blow-up time. Some authors studied numerical blow-up time and its convergence. As for the semilinear parabolic equation ut = u xx + u p , Nakagawa [35] (p = 2) and Chen [13] (p > 1) studied a finite difference scheme for this equation and proved the convergence of the blow-up time of approximate solutions to that of the real solution. They assumed that L q (q = I or 2) norm of the solution diverges at blow-up time. This assumption, however, does not always hold [17]. Recently, Abia, Lopez-Marcos and Martinez [1] considered a one dimensional semi-discrete problem for this equation. Here a semi-discrete problem means the syste...

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Section: So the Inequality (Ii) Holds With G(s) = ~~S~a-supporting

confidence: 53%

Section: §1 Introduction and Main Resultsmentioning

confidence: 99%

On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations

Ushijima¹

2000

Publ. Res. Inst. Math. Sci.

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“…the blow-up time, useful for the point of view of applications. Similar works that investigate the convergence of a numerical solution during blow-up have been done, for a parabolic problem, in [2,3,11]. In our case, for example, for f being an increasing function a discontinuity in the initial data may cause blow-up of the solution [15].…”

Section: Introductionmentioning

confidence: 52%

Numerical Solution of a Nonlocal Problem Modelling Ohmic Heating of Foods

Nikolopoulos

2009

Comput. Methods Appl. Math.

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-An upwind and a Lax-Wendroff scheme are introduced for the solution of a one dimensional non-local problem modelling Ohmic heating of Foods. The schemes are studied regarding their consistency, stability and the rate of convergence for the cases that the problem attains a global solution in time. A high resolution scheme is also introduced and it is shown that it is total-variation-stable. Finally some numerical experiments are presented in support of the theoretical results.

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“…We consider the following initial-boundary value problem for a semilinear heat equation of the form u t (x, t) = u xx (x, t) + γf (u(0, t)), (x, t) ∈ (−l, l) × (0, T ), (1) u(−l, t) = 0, u(l, t) = 0, t ∈ (0, T ), (2) u(x, 0) = u 0 (x) ≥ 0, x ∈ (−l, l), (3) which models the temperature distribution of a large number of physical phenomena from physics, chemistry and biology. The particularity of the problem described in (1)- (3) is that it represents a model in physical phenomena where the reaction is driven by the temperature at a single site.…”

Section: Introductionmentioning

confidence: 99%

“…The particularity of the problem described in (1)- (3) is that it represents a model in physical phenomena where the reaction is driven by the temperature at a single site. This kind of phenomena is observed in biological systems and in chemical reaction diffusion processes in which the reaction takes place only at some local sites.…”

Section: Introductionmentioning

confidence: 99%

Blow-Up for Discretization of a Localized Semilinear Heat Equation

Boni¹,

Nachid²,

Nabongo³

2010

Annals of the Alexandru Ioan Cuza University - Mathematics

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Abstract. This paper concerns the study of the numerical approximation for the following initial-boundary value problem:and γ is a positive parameter. Under some assumptions, we prove that the solution of a discrete form of the above problem blows up in a finite time and estimate its numerical blow-up time. We also show that the numerical blow-up time in certain cases converges to the real one when the mesh size tends to zero. Finally, we give some numerical experiments to illustrate our analysis.

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On the blow-up time convergence of semidiscretizations of reaction-diffusion equations

Cited by 49 publications

References 10 publications

On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations

On the Approximation of Blow-up Time for Solutions of Nonlinear Parabolic Equations

Numerical Solution of a Nonlocal Problem Modelling Ohmic Heating of Foods

Blow-Up for Discretization of a Localized Semilinear Heat Equation

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