A numerical method of estimating blow-up rates for nonlinear evolution equations by using rescaling algorithm

Anada, Koichi; Ishiwata, Tetsuya; Ushijima, Takeo

doi:10.1007/s13160-017-0273-9

Cited by 10 publications

(10 citation statements)

References 15 publications

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“…[23,32,60]) or numerical simulations (e.g. [1,14,15,63]), many of which would be sink-type through related numerical simulations and computer-assisted proofs (e.g. [45,48,49]).…”

Section: Saddle-type Blow-up Solutionsmentioning

confidence: 99%

“…[23,32,51,60] from theoretical viewpoints and e.g. [1,5,14,15,63] from numerical viewpoints). Fundamental questions for blow-up problem are whether or not a solution blows up and, if it does, when, where, and how it blows up.…”

Section: Introductionmentioning

confidence: 99%

“…In the present study, invariant sets at infinity are restricted to equilibria. In order to extract asymptotic behavior 1 The above ideas themselves are applied to describe dynamics around bounded invariant sets in several preceding works (e.g. [20]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Lessard¹,

Matsue²,

Takayasu³

2021

Preprint

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In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial conditions are studied. Combining dynamical systems machinery (e.g. phase space compactifications, time-scale desingularizations of vector fields) with tools from computer-assisted proofs (e.g. rigorous integrators, parameterization method for invariant manifolds), these unstable blow-up solutions are obtained as trajectories on stable manifolds of hyperbolic (saddle) equilibria at infinity. In this process, important features are obtained: smooth dependence of blow-up times on initial points near blow-up, level set distribution of blow-up times, singular behavior of blow-up times on unstable blow-up solutions, organization of the phase space via separatrices (stable manifolds). In particular, we show that unstable blow-up solutions themselves, and solutions defined globally in time connected by those blow-up solutions can separate initial conditions into two regions where solution trajectories are either globally bounded or blow-up, no matter how the large initial points are.

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“…[23,32,60]) or numerical simulations (e.g. [1,14,15,63]), many of which would be sink-type through related numerical simulations and computer-assisted proofs (e.g. [45,48,49]).…”

Section: Saddle-type Blow-up Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Lessard¹,

Matsue²,

Takayasu³

2021

Preprint

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show abstract

“…Later, assuming that the initial data satisfy (6.1), Cho [10] and Lin [18] used the algorithm proposed in [8] to compute a blow-up solution and showed that the blow-up rate for the numerical solution is also of Type I. On the other hand, Anada et al [2] used the rescaling algorithm of [3] to compute blow-up solutions and their blow-up rates. They showed that Type II blow-up for several model problems can be reproduced by their algorithm.…”

Section: Computation Of a Minimal Blow-up Solutionmentioning

confidence: 99%

Finite difference schemes for an axisymmetric nonlinear heat equation with blow-up

Cho¹,

Okamoto²

2020

etna

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We study finite difference schemes for axisymmetric blow-up solutions of a nonlinear heat equation in higher spatial dimensions. The phenomenology of blow-up in higher-dimensional space is much more complex than that in one space dimension. To obtain a more complete picture for such phenomena from computational results, it is useful to know the technical details of the numerical schemes for higher spatial dimensions. Since first-order differentiation appears in the differential equation, we pay special attention to it. A sufficient condition for stability is derived. In addition to the convergence of the numerical blow-up time, certain blow-up behaviors, such as blow-up sets and blow-up in the L p -norm, are taken into consideration. It is sometimes experienced that a certain property of solutions of a partial differential equation may be lost by a faithfully constructed convergent numerical scheme. The phenomenon of one-point blow-up is a typical example in the numerical analysis of blow-up problems. We prove that our scheme can preserve such a property. It is also remarkable that the L p -norm (1 ≤ p < ∞) of the solution of the nonlinear heat equation may blow up simultaneously with the L ∞ -norm or remains bounded in [0, T ), where T denotes the blow-up time of the L ∞ -norm. We propose a systematic way to compute numerical evidence of the L p -norm blow-up. The computational results are also analyzed. Moreover, we prove an abstract theorem which shows the relationship between the numerical L p -norm blow-up and the exact L p -norm blow-up. Numerical examples for higher-dimensional blow-up solutions are presented and discussed.

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“…Yuan and Hu [12] proposed a new three-term conjugate gradient algorithm under the Yuan-Wei-Lu line search technique to solve large-scale unconstrained optimization problems and nonlinear equations. e numerical method is used to solve systems of equations in paper [13].…”

Section: Introductionmentioning

confidence: 99%

A Velocity-Combined Local Best Particle Swarm Optimization Algorithm for Nonlinear Equations

Lian

Wang

Chen

2020

Mathematical Problems in Engineering

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Many people use traditional methods such as quasi-Newton method and Gauss–Newton-based BFGS to solve nonlinear equations. In this paper, we present an improved particle swarm optimization algorithm to solve nonlinear equations. The novel algorithm introduces the historical and local optimum information of particles to update a particle’s velocity. Five sets of typical nonlinear equations are employed to test the quality and reliability of the novel algorithm search comparing with the PSO algorithm. Numerical results show that the proposed method is effective for the given test problems. The new algorithm can be used as a new tool to solve nonlinear equations, continuous function optimization, etc., and the combinatorial optimization problem. The global convergence of the given method is established.

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A numerical method of estimating blow-up rates for nonlinear evolution equations by using rescaling algorithm

Cited by 10 publications

References 15 publications

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

Finite difference schemes for an axisymmetric nonlinear heat equation with blow-up

A Velocity-Combined Local Best Particle Swarm Optimization Algorithm for Nonlinear Equations

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