Numerical existence and uniqueness proof for solutions of nonlinear hyperbolic equations

Minamoto, Teruya

doi:10.1016/s0377-0427(00)00563-x

Cited by 5 publications

(9 citation statements)

References 10 publications

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“…Although an application result [70] is ordinary differential equations, his procedure describes on general Banach spaces and operators, and has a common point from our method. We also cite some estimations of L −1 for parabolic and hyperbolic equations by Minamoto and the first authors of the present paper [16,17,[19][20][21].…”

Section: In-linz -Infinite Dimensional Newton Methods With Linearized mentioning

confidence: 99%

“…where G is defined in (21) and • E is the Euclidean norm. Then a different verification condition from (24) in Theorem 2 for finite dimensional part can be obtained.…”

Section: Improvement Of Computation For Finite Dimensional Partmentioning

confidence: 99%

“…We have some alternatives for the choice for w h . For example, if we want to have the same Newton matrix G in (21), namely, to get g (w h )w = f (u h )w, we can take w h = 0 in (50) or…”

Section: Remark 11mentioning

confidence: 99%

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Numerical verification methods for solutions of semilinear elliptic boundary value problems

Nakao

Watanabe

2011

NOLTA

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This article describes a survey on numerical verification methods for second-order semilinear elliptic boundary value problems introduced by authors and their colleagues. Here "numerical verification" means a computer-assisted numerical method for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. Three kinds of methods based on the infinite dimensional fixed-point theorems using Newton-like operator will be presented. In each verification method, a projection into a finite dimensional subspace and constructive error estimates of the projection play an important and essential role. It is shown that these methods are really useful for actual problems by illustrating numerical examples.

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Section: In-linz -Infinite Dimensional Newton Methods With Linearized mentioning

confidence: 99%

“…where G is defined in (21) and • E is the Euclidean norm. Then a different verification condition from (24) in Theorem 2 for finite dimensional part can be obtained.…”

Section: Improvement Of Computation For Finite Dimensional Partmentioning

confidence: 99%

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Numerical verification methods for solutions of semilinear elliptic boundary value problems

Nakao

Watanabe

2011

NOLTA

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“…The propagation of velocity is also changed by the discretization. Furthermore, in the viewpoint of verified numerical computations, there are few results [14,15,16,20] for initial-boundary value problems of hyperbolic PDEs. Verified numerical computations for PDEs have been established by Nakao [19] and Plum [24] independently.…”

Section: Introductionmentioning

confidence: 99%

Rigorous numerical computations for 1D advection equations with variable coefficients

Takayasu

Yoon

Endo

2019

Japan J. Indust. Appl. Math.

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This paper provides a methodology of verified computing for solutions to 1-dimensional advection equations with variable coefficients. The advection equation is typical partial differential equations (PDEs) of hyperbolic type. There are few results of verified numerical computations to initial-boundary value problems of hyperbolic PDEs. Our methodology is based on the spectral method and semigroup theory. The provided method in this paper is regarded as an efficient application of semigroup theory in a sequence space associated with the Fourier series of unknown functions. This is a foundational approach of verified numerical computations for hyperbolic PDEs. Numerical examples show that the rigorous error estimate showing the well-posedness of the exact solution is given with high accuracy and high speed.

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“…Simple elliptic problems with various boundary conditions have been studied extensively, and have been put into formulations that satisfy certain conditions. More complex problems have been studied case by case, like the Navier-Stokes equations, Burger's equations, and other linear/nonlinear stationary or time-dependent problems, and/or are still active research topics [131].…”

Section: Existence and Uniqueness Of A Solutionmentioning

confidence: 99%

Applied Mathematics of Space-time & Space+time: Problems in General Relativity and Cosmology

Cattoën¹

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<p>Cosmography is the part of cosmology that proceeds by making minimal dynamic assumptions. That is, one does not assume the Friedmann equations (Einstein equations) unless and until absolutely necessary. On the other hand, cosmodynamics is the part of cosmology that relates the geometry to the density and pressure using the Friedmann equations. In both frameworks, we consider the amount of information and the nature of the constraints we can obtain from the Hubble flow in a FLRW universe. Indeed, the cosmological parameters contained in the Hubble relation between distance and redshift provide information on the behaviour of the universe (expansion, acceleration etc...). In the first framework, it is possible to concentrate more directly on the observational situation in a model-independent manner. We perform a number of inter-related cosmographic fits to supernova datasets, and pay particular attention to the extent to which the choice of distance scale and manner of representing the redshift scale affect the cosmological parameters. In the second framework, we use the class of w-parameter models which has become increasingly popular in the last decade. We explore the extent to which a constraint on the w-parameter leads to useful and non-trivial constraints on the Hubble flow in terms of cosmological parameters H(z), density p(z), density parameter O(z), distance scales d(z), and lookback time T(z). On another front, Numerical Relativity has experienced many breakthroughs since 2005, with full inspiral-merger-ringdown simulations now possible. One of the main goals is to provide very accurate templates of gravitational waves for ground-based and space-based interferometers. We explore the potential of a very recent and accurate numerical method, the Spectral Element Method (SEM), for Numerical Relativity, by treating a singular Schwarszchild black hole evolution as a test case. Spectral elements combine the theory of spectral and pseudo-spectral methods for high order polynomials and the variational formulation of finite elements and the associated geometric flexibility. We use the BSSN formulation of the Einstein equations with the method of the moving punctures. After applying the variational formulation to the BSSN system, we present several possible weak forms of this system and its spectral element discretization in space. We use a Runge-Kutta fourth order time discretization. The accuracy of high order methods can deteriorate in the presence of discontinuities or sharp gradients. We show that we can treat the element that contains the puncture with a filtering method to avoid artificial and spurious oscillations. These might form and propagate into the domain coming from discontinuous initial data from the BSSN system.</p>

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Numerical existence and uniqueness proof for solutions of nonlinear hyperbolic equations

Cited by 5 publications

References 10 publications

Numerical verification methods for solutions of semilinear elliptic boundary value problems

Numerical verification methods for solutions of semilinear elliptic boundary value problems

Rigorous numerical computations for 1D advection equations with variable coefficients

Applied Mathematics of Space-time & Space+time: Problems in General Relativity and Cosmology

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