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Michela Procesi - One of the best experts on this subject based on the ideXlab platform.

  • reducibility of first order linear operators on tori via moser s theorem
    Journal of Functional Analysis, 2019
    Co-Authors: Roberto Feola, Filippo Giuliani, Riccardo Montalto, Michela Procesi
    Abstract:

    Abstract In this paper we prove reducibility of a class of first order, quasi-linear, quasi-periodic time dependent PDEs on the torus ∂ t u + ζ ⋅ ∂ x u + a ( ω t , x ) ⋅ ∂ x u = 0 , x ∈ T d , ζ ∈ R d , ω ∈ R ν . As a consequence we deduce a stability result on the associated Cauchy problem in Sobolev spaces. By the identification between first order operators and Vector Fields this problem can be formulated as the problem of finding a change of coordinates which conjugates a weakly perturbed Constant Vector Field on T ν + d to a Constant diophantine flow. For this purpose we generalize Moser's straightening theorem: considering smooth perturbations we prove that the corresponding straightening torus diffeomorphism is smooth, under the assumption that the perturbation is small only in some given Sobolev norm and that the initial frequency belongs to some Cantor-like set. In view of applications in KAM theory for PDEs we provide also tame estimates on the change of variables.

  • reducibility of first order linear operators on tori via moser s theorem
    arXiv: Analysis of PDEs, 2018
    Co-Authors: Roberto Feola, Filippo Giuliani, Riccardo Montalto, Michela Procesi
    Abstract:

    In this paper we prove reducibility of classes of linear first order operators on tori by applying a generalization of Moser's theorem on straightening of Vector Fields on a torus. We consider Vector Fields which are a $C^\infty$ perturbations of a Constant Vector Field, and prove that they are conjugated --by a $C^\infty$ torus diffeomorphism-- to a Constant diophantine flow, provided that the perturbation is small in some given $H^{s_1}$ norm and that the initial frequency is in some Cantor-like set. Actually in the classical results of this type the regularity of the change of coordinates which straightens the perturbed Vector Field coincides with the class of regularity in which the perturbation is required to be small. This improvement is achieved thanks to ideas and techniques coming from the Nash-Moser theory.

Phillip Colella - One of the best experts on this subject based on the ideXlab platform.

  • a freestream preserving fourth order finite volume method in mapped coordinates with adaptive mesh refinement
    Computers & Fluids, 2015
    Co-Authors: Stephen M. Guzik, Peter Mccorquodale, Landon D Owen, Phillip Colella
    Abstract:

    Abstract A fourth-order accurate finite-volume method is presented for solving time-dependent hyperbolic systems of conservation laws on mapped grids that are adaptively refined in space and time. Novel considerations for formulating the semi-discrete system of equations in computational space are combined with detailed mechanisms for accommodating the adapting grids. These considerations ensure that conservation is maintained and that the divergence of a Constant Vector Field is always zero (freestream-preservation property). The solution in time is advanced with a fourth-order Runge–Kutta method. A series of tests verifies that the expected accuracy is achieved in smooth flows and the solution of a Mach reflection problem demonstrates the effectiveness of the algorithm in resolving strong discontinuities.

  • A Freestream-Preserving High-Order Finite-Volume Method for Mapped Grids with Adaptive-Mesh Refinement
    50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2012
    Co-Authors: Stephen M. Guzik, Peter Mccorquodale, Phillip Colella
    Abstract:

    A fourth-order accurate finite-volume method is presented for solving time-dependent hyperbolic systems of conservation laws on mapped grids that are adaptively refined in space and time. Novel considerations for formulating the semi-discrete system of equations in computational space combined with detailed mechanisms for accommodating the adapting grids ensure that conservation is maintained and that the divergence of a Constant Vector Field is always zero (freestream-preservation property). Advancement in time is achieved with a fourth-order Runge-Kutta method.

Christophe Prange - One of the best experts on this subject based on the ideXlab platform.

  • Asymptotic Analysis of Boundary Layer Correctors in Periodic Homogenization
    SIAM Journal on Mathematical Analysis, 2013
    Co-Authors: Christophe Prange
    Abstract:

    This paper is devoted to the asymptotic analysis of boundary layers in periodic homogenization. We investigate the behavior of the boundary layer corrector, defined in the half-space $\Omega_{n,a}:=\{y\cdot n-a>0\}$, far away from the boundary and prove the convergence toward a Constant Vector Field, the boundary layer tail. This problem happens to depend strongly on the way the boundary $\partial\Omega_{n,a}$ intersects the underlying microstructure. Our study complements the previous results obtained on the one hand for $n\in\mathbb R\mathbb Q^d$ and on the other hand for $n\notin\mathbb R\mathbb Q^d$ satisfying a small divisors assumption. We tackle the case of arbitrary $n\notin\mathbb R\mathbb Q^d$ using ergodicity of the boundary layer along $\partial\Omega_{n,a}$. Moreover, we get an asymptotic expansion of Poisson's kernel $P=P(y,\tilde{y})$, associated to the elliptic operator $-\nabla\cdot A(y)\nabla\cdot$ and $\Omega_{n,a}$, for $|y-\tilde{y}|\rightarrow\infty$. Finally, we show that, in genera...

  • Asymptotic analysis of boundary layer correctors in periodic homogenization
    arXiv: Analysis of PDEs, 2012
    Co-Authors: Christophe Prange
    Abstract:

    This paper is devoted to the asymptotic analysis of boundary layers in periodic homogenization. We investigate the behaviour of the boundary layer corrector, defined in the half-space $\Omega_{n,a}:=\{y\cdot n-a>0\}$, far away from the boundary and prove the convergence towards a Constant Vector Field, the boundary layer tail. This problem happens to depend strongly on the way the boundary $\partial\Omega_{n,a}$ intersects the underlying microstructure. Our study complements the previous results obtained on the one hand for $n\in\mathbb R\mathbb Q^d$, and on the other hand for $n\notin\mathbb R\mathbb Q^d$ satisfying a small divisors assumption. We tackle the case of arbitrary $n\notin\mathbb R\mathbb Q^d$ using ergodicity of the boundary layer along $\partial\Omega_{n,a}$. Moreover, we get an asymptotic expansion of Poisson's kernel $P=P(y,\tilde{y})$, associated to the elliptic operator $-\nabla\cdot A(y)\nabla\cdot$ and $\Omega_{n,a}$, for $|y-\tilde{y}|\rightarrow\infty$. Finally, we show that, in general, convergence towards the boundary layer tail can be arbitrarily slow, which makes the general case very different from the rational or the small divisors one.

Roberto Feola - One of the best experts on this subject based on the ideXlab platform.

  • reducibility of first order linear operators on tori via moser s theorem
    Journal of Functional Analysis, 2019
    Co-Authors: Roberto Feola, Filippo Giuliani, Riccardo Montalto, Michela Procesi
    Abstract:

    Abstract In this paper we prove reducibility of a class of first order, quasi-linear, quasi-periodic time dependent PDEs on the torus ∂ t u + ζ ⋅ ∂ x u + a ( ω t , x ) ⋅ ∂ x u = 0 , x ∈ T d , ζ ∈ R d , ω ∈ R ν . As a consequence we deduce a stability result on the associated Cauchy problem in Sobolev spaces. By the identification between first order operators and Vector Fields this problem can be formulated as the problem of finding a change of coordinates which conjugates a weakly perturbed Constant Vector Field on T ν + d to a Constant diophantine flow. For this purpose we generalize Moser's straightening theorem: considering smooth perturbations we prove that the corresponding straightening torus diffeomorphism is smooth, under the assumption that the perturbation is small only in some given Sobolev norm and that the initial frequency belongs to some Cantor-like set. In view of applications in KAM theory for PDEs we provide also tame estimates on the change of variables.

  • reducibility of first order linear operators on tori via moser s theorem
    arXiv: Analysis of PDEs, 2018
    Co-Authors: Roberto Feola, Filippo Giuliani, Riccardo Montalto, Michela Procesi
    Abstract:

    In this paper we prove reducibility of classes of linear first order operators on tori by applying a generalization of Moser's theorem on straightening of Vector Fields on a torus. We consider Vector Fields which are a $C^\infty$ perturbations of a Constant Vector Field, and prove that they are conjugated --by a $C^\infty$ torus diffeomorphism-- to a Constant diophantine flow, provided that the perturbation is small in some given $H^{s_1}$ norm and that the initial frequency is in some Cantor-like set. Actually in the classical results of this type the regularity of the change of coordinates which straightens the perturbed Vector Field coincides with the class of regularity in which the perturbation is required to be small. This improvement is achieved thanks to ideas and techniques coming from the Nash-Moser theory.

Johansson, Peter H. - One of the best experts on this subject based on the ideXlab platform.

  • Barycentric interpolation on Riemannian and semi-Riemannian spaces
    'Oxford University Press (OUP)', 2019
    Co-Authors: Pihajoki Pauli, Mannerkoski Matias, Johansson, Peter H.
    Abstract:

    Interpolation of data represented in curvilinear coordinates and possibly having some non-trivial, typically Riemannian or semi-Riemannian geometry is a ubiquitous task in all of physics. In this work, we present a covariant generalization of the barycentric coordinates and the barycentric interpolation method for Riemannian and semi-Riemannian spaces of arbitrary dimension. We show that our new method preserves the linear accuracy property of barycentric interpolation in a coordinate-invariant sense. In addition, we show how the method can be used to interpolate constrained quantities so that the given constraint is automatically respected. We showcase the method with two astrophysics related examples situated in the curved Kerr space-time. The first problem is interpolating a locally Constant Vector Field, in which case curvature effects are expected to be maximally important. The second example is a general relativistic magnetohydrodynamics simulation of a turbulent accretion flow around a black hole, wherein high intrinsic variability is expected to be at least as important as curvature effects.Peer reviewe

  • Barycentric interpolation on Riemannian and semi-Riemannian spaces
    'Oxford University Press (OUP)', 2019
    Co-Authors: Pihajoki Pauli, Mannerkoski Matias, Johansson, Peter H.
    Abstract:

    Interpolation of data represented in curvilinear coordinates and possibly having some non-trivial, typically Riemannian or semi-Riemannian geometry is an ubiquitous task in all of physics. In this work we present a covariant generalization of the barycentric coordinates and the barycentric interpolation method for Riemannian and semi-Riemannian spaces of arbitrary dimension. We show that our new method preserves the linear accuracy property of barycentric interpolation in a coordinate-invariant sense. In addition, we show how the method can be used to interpolate constrained quantities so that the given constraint is automatically respected. We showcase the method with two astrophysics related examples situated in the curved Kerr spacetime. The first problem is interpolating a locally Constant Vector Field, in which case curvature effects are expected to be maximally important. The second example is a General Relativistic Magnetohydrodynamics simulation of a turbulent accretion flow around a black hole, wherein high intrinsic variability is expected to be at least as important as curvature effects.Comment: 10 pages, 3 figures. Revised version with small additions, accepted to MNRA