Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach

Marica, Aurora; Zuazua, Enrique

doi:10.1007/s10208-014-9232-x

Cited by 14 publications

(26 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This was done in [19] for the discrete wave equation in an infinite lattice (i.e. with no boundary) and, more recently, this theory was extended to non-uniform meshes obtained by means of a C 2 diffeomorphism of the uniform mesh (jh) j∈Z in [20].…”

Section: A Dynamical System Approachmentioning

confidence: 99%

“…To be more precise, in [20], solutions of (1.5) were constructed, with frequencies of the order of 1/h, localized around the rays of geometric optics given by the projection on the physical space of the bicharacteristic curves provided by the Hamiltonian 10) or equivalently…”

Section: A Dynamical System Approachmentioning

confidence: 99%

“…More recently, in [20], the problem of the propagation of high-frequency numerical solutions was considered in non-uniform meshes built through a diffeomorphic deformation of a uniform one. Using microlocal tools, the notions of numerical symbol and bicharacteristic rays were introduced and it was shown both analytically and through numerical simulations, that high-frequency numerical wave packets follow, in space-time, the path indicated by the bicharacteristics.…”

Section: Introductionmentioning

confidence: 99%

“…On a non-uniform mesh, the situation is even more intricate. As indicated in [27] and [20], for some numerical grids, the rays of Geometric Optics may present internal reflections generated by the mesh.…”

Section: Introductionmentioning

confidence: 99%

“…The analysis in [20] shows that the rays corresponding to frequencies of the order of 1/h obey the following Hamiltonian flow:…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis

2015

Self Cite

View full text Add to dashboard Cite

We build non-uniform numerical meshes for the finite difference and finite element approximations of the 1d wave equation, ensuring that all numerical solutions reach the boundary, as continuous solutions do, in the sense that the full discrete energy can be observed by means of boundary measurements, uniformly with respect to the mesh-size. The construction of the nonuniform mesh is achieved by means of a concave diffeomorphic transformation of a uniform grid into a non-uniform one, making the mesh finer and finer when approaching the right boundary. For uniform meshes it is known that high-frequency numerical wave packets propagate very slowly without never getting to the boundary. Our results show that this pathology can be avoided by taking suitable non-uniform meshes. This also allows to build convergent numerical algorithms for the approximation of boundary controls of the wave equation.

show abstract

Section: A Dynamical System Approachmentioning

confidence: 99%

Section: A Dynamical System Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The analysis in [20] shows that the rays corresponding to frequencies of the order of 1/h obey the following Hamiltonian flow:…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis

2015

Self Cite

View full text Add to dashboard Cite

show abstract

Boundary Stabilization of Numerical Approximations of the 1-D Variable Coefficients Wave Equation: A Numerical Viscosity Approach

Marica

Zuazua

2014

Lecture Notes in Computational Science and Engineering

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper, we consider the boundary stabilization problem associated to the 1 − d wave equation with both variable density and diffusion coefficients and to its finite difference semi-discretizations. It is well-known that, for the finite difference semi-discretization of the constant coefficients wave equation on uniform meshes [24] or on some non-uniform meshes [20], the discrete decay rate fails to be uniform with respect to the mesh-size parameter. We prove that, under suitable regularity assumptions on the coefficients and after adding an appropriate artificial viscosity to the numerical scheme, the decay rate is uniform as the mesh-size tends to zero. This extends previous results in [24] on the constant coefficient wave equation. The methodology of proof consists in applying the classical multiplier technique at the discrete level, with a multiplier adapted to the variable coefficients. PreliminariesLet us consider the following initial boundary value problem associated to the 1 − d wave equation with variable coefficients and with a damping mechanism acting on the right endpoint of the space interval:Here we have taken the dissipative boundary condition σ(1)v x (1, t) + v t (1, t) = 0, but similar results can be proved for more general feedback terms as, for instance,with k > 0 and k = 1. In fact, problem (1.1) with the second dissipative condition can be reduced to (1.1) by scaling the time variable.The energy corresponding to the solution of (1.1),obeys the following dissipation law:When the variable coefficients ρ and σ belong to the BV (0, 1) class of functions with bounded variation, the following stabilization property holds, ensuring the exponential decay of the energy E ρ,σ (v(·, t), v t (·, t))The main idea of this work originated when the first author has visited BCAM in December 2012.

show abstract

Observability Inequalities for Hermite Bi-cubic Orthogonal Spline Collocation Methods of 2-D Integro-differential Equations in the Square Domains

2020

Appl Math Optim

View full text Add to dashboard Cite

Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach

Cited by 14 publications

References 46 publications

Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis

Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis

Boundary Stabilization of Numerical Approximations of the 1-D Variable Coefficients Wave Equation: A Numerical Viscosity Approach

Observability Inequalities for Hermite Bi-cubic Orthogonal Spline Collocation Methods of 2-D Integro-differential Equations in the Square Domains

Contact Info

Product

Resources

About