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This article is devoted to the explanation of the onset of localization and the formation of shear bands in high strain-rate plasticity of metals. We employ the Arrhenius constitutive model and show Hadamard instability for the linearized problem. For the nonlinear model, using an asymptotic procedure motivated by the theory of relaxation and the… (More)
The Upwind Source at Interface (U.S.I.) method for hyperbolic conservation laws with source term introduced by Perthame and Simeoni is essentially first order accurate. Under appropriate hypotheses of consistency on the finite volume discretization of the source term, we prove L p-error estimates , 1 ≤ p <+∞, in the case of a uniform spatial mesh, for which… (More)
We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock… (More)
Hyperbolic conservation laws with source terms arise in many applications, especially as a model for geophysical flows because of the gravity, and their numerical approximation leads to specific difficulties. In the context of finite volume schemes, many authors have proposed to Upwind Sources at Interfaces, i.e. the " U. S. I. " method, while a… (More)
We extend the framework of the finite volume method to dispersive unidi-rectional water wave propagation in one space dimension. In particular we consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants… (More)
We consider finite volume relaxation schemes for multidimensional scalar conservation laws. These schemes are constructed by appropriate dis-cretization of a relaxation system and it is shown to converge to the entropy solution of the conservation law with a rate of h 1/4 in L ∞ ([0, T ], L 1 loc (R d)) .
We develop a quantitative criterion determining the onset of localization and shear band formation at high-strain rate deformations of metals. We introduce an asymptotic procedure motivated by the theory of relaxation and the Chapman-Enskog expansion and derive an effective equation for the evolution of the strain rate, consisting of a second-order… (More)