2,376 research outputs found
Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
We consider two physically and mathematically distinct regularization
mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the
combination of diffusion and dispersion are known to give rise to monotonic and
oscillatory traveling waves that approximate shock waves. The zero-diffusion
limits of these traveling waves are dynamically expanding dispersive shock
waves (DSWs). A richer set of wave solutions can be found when the flux is
non-convex. This review compares the structure of solutions of Riemann problems
for a conservation law with non-convex, cubic flux regularized by two different
mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation;
and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation.
In the first case, the possible dynamics involve two qualitatively different
types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the
second case, in addition to RWs, there are traveling wave solutions
approximating both classical (Lax) and non-classical (undercompressive) shock
waves. Despite the singular nature of the zero-diffusion limit and rather
differing analytical approaches employed in the descriptions of dispersive and
diffusive-dispersive regularization, the resulting comparison of the two cases
reveals a number of striking parallels. In contrast to the case of convex flux,
the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is
identified as an undercompressive DSW. Other prominent features, such as
shock-rarefactions, also find their purely dispersive counterparts involving
special contact DSWs, which exhibit features analogous to contact
discontinuities. This review describes an important link between two major
areas of applied mathematics, hyperbolic conservation laws and nonlinear
dispersive waves.Comment: Revision from v2; 57 pages, 19 figure
Asymptotic-induced numerical methods for conservation laws
Asymptotic-induced methods are presented for the numerical solution of hyperbolic conservation laws with or without viscosity. The methods consist of multiple stages. The first stage is to obtain a first approximation by using a first-order method, such as the Godunov scheme. Subsequent stages of the method involve solving internal-layer problems identified by using techniques derived via asymptotics. Finally, a residual correction increases the accuracy of the scheme. The method is derived and justified with singular perturbation techniques
Acoustic Energy and Momentum in a Moving Medium
By exploiting the mathematical analogy between the propagation of sound in a
non-homogeneous potential flow and the propagation of a scalar field in a
background gravitational field, various wave ``energy'' and wave ``momentum''
conservation laws are established in a systematic manner. In particular the
acoustic energy conservation law due to Blokhintsev appears as the result of
the conservation of a mixed co- and contravariant energy-momentum tensor, while
the exchange of relative energy between the wave and the mean flow mediated by
the radiation stress tensor, first noted by Longuet-Higgins and Stewart in the
context of ocean waves, appears as the covariant conservation of the doubly
contravariant form of the same energy-momentum tensor.Comment: 25 Pages, Late
Sensitivity analysis of 1-d steady forced scalar conservation laws
We analyze 1 - d forced steady state scalar conservation laws. We first show the existence and uniqueness of entropy solutions as limits as t→ ∞ of the corresponding solutions of the scalar evolutionary hyperbolic conservation law. We then linearize the steady state equation with respect to perturbations of the forcing term. This leads to a linear first order differential equation with, possibly, discontinuous coefficients. We show the existence and uniqueness of solutions in the context of duality solutions. We also show that this system corresponds to the steady state version of the linearized evolutionary hyperbolic conservation law. This analysis leads us to the study of the sensitivity of the shock location with respect to variations of the forcing term, an issue that is relevant in applications to optimal control and parameter identification problems
Cascades and Dissipative Anomalies in Compressible Fluid Turbulence
We investigate dissipative anomalies in a turbulent fluid governed by the
compressible Navier-Stokes equation. We follow an exact approach pioneered by
Onsager, which we explain as a non-perturbative application of the principle of
renormalization-group invariance. In the limit of high Reynolds and P\'eclet
numbers, the flow realizations are found to be described as distributional or
"coarse-grained" solutions of the compressible Euler equations, with standard
conservation laws broken by turbulent anomalies. The anomalous dissipation of
kinetic energy is shown to be due not only to local cascade, but also to a
distinct mechanism called pressure-work defect. Irreversible heating in
stationary, planar shocks with an ideal-gas equation of state exemplifies the
second mechanism. Entropy conservation anomalies are also found to occur by two
mechanisms: an anomalous input of negative entropy (negentropy) by
pressure-work and a cascade of negentropy to small scales. We derive
"4/5th-law"-type expressions for the anomalies, which allow us to characterize
the singularities (structure-function scaling exponents) required to sustain
the cascades. We compare our approach with alternative theories and empirical
evidence. It is argued that the "Big Power-Law in the Sky" observed in electron
density scintillations in the interstellar medium is a manifestation of a
forward negentropy cascade, or an inverse cascade of usual thermodynamic
entropy
An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks.
We introduce a new optimization strategy to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the 1 − d inviscid Burgers equation. We first prove the existence of minimizers and, by a -convergence argument, the convergence of discrete minima obtained by means of numerical approximation schemes satisfying the so called onesided Lipschitz condition (OSLC). Then we address the problem of developing efficient descent algorithms. We first consider and compare the existing two possible approaches: the so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. When optimal solutions have shock discontinuities, both approaches produce highly oscillating minimizing sequences and the effective descent rate is very weak. As a solution we propose a new method, that we shall call alternating descent method, that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions. This method distinguishes and alternates the descent directions that move the shock and those that perturb the profile of the solution away of it producing very efficient and fast descent algorithms
Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence
We develop first-principles theory of relativistic fluid turbulence at high
Reynolds and P\'eclet numbers. We follow an exact approach pioneered by
Onsager, which we explain as a non-perturbative application of the principle of
renormalization-group invariance. We obtain results very similar to those for
non-relativistic turbulence, with hydrodynamic fields in the inertial-range
described as distributional or "coarse-grained" solutions of the relativistic
Euler equations. These solutions do not, however, satisfy the naive
conservation-laws of smooth Euler solutions but are afflicted with dissipative
anomalies in the balance equations of internal energy and entropy. The
anomalies are shown to be possible by exactly two mechanisms, local cascade and
pressure-work defect. We derive "4/5th-law"-type expressions for the anomalies,
which allow us to characterize the singularities (structure-function scaling
exponents) required for their non-vanishing. We also investigate the Lorentz
covariance of the inertial-range fluxes, which we find is broken by our
coarse-graining regularization but which is restored in the limit that the
regularization is removed, similar to relativistic lattice quantum field
theory. In the formal limit as speed of light goes to infinity, we recover the
results of previous non-relativistic theory. In particular, anomalous heat
input to relativistic internal energy coincides in that limit with anomalous
dissipation of non-relativistic kinetic energy
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