Entropy Solutions Research Articles

Convergence, bounded variation properties and Kruzhkov solution of a fully discrete Lagrangian–Eulerian scheme via weak asymptotic analysis for 1D hyperbolic problems

AbstractWe design and implement an effective fully discrete Lagrangian–Eulerian scheme for a class of scalar, local and nonlocal models, and systems of hyperbolic problems in 1D. We propose statements, via a weak asymptotic analysis, which include existence, uniqueness, regularity, and numerical approximations of entropy‐weak solutions computed with the scheme for the corresponding nonlinear initial value problem for the local scalar case. We study both convergence and weak bounded variation (BV) properties of the scheme to the entropy solution (for the local and scalar case) in the sense of Kruzhkov. The approach is based on the improved concept of no‐flow curves, as introduced by the authors, and we highlight the strengths of the method: (i) the scheme for systems of hyperbolic problems does not require computation of the eigenvalues (exact or approximate) either to the numerical flux function or the weak CFL stability condition (wCFL) and (ii) we prove the properties: positivity‐preserving, total variation nonincreasing, and maximum principle subject to the wCFL. We present numerical experiments to evaluate the shock capturing capabilities of the scheme in resolving the main features for hyperbolic problems: shock waves, rarefaction waves, contact discontinuities, positivity‐preserving properties, and nonlinear wave formations and interactions.

Self‐similar viscosity approach to the Riemann problem for a strictly hyperbolic system of conservation laws

Here, we study the Riemann problem for a strictly hyperbolic system of conservation laws, which occurs in gas dynamics and nonlinear elasticity. We establish the existence and uniqueness of the solution of Riemann problem containing delta shock wave by employing self‐similar vanishing viscosity approach. We prove that delta shock is stable under self‐similar viscosity perturbation, which ensures that delta shock wave is a unique entropy solution.

Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the “meaningful objects” are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting. It implies that a weak solution indeed satisfies the balance law. In fact, it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary. It should be emphasized that the weak solutions considered here need not be entropy solutions. Furthermore, the assumption imposed on the flux f(u) is quite minimal—just that it is locally bounded.

Analysis of an alternative Navier–Stokes system: Weak entropy solutions and a convergent numerical scheme

We consider an alternative Navier–Stokes model for compressible viscous ideal gases, originally proposed in [M. Svärd, A new Eulerian model for viscous and heat conducting compressible flows, Physica A 506 (2018) 350–375.] We derive a priori estimates that are sufficiently strong to support a weak entropy solution of the system. Guided by these estimates, we propose a finite-volume scheme, derive the analogous estimates and demonstrate grid convergence towards a weak entropy solution. Furthermore, this existence proof is valid for “large” initial data and no a priori assumptions on the solution are needed.

The one-sided lipschitz condition in the follow-the-leader approximation of scalar conservation laws

We consider the follow-the-leader particle approximation scheme for a [Formula: see text] scalar conservation law with non-negative compactly supported [Formula: see text] initial datum and with a [Formula: see text] concave flux, which is known to provide convergence towards the entropy solution [Formula: see text] to the corresponding Cauchy problem. We provide two novel contributions to this theory. First, we prove that the one-sided Lipschitz condition satisfied by the approximate density [Formula: see text] is a “discrete version of an entropy condition”; more precisely, under fairly general assumptions on [Formula: see text] (which imply concavity of [Formula: see text]) we prove that the continuum version [Formula: see text] of said condition allows to select a unique weak solution, despite [Formula: see text] is apparently weaker than the classical Oleinik–Hoff one-sided Lipschitz condition [Formula: see text]. Said result relies on an improved version of Hoff’s uniqueness. A byproduct of it is that the entropy condition is encoded in the particle scheme prior to the many-particle limit, which was never proven before. Second, we prove that in case [Formula: see text] the one-sided Lipschitz condition can be improved to a discrete version of the classical (and “sharp”) Oleinik–Hoff condition. In order to make the paper self-contained, we provide proofs (in some cases “alternative” ones) of all steps of the convergence of the particle scheme.

Simultaneous Development of Shocks and Cusps for 2D Euler with Azimuthal Symmetry from Smooth Data

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called $$C^{\frac{1}{3}} $$ pre-shock. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that a discontinuous shock instantaneously develops after the pre-shock. This regular shock solution is shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. These surfaces have been termed weak discontinuities by Landau & Lifschitz [12, Chapter IX, §96], who conjectured some type of singular behavior of derivatives along such surfaces. We prove that along the slowest surface, all fluid variables except the entropy have $$C^{1, {\frac{1}{2}} }$$ one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a weak rarefaction wave. Along the surface moving with the fluid velocity, density and entropy form $$C^{1, {\frac{1}{2}} }$$ one-sided cusps while the pressure and normal velocity remain $$C^2$$ ; as such, we term this surface a weak contact discontinuity.

Front-tracking transition system model for traffic state reconstruction, model learning, and control with application to stop-and-go wave dissipation

Connected and Autonomous Vehicles is a technology that will be disruptive for all layers of traffic control. The Lagrangian, in-the-flow nature of their operation offers untapped new potentials for sensing and actuation, but also presents new fundamental challenges. In order to use these vehicles for traffic state reconstruction and control, we need suitable traffic models, which should be computationally efficient and able to represent complex traffic phenomena. To this end, we propose the Front-tracking Transition System Model, a cell-free modelling approach that can incorporate Lagrangian measurements, and has a structure that yields itself to on-line model learning and control. The model is formulated as a transition system, and based on the front-tracking method for finding entropy solutions to the Lighthill–Whitham–Richards model. We characterize the solution of this model and show that it corresponds to the solution of the underlying PDE traffic model. Algorithms for traffic state reconstruction and model learning are proposed, exploiting the model structure. The model is then used to design a prediction-based control law for stop-and-go wave dissipation using randomly arriving Connected and Autonomous Vehicles. The proposed control framework is able to estimate the traffic state and model, adapt to changes in the traffic dynamics, and achieve a reduction in vehicles’ Total Time Spent.

A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels

We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in $C(L^{1}_{\text{loc}})$ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference between the solution and the nonlocal term.

Entropy solutions for elliptic Schrödinger type equations under Fourier boundary conditions

Entropy solutions for elliptic Schrödinger type equations under Fourier boundary conditions

Decay of entropy solutions of a scalar conservation law beyond a stationary ergodic setting

We prove the asymptotic decay of the Cauchy problem for a multidimensional scalar conservation law with initial data being a Besicovitch ergodic functions or the composition of stationary functions with stochastic deformations.

On decay of entropy solutions to degenerate nonlinear parabolic equations with perturbed periodic initial data

Under a precise nonlinearity-diffusivity assumption we establish the decay of entropy solutions of a degenerate nonlinear parabolic equation with initial data being a sum of periodic function and a function vanishing at infinity in the appropriate sense.

The existence and uniqueness result of entropy solutions for a p(·)-Laplace operator problem in weighted Sobolev spaces

The paper is concerned with the existence and uniqueness result of entropy solutions for a degenerate elliptic p(·)-Laplacian problem by using the regularization approach and theory of weighted Sobolev spaces with variable exponents.

Parabolic inequalities in inhomogeneous Orlicz-Sobolev spaces with gradients constraints and L 1-data

Abstract This work is devoted to the study of a new class of parabolic problems in inhomogeneous Orlicz spaces with gradient constraints and L 1-data. One proves the existence of the solution by studying the asymptotic behaviour as p goes to ∞, of a sequence of entropy solutions (up ) of some nonlinear parabolic equation in inhomogeneous Orlicz-Sobolev spaces with L 1-data involving the parameter p.

On the Theory of Entropy Solutions of Nonlinear Degenerate Parabolic Equations

We consider a second-order nonlinear degenerate parabolic equation in the case, where the flux vector and the nonstrictly increasing diffusion function are merely continuous. In the case of zero diffusion, this equation degenerates into a first order quasilinear equation (conservation law). It is known that in the general case under consideration an entropy solution (in the sense of Kruzhkov–Carrillo) of the Cauchy problem can be nonunique. Therefore, it is important to study special entropy solutions of the Cauchy problem and to find additional conditions on the input data of the problem that are sufficient for uniqueness. In this paper, we obtain some new results in this direction. Namely, the existence of the largest and the smallest entropy solutions of the Cauchy problem is proved. With the help of this result, the uniqueness of the entropy solution with periodic initial data is established. More generally, the comparison principle is proved for entropy sub- and supersolutions, in the case where at least one of the initial functions is periodic. The obtained results are a generalization of the results known for conservation laws to the parabolic case.

Numerical analysis of nonlinear parabolic problems with variable exponent and L^1 data

In this paper, we make the numerical analysis of the mild solution which is also an entropy solution of parabolic problem involving the \(p(x)-\)Laplacian operator with \(L^1-\) data.

Game Theory and an Improved Maximum Entropy-Attribute Measure Interval Model for Predicting Rockburst Intensity

To improve the accuracy of predicting rockburst intensity, game theory and an improved maximum entropy-attribute measure interval model were established. First, by studying the mechanism of rockburst and typical cases, rock uniaxial compressive strength σc, rock compression-tension ratio σc/σt, rock shear compression ratio σθ/σc, rock elastic deformation coefficient Wet, and rock integrity coefficient Kv were selected as indexes for predicting rockburst intensity. Second, by combining the maximum entropy principle with the attribute measure interval and using the minimum distance Di−k between sample and class as the guide, the entropy solution of the attribute measure was obtained, which eliminates the greyness and ambiguity of the rockburst indexes to the maximum extent. Third, using the compromise coefficient to integrate the comprehensive attribute measure, which avoids the ambiguity about the number of attribute measure intervals. Fourth, from the essence of measurement theory, the Euclidean distance formula was used to improve the attribute identification mode, which overcomes the effect of the confidence coefficient taking on the results. Moreover, in order to balance the shortcomings of the subjective weights of the Analytic Hierarchy Process and the objective weights of the CRITIC method, game theory was used for the combined weights, which balances experts’ experience and the amount of data information. Finally, 20 sets of typical cases for rockburst in the world were selected as samples. On the one hand, the reasonableness of the combined weights of indexes was analyzed; on the other hand, the results of this paper’s model were compared with the three analytical models for predicting rockburst, and this paper’s model had the lowest number of misjudged samples and an accuracy rate of 80%, which was better than other models, verifying the accuracy and applicability.

Numerical approximation of singular forward-backward SDEs

In this work, we study the numerical approximation of a class of singular fully coupled forward backward stochastic differential equations. These equations have a degenerate forward component and non-smooth terminal condition. They are used, for example, in the modelling of carbon market [1] and are linked to scalar conservation law perturbed by a diffusion. Classical FBSDEs methods fail to capture the correct entropy solution to the associated quasi-linear PDE. We introduce a splitting approach that circumvent this difficulty by treating differently the numerical approximation of the diffusion part and the non-linear transport part. Under the structural condition guaranteeing the well-posedness of the singular FBSDEs [2], we show that the splitting method is convergent with a rate 1/2. We implement the splitting scheme combining non-linear regression based on deep neural networks and conservative finite difference schemes. The numerical tests show very good results in possibly high dimensional framework.

Hydrodynamics for One-Dimensional ASEP in Contact with a Class of Reservoirs

We study the hydrodynamic behaviour of the asymmetric simple exclusion process on the lattice of size n. In the bulk, the exclusion dynamics performs rightward flux. At the boundaries, the dynamics is attached to reservoirs. We investigate two types of reservoirs: (1) the reservoirs that are weakened by $$n^\theta $$ for some $$\theta <0$$ and (2) the reservoirs that create particles only at the right boundary and annihilate particles only at the left boundary. We prove that the spatial density of particles, under the hyperbolic time scale, evolves with the entropy solution to a scalar conservation law on [0, 1] with boundary conditions. The boundary conditions are characterised by the boundary traces [3, 17, 20] at $$x=0$$ and $$x=1$$ which take values from $$\{0,1\}$$ .

A weighted fractional problem involving a singular nonlinearity and an L 1 datum

In this article, we show the existence of a unique entropy solution to the following problem: where the domain is bounded and contains the origin, , , , sp<N, , for q>1 and h is a general singular function with singularity at 0. Further, the fractional p-Laplacian with weight α is given by

Deep learning via message passing algorithms based on belief propagation

Message-passing algorithms based on the belief propagation (BP) equations constitute a well-known distributed computational scheme. They yield exact marginals on tree-like graphical models and have also proven to be effective in many problems defined on loopy graphs, from inference to optimization, from signal processing to clustering. The BP-based schemes are fundamentally different from stochastic gradient descent (SGD), on which the current success of deep networks is based. In this paper, we present and adapt to mini-batch training on GPUs a family of BP-based message-passing algorithms with a reinforcement term that biases distributions towards locally entropic solutions. These algorithms are capable of training multi-layer neural networks with performance comparable to SGD heuristics in a diverse set of experiments on natural datasets including multi-class image classification and continual learning, while being capable of yielding improved performances on sparse networks. Furthermore, they allow to make approximate Bayesian predictions that have higher accuracy than point-wise ones.