Solutions Of Conservation Laws Research Articles

Delta Shocks as Solutions of Conservation Laws with Discontinuous Moving Source

Delta Shocks as Solutions of Conservation Laws with Discontinuous Moving Source

Adaptive Model Reduction of High-Order Solutions of Compressible Flows via Optimal Transport

The solution of conservation laws with parametrised shock waves presents challenges for both high-order numerical methods and model reduction techniques. We introduce an r-adaptivity scheme based on optimal transport and apply it to develop reduced order models for compressible flows. The optimal transport theory allows us to compute high-order r-adaptive meshes from a starting reference mesh by solving the Monge–Ampère equation. A high-order discretization of the conservation laws enables high-order solutions to be computed on the resulting r-adaptive meshes. Furthermore, the Monge–Ampère solutions contain mappings that are used to reduce the spatial locality of the resulting solutions and make them more amenable to model reduction. We use a non-intrusive model reduction method to construct reduced order models of both the mesh and the solution. The procedure is demonstrated on three supersonic and hypersonic test cases, with the hybridisable discontinuous Galerkin method being used as the full order model.

On a new centered strategy to control the accuracy of weighted essentially non oscillatory algorithm for conservation laws close to discontinuities

AbstractIn this paper we try to redesign an algorithm analyzed in previous works by Amat et al. in order to obtain a centered strategy aimed to approximate the solution of linear and nonlinear systems of conservation laws. We perform the analysis of the new algorithm in order to explain why the new strategy works close to shocks in the solution of conservation laws. The objective is to force the use of the most centered stencil of those used whenever the data are affected by the presence of a high gradient.

Solving 1D Conservation Laws Using Pontryagin’s Minimum Principle

This paper discusses a connection between scalar convex conservation laws and Pontryagin's minimum principle. For flux functions for which an associated optimal control problem can be found, a minimum value solution of the conservation law is proposed. For scalar space-independent convex conservation laws such a control problem exists and the minimum value solution of the conservation law is equivalent to the entropy solution. This can be seen as a generalization of the Lax---Oleinik formula to convex (not necessarily uniformly convex) flux functions. Using Pontryagin's minimum principle, an algorithm for finding the minimum value solution pointwise of scalar convex conservation laws is given. Numerical examples of approximating the solution of both space-dependent and space-independent conservation laws are provided to demonstrate the accuracy and applicability of the proposed algorithm. Furthermore, a MATLAB routine using Chebfun is provided (along with demonstration code on how to use it) to approximately solve scalar convex conservation laws with space-independent flux functions.

Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes

Flow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pressure exists and is a weak solution of the corresponding scalar conservation law with discontinuous coefficient. A robust numerical scheme for approximating the resulting limit solutions is introduced. Numerical experiments show that the scheme is able to approximate interesting solution features such as propagating non-classical shock waves as well as discontinuous standing waves efficiently.

Alternating Evolution Schemes for Hyperbolic Conservation Laws

The alternating evolution (AE) system of Liu [J. Hyperbolic Differ. Equ., 5 (2008), pp. 421-447], $\partial_t u+ \partial_x f(v) =\frac{1}{\epsilon}(v-u)$, $\partial_t v + \partial_x f(u) =\frac{1}{\epsilon}(u-v)$, serves as a refined description of systems of hyperbolic conservation laws $\partial_t \phi+ \partial_x f(\phi)=0$, $\phi(x, 0)=\phi_0(x)$. The solution of conservation laws is precisely captured when two components take the same initial value as $\phi_0$, or is approached by two components exponentially fast when $\epsilon \downarrow 0$ if two initial states are sufficiently close. This nice property enables us to construct novel shock capturing schemes by sampling the AE system on alternating grids. In this paper we develop a class of local AE schemes by taking advantage of the AE system. Our approach is based on an average of the AE system over a hypercube centered at $x$ with vertices at $x\pm\Delta x$. The numerical scheme is then constructed by sampling the averaged system over alternating grids. Higher order accuracy is achieved by a combination of high order nonoscillatory polynomial reconstruction from the obtained averages and a matching Runge-Kutta solver in time discretization. Local AE schemes are made possible by letting the scale parameter $\epsilon$ reflect the local distribution of nonlinear waves. The AE schemes have the advantage of easier formulation and implementation, and efficient computation of the solution. For the first and second order local AE schemes applied to one-dimensional scalar conservation laws, we prove the numerical stability in the sense of satisfying the maximum principle and total variation diminishing property. The formulation procedure of AE schemes in multiple dimensions is given, followed by both the first and second order AE schemes for two-dimensional conservation laws. Numerical experiments for both scalar conservation laws and compressible Euler equations are presented to demonstrate the high order accuracy and capacity of these AE schemes.

On the well-posedness of entropy solutions for conservation laws with source terms

In this paper we study the initial boundary value problems for scalar conservation laws with source terms possessing limited regularity. We first define a strong trace of large class of entropy solutions of scalar conservation laws with source terms at the boundary $(0,T)\times\{0\}$ reached by $L^1$ in order to find a good boundary condition and we prove the well-posedness for scalar conservation laws with source terms. The proof is based on the kinetic formulation and the compensated compactness method.

Gearhart–Prüss Theorem and Linear Stability for Riemann Solutions of Conservation Laws

We study the spectral and linear stability of Riemann solutions with multiple Lax shocks for systems of conservation laws. Using a self-similar change of variables, Riemann solutions become stationary solutions for the system u t + (Df(u) − x I)u x = 0. In the space of O((1 + |x|)−η) functions, we show that if $${\mathfrak{R}}\lambda > -\eta$$ , then λ is either an eigenvalue or a resolvent point. Eigenvalues of the linearized system are zeros of the determinant of a transcendental matrix. On some vertical lines in the complex plane, called resonance lines, the determinant can be arbitrarily small but nonzero. A C 0 semigroup is constructed. Using the Gearhart–Pruss Theorem, we show that the solutions are O(e γ t ) if γ is greater than the real parts of the eigenvalues and the coordinates of resonance lines. We study examples where Riemann solutions have two or three Lax-shocks.

OPTIMAL ENTROPY SOLUTIONS FOR CONSERVATION LAWS WITH DISCONTINUOUS FLUX-FUNCTIONS

We deal with a single conservation law in one space dimension whose flux function is discontinuous in the space variable and we introduce a proper framework of entropy solutions. We consider a large class of fluxes, namely, fluxes of the convex-convex type and of the concave-convex (mixed) type. The alternative entropy framework that is proposed here is based on a two step approach. In the first step, infinitely many classes of entropy solutions are defined, each associated with an interface connection. We show that each of these class of entropy solutions form a contractive semigroup in L1 and is hence unique. Godunov type schemes based on solutions of the Riemann problem are designed and shown to converge to each class of these entropy solutions. The second step is to choose one of these classes of solutions. This choice depends on the Physics of the problem being considered and we concentrate on the model of two-phase flows in a heterogeneous porous medium. We define an optimization problem on the set of admissible interface connections and show the existence of an unique optimal connection and its corresponding optimal entropy solution. The optimal entropy solution is consistent with the expected solutions for two-phase flows in heterogeneous porous media.

Compactness of Velocity Averages

We extend previous results on compactness of velocity averages. The extension provides a more precise characterization of regularity spaces for solutions of certain transport equations. This provides regularity for solutions of conservation laws via the kinetic formulation.

Vanishing Curvature Viscosity for Front Propagation

In this paper we study the front propagation with constant speed and small curvature viscosity. We first investigate two related problems of conservation laws, one of which is on the nonlinear viscosity methods for the conservation laws, and the other one is on the structure of solutions to conservation laws with L1 initial data. We show that the nonlinear viscosity methods approaching the piecewise smooth solutions with finitely many discontinuity for convex conservation laws have the first-order rate of L1-convergence. The solutions of conservation laws with L1 initial data are shown to be bounded after t>0 if all singular points of initial data are from shocks. These results suggest that the front propagation with constant speed and a small curvature viscosity will approach the front movements with a constant speed, as the small parameter goes to zero. After the front breaks down, the cusps will disappear promptly and corners will be formed.

A Hybrid Adaptive ENO Scheme

This paper describes a new hybrid adaptive ENO scheme for partial differential equation in conservative form. The scheme is hybrid because it combines finite difference approximations for points away from the shock and ENO approximations near the shock. The method is adaptive in two ways. The first is that this method changes the computational grid in order to maintain a uniform approximation. Second, the method determines where to use the conservative ENO approximation to the derivative. The combination of these produces a quick algorithm for the solution of conservation laws.

Quadratic systems of conservation laws with generic behavior at infinity

We identify quadratic systems of conservation laws with “generic” behavior at infinity, where the genericity conditions derive naturally when studying weak solutions of conservation laws. Namely, we identify quadratic models for which the vector field associated with the viscosity admissibility criterion has properties at infinity that are true for an open and dense subset of the set of all planar quadratic vector fields in the metric associated with the Euclidian space of coefficients. We determine the boundaries of the regions containing, “generic” models in the parameter space of coefficients of quadratic models. We show that when crossing the boundaries of nongeneric models transversally, the Poincare compactification of the corresponding vector field undergoes either a saddle node or a transcritical bifurcation at infinity. For quadratic models with a bounded elliptic region we calculate the loci of nongeneric models assuming the viscosity matrix to be the identity. We obtain a two-parameter normal form for such models and show that the boundaries that determine generic models in the two-dimensional parameter space correspond to the Schaeffer-Shearer classification of models with an isolated umbilic point. Since the loci of nongeneric models are invariant under the equivalence transformations that preserve weak solutions of conservation laws, understanding their behavior at infinity promises to provide an insight into a general classification of quadratic conservation laws.

Asymptotic stability of Riemann waves for conservation laws

We are concerned with the asymptotic behavior of entropy solutions of conservation laws. A new notion about the asymptotic stability of Riemann solutions is introduced, and corresponding analytical frameworks are developed. The correlation between the asymptotic problem and many important topics in conservation laws and nonlinear analysis is recognized and analyzed, such as zero dissipation limits, uniqueness of entropy solutions, entropy analysis, and divergence-measure fields in \(L^\infty\). Then this theory is applied to understanding the asymptotic behavior of entropy solutions for many important systems of conservation laws.

L1Nonlinear Stability of Traveling Waves for a Hyperbolic System with Relaxation

where (x, t) # R_R+ . The unknowns u, v belong to R, the function f = f (u) is in C(R), and :>0 is a fixed constant. Relaxation systems often arise in many physical situations. Let us recall, for example, gases not in thermodynamic equilibrium [VK], kinetic theory [Ce, CIP, PI], chromatography [RAA], river flows, traffic flows, and more general waves [Wh]. More recently the study of a special class of hyperbolic systems with relaxation was developed in view of the numerical approximation of discontinuous solutions of conservation laws [JX, AN]. The 2_2 relaxation hyperbolic systems of conservation laws were first analyzed by Liu [Li2], who justified some nonlinear stability criteria for diffusion waves, expansion waves and traveling waves. article no. 0180

A note on the effect of artificial viscosity on solutions of conservation laws

We consider central difference schemes with artificial viscosity terms for nonlinear hyperbolic systems. We analyze the influence of artificial viscosity on solutions with shocks of nonlinear hyperbolic problems in one dimension. Both stationary and moving shocks are considered. The analysis shows that for the Euler equations one can obtain well behaved (sharp) shock layers when a scalar viscosity coefficient is used. This is not true for a general system. Numerical computations of Burgers' equation and the Euler equations are presented. They support the results from the linear theory.

Unbounded solutions for conservation laws with source

Unbounded solutions for conservation laws with source

Convergence of hybrid MUSCL-type schemes

Fully discrete 2nd order accurate TVNI Hybrid MUSCL-type schemes are presented. The entropy inequality of the schemes is proved. Hence, the approximation solutions converge to the entropy solution of conservation laws.

Periodic solutions of conservation laws

Periodic solutions of conservation laws

Periodic solutions of conservation laws

Periodic solutions of conservation laws