One-dimensional Scalar Conservation Laws Research Articles

Minimal entropy conditions for scalar conservation laws with general convex fluxes

We are concerned with the minimal entropy conditions for one-dimensional scalar conservation laws with general convex flux functions. For such scalar conservation laws, we prove that a single entropy-entropy flux pair ( η ( u ) , q ( u ) ) (\eta (u),q(u)) with η ( u ) \eta (u) of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in L l o c ∞ L^\infty _{\mathrm { loc}} that satisfy the inequality: η ( u ) t + q ( u ) x ≤ μ \eta (u)_t+q(u)_x\leq \mu in the distributional sense for some non-negative Radon measure μ \mu . Furthermore, we extend this result to the class of weak solutions in L l o c p L^p_{\mathrm {loc}} , based on the asymptotic behavior of the flux function f ( u ) f(u) and the entropy function η ( u ) \eta (u) at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness.

A Hilliges-Weidlich-type scheme for a one-dimensional scalar conservation law with nonlocal flux

<abstract><p>The simulation model proposed in [M. Hilliges and W. Weidlich. A phenomenological model for dynamic traffic flow in networks. <italic>Transportation Research Part B: Methodological</italic>, <bold>29</bold> (6): 407–431, 1995] can be understood as a simple method for approximating solutions of scalar conservation laws whose flux is of density times velocity type, where the density and velocity factors are evaluated on neighboring cells. The resulting scheme is monotone and converges to the unique entropy solution of the underlying problem. The same idea is applied to devise a numerical scheme for a class of one-dimensional scalar conservation laws with nonlocal flux and initial and boundary conditions. Uniqueness of entropy solutions to the nonlocal model follows from the Lipschitz continuous dependence of a solution on initial and boundary data. By various uniform estimates, namely a maximum principle and bounded variation estimates, along with a discrete entropy inequality, the sequence of approximate solutions is shown to converge to an entropy weak solution of the nonlocal problem. The improved accuracy of the proposed scheme in comparison to schemes based on the Lax-Friedrichs flux is illustrated by numerical examples. A second-order scheme based on MUSCL methods is presented.</p></abstract>

L$^2$ Error Estimate to Smooth Solutions of High Order Runge--Kutta Discontinuous Galerkin Method for Scalar Nonlinear Conservation Laws with and without Sonic Points

In this paper we shall establish an a priori L$^2$-norm error estimate of the fourth order Runge--Kutta discontinuous Galerkin method for solving sufficiently smooth solutions of one-dimensional scalar nonlinear conservation laws. The optimal order of accuracy in time is obtained under the standard Courant--Friedrichs--Lewy condition, and the quasi-optimal and/or optimal order of accuracy in space is achieved for many widely used numerical fluxes, no matter whether the solution contains sonic points or not. The main tools used in this paper are the matrix transferring process and the generalized Gauss--Radau projection of the reference functions, depending on the relative upwind effect. Finally some numerical experiments are given to support our theoretical results.

A Semi-Lagrangian Godunov-Type Method without Numerical Viscosity for Shocks

One of the most important and complex effects in compressible fluid flow simulation is a shock-capturing mechanism. Numerous high-resolution Euler-type methods have been proposed to resolve smooth flow scales accurately and to capture the discontinuities simultaneously. One of the disadvantages of these methods is a numerical viscosity for shocks. In the shock, the flow parameters change abruptly at a distance equal to the mean free path of a gas molecule, which is much smaller than the cell size of the computational grid. Due to the numerical viscosity, the aforementioned Euler-type methods stretch the parameter change in the shock over few grid cells. We introduce a semi-Lagrangian Godunov-type method without numerical viscosity for shocks. Another well-known approach is a method of characteristics that has no numerical viscosity and uses the Riemann invariants or solvers for water hammer phenomenon modeling, but in its formulation the convective terms are typically neglected. We use a similar approach to solve the one-dimensional adiabatic gas dynamics equations, but we split the equations into parts describing convection and acoustic processes separately, with corresponding different time steps. When we are looking for the solution to the one-dimensional problem of the scalar hyperbolic conservation law by the proposed method, we additionally use the iterative Godunov exact solver, because the Riemann invariants are non-conserved for moderate and strong shocks in an ideal gas. The proposed method belongs to a group of particle-in-cell (PIC) methods; to the best of the author’s knowledge, there are no similar PIC numerical schemes using the Riemann invariants or the iterative Godunov exact solver. This article describes the application of the aforementioned method for the inviscid Burgers’ equation, adiabatic gas dynamics equations, and the one-dimensional scalar hyperbolic conservation law. The numerical analysis results for several test cases (e.g., the standard shock-tube problem of Sod, the Riemann problem of Lax, the double expansion wave problem, the Shu–Osher shock-tube problem) are compared with the exact solution and Harten’s data. In the shock for the proposed method, the flow properties change instantaneously (with an accuracy dependent on the grid cell size). The iterative Godunov exact solver determines the accuracy of the proposed method for flow discontinuities. In calculations, we use the iteration termination condition less than 10−5 to find the pressure difference between the current and previous iterations.

A semi-discrete Lagrangian–Eulerian scheme for hyperbolic-transport models

In this work, we introduce a new semi-discrete scheme based on the so-called no-flow curves and its numerical analysis for solving initial value problems that involve one-dimensional (1D) scalar hyperbolic conservation laws and of the form, ut+H(u)x=0, x∈R,t>0, u(x,0)=u0(x). In addition, we present a two-dimensional (2D) version of the semi-discrete Lagrangian–Eulerian scheme to show that the proposed method can be applied to multidimensional problems. From an improved 1D weak numerical asymptotic analysis, we found that the solutions provided by the novel semi-discrete scheme satisfy a maximum principle property and a Kruzhkov entropy condition. We also highlight the possibility of using the no-flow curves as a new desingularization analysis technique for the construction of computationally stable numerical fluxes in the locally conservative form for nonlinear hyperbolic problems. We provide nontrivial 1D and 2D numerical examples with nonlinear wave interaction to illustrate the effectiveness and capabilities of the proposed approach and verify the theory. According to the results, the scheme handles discontinuous solutions (shocks) with low numerical dissipation quite well and shows a very good resolution of rarefaction waves with no spurious glitch effect in the vicinity of the sonic points. We also consider a test case for nonstrictly hyperbolic conservation laws with a resonance point (coincidence of eigenvalues) modeling three-phase flow in a porous media transport problem.

One-Dimensional Scalar Conservation Laws with Regulated Data

We show that the Kružkov solution of the Cauchy problem for a scalar conservation law in one spatial dimension propagates regulated initial data into regulated solutions at later times. The proof i...

Discontinuous Galerkin Methods for Nonlinear Scalar Conservation Laws: Generalized Local Lax--Friedrichs Numerical Fluxes

In this paper, we study the discontinuous Galerkin (DG) method with a class of generalized numerical fluxes for one-dimensional scalar nonlinear conservation laws. The generalized local Lax--Friedrichs (GLLF) fluxes with two weights, which may not be monotone, are proposed and analyzed. Under a condition for the weights, we first show the monotonicity for the flux and thus the $L^2$ stability of the scheme. Then, by constructing and analyzing a special piecewise global projection which commutes with the time derivative operator, we are able to show optimal error estimates for the DG scheme with GLLF fluxes. The result is sharp for monotone numerical fluxes, for which only suboptimal estimates can be proved in previous work. Moreover, optimal error estimates are still valid for fluxes that are not monotone, allowing us to choose some suitable weights to achieve less numerical dissipation and thus to better resolve shocks. Numerical experiments are provided to show the sharpness of theoretical results.

Entropy inequalities for fully-discrete E-schemes

We consider the numerical solution of one-dimensional scalar conservation laws. In particular, we present some very simple, yet appropriate, discrete entropy fluxes for the class of fully-discrete E-schemes. We show that these provide the required entropy inequalities under sharp CFL conditions.

On Riemann solutions under different initial periodic perturbations at two infinities for 1-d scalar convex conservation laws

This paper is concerned with the large time behaviors of the entropy solutions to one-dimensional scalar convex conservation laws, of which the initial data are assumed to approach two arbitrary L∞ periodic functions as x→−∞ and x→+∞, respectively. We show that the solutions approach the Riemann solutions at algebraic rates as time increases. Moreover, a new discovery in this paper is that the difference between the two periodic perturbations at two infinities may generate a constant shift on the background shock wave, which is different from the result in [11], where the two periodic perturbations are the same.

Total variation bounded flux limiters for high order finite difference schemes solving one-dimensional scalar conservation laws

Total variation bounded flux limiters for high order finite difference schemes solving one-dimensional scalar conservation laws

Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux

We introduce and analyze a class of models with nonlocal point constraints for traffic flow through bottlenecks, such as exit doors in the context of pedestrians traffic and toll gates in vehicular traffic. Constraints are defined based on data collected from non-local in space and/or in time observations of the flow. We propose a theoretical analysis and discretization framework that permits to include different data acquisition strategies; a numerical comparison is provided. Nonlocal constraint allows to model, e.g., the irrational behavior (“panic”) near the exit observed in dense crowds and the capacity drop at tollbooth in vehicular traffic.Existence and uniqueness of solutions are shown under suitable and “easy to check” assumptions on the constraint operator. A numerical scheme for the problem, based on finite volume methods, is designed, its convergence is proved and its validation is done with an explicit solution. Numerical examples show that nonlocally constrained models are able to reproduce important features in traffic flow such as self-organization.

Smoothing effect in BVΦ for entropy solutions of scalar conservation laws

This paper deals with a sharp smoothing effect for entropy solutions of one-dimensional scalar conservation laws with a degenerate convex flux. We briefly explain why degenerate fluxes are related with the optimal smoothing effect conjectured by Lions, Perthame, Tadmor for entropy solutions of multidimensional conservation laws. It turns out that generalized spaces of bounded variation BVΦ are particularly suitable –better than Sobolev spaces– to quantify the regularizing effect and to obtain traces as in BV. The function Φ in question is linked to the degeneracy of the flux. Up to the present, the Lax–Oleĭnik formula has provided optimal results for a uniformly convex flux. This formula is validated in this paper for the more general class of C1 strictly convex fluxes –which contains degenerate convex fluxes– and enables the BVΦ smoothing effect in this class. We give a complete proof that for a C1 strictly convex flux the Lax–Oleĭnik formula provides the unique entropy solution, namely the Kruzhkov solution.

Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement

In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the alpha -th order (1 le alpha le {k+1}) divided difference of the DG error in the L^2 norm is of order {k + frac{3}{2} - frac{alpha }{2}} when upwind fluxes are used, under the condition that |f'(u)| possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order {2k + frac{3}{2} - frac{alpha }{2}} in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least ({frac{3}{2}k+1})th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order {k+1} in the L^2 norm for the divided differences of DG errors and thus ({2k+1})th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results.

Fractional BV spaces and applications to scalar conservation laws

We obtain new fine properties of entropy solutions to scalar nonlinear conservation laws. For this purpose, we study the "fractional BV spaces" denoted by BVs(ℝ) (for 0 < s ≤ 1), which were introduced by Love and Young in 1937 and closely related to the critical Sobolev space Ws,1/s(ℝ). We investigate these spaces in connection with one-dimensional scalar conservation laws. The BVsspaces allow one to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with BVsinitial data. Furthermore, for the first time, we get the maximal Ws,psmoothing effect conjectured by Lions, Perthame and Tadmor for all nonlinear (possibly degenerate) convex fluxes.

An inviscid regularization of hyperbolic conservation laws

This article examines the utilization of a spatial averaging technique to the nonlinear terms of the partial differential equations as an inviscid shock-regularization of hyperbolic conservation laws. A central motivation is to promote the idea of applying filtering techniques such as the observable divergence method, rather than viscous regularization, as an alternative to the simulation of shocks and turbulence in inviscid flows while, on the other hand, generalizing and unifying previous mathematical and numerical analysis of the method applied to the one-dimensional Burgers’ and Euler equations. This article primarily concerns the mathematical analysis of the technique and examines two fundamental issues. The first is on the global existence and uniqueness of classical solutions for the regularization under the more general setting of quasilinear, symmetric hyperbolic systems in higher dimensions. The second issue examines one-dimensional scalar conservation laws and shows that the inviscid regularization method captures the unique entropy or physically relevant solution of the original, non-averaged problem as filtering vanishes.

On the existence, uniqueness and stability of entropy solutions to scalar conservation laws

We consider one-dimensional scalar conservation laws with and without viscosity where the flux function F(x,t,u) is only assumed to be absolutely continuous in x, locally integrable in t and continuous in u. The existence and uniqueness of entropy solutions for the associated initial-value problem are obtained through the vanishing viscosity method and the doubling variables technique. We also prove the stability of entropy solutions in C([0,T];Lloc1(R)) and in C([0,T];L1(R)) with respect to both initial data and flux functions.

High-resolution numerical algorithm for one-dimensional scalar conservation laws with a constrained solution

The CABARET computational algorithm is generalized to one-dimensional scalar quasilinear hyperbolic partial differential equations with allowance for inequality constraints on the solution. This generalization can be used to analyze seepage of liquid radioactive wastes through the unsaturated zone.

An invariant in shock clustering and Burgers turbulence

One-dimensional scalar conservation laws with convex flux and Markov initial data are now known to yield a completely integrable Hamiltonian system. In this paper, we rederive the analogue of Loitsiansky's invariant in hydrodynamic turbulence from the perspective of integrable systems. Other relevant physical notions such as energy dissipation and spectrum are also discussed.

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 maximum-principle-satisfying high order schemes for scalar conservation laws

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one-dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. It is straightforward to extend the method to two and higher dimensions on rectangular meshes. We also show that the same limiter can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported.