Nonlinear Scalar Conservation Laws Research Articles

High Frequency Waves and the Maximal Smoothing Effect for Nonlinear Scalar Conservation Laws

The article first studies the propagation of well prepared high frequency waves with small amplitude $\varepsilon$ near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, and Tadmor [J. Amer. Math. Soc., 7 (1994), pp. 169--192] about the maximal regularizing effect for nonlinear conservation laws. For this purpose, a definition of smooth nonlinear flux is stated and compared to classical definitions. Then it is proved that the uniform smoothness expected by these authors in Sobolev spaces cannot be exceeded for all smooth nonlinear fluxes.

A Skew-Symmetric Discontinuous Galerkin Spectral Element Discretization and Its Relation to SBP-SAT Finite Difference Methods

This paper shows that the discontinuous Galerkin collocation spectral element method with Gauss--Lobatto points (DGSEM-GL) satisfies the discrete summation-by-parts (SBP) property and can thus be classified as an SBP-SAT (simultaneous approximation term) scheme with a diagonal norm operator. In the same way, SBP-SAT finite difference schemes can be interpreted as discontinuous Galerkin-type methods with a corresponding weak formulation based on an inner-product formulation common in the finite element community. This relation allows the use of matrix-vector notation (common in the SBP-SAT finite difference community) to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for DGSEM-GL and diagonal norm SBP-SAT. Based on this result, a skew-symmetric energy stable discretely conservative DGSEM-GL formulation (applicable to general diagonal norm SBP-SAT schemes) for the nonlinear Burgers equation is constructed.

Numerical smoothness and error analysis for RKDG on the scalar nonlinear conservation laws

The new concept of numerical smoothness is applied to the RKDG (Runge–Kutta/Discontinuous Galerkin) methods for scalar nonlinear conservations laws. The main result is an a posteriori error estimate for the RKDG methods of arbitrary order in space and time, with optimal convergence rate. In this paper, the case of smooth solutions is the focus point. However, the error propagation analysis framework is prepared to deal with discontinuous solutions in the future.

OPTIMAL DECAY RATES TO CONSERVATION LAWS WITH DIFFUSION-TYPE TERMS OF REGULARITY-GAIN AND REGULARITY-LOSS

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index s ∈ ℝ over the whole space ℝn for any spatial dimension n ≥ 1. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for s < 1 and s > 1, respectively. For all s ∈ ℝ, we not only obtain the Lp–Lq time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of small-amplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising from physics.

Convergence of time-space adaptive algorithms for nonlinear conservation laws

A family of explicit adaptive algorithms is designed to solve nonlinear scalar one-dimensional conservation laws. Based on the Godunov scheme on a uniform grid, a first strategy uses the multiresolution analysis of the solution to design an adaptive grid that evolves in time according to the time-dependent local smoothness. The method is furthermore enhanced by a local time-stepping strategy. Both numerical schemes are shown to converge towards the unique entropy solution.

Superconvergence of Discontinuous Galerkin Methods for Scalar Nonlinear Conservation Laws in One Space Dimension

In this paper, an analysis of the superconvergence property of the semidiscrete discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves $\left(k + \frac32\right)$th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree $k$ ($k \ge 1$), under the condition that $|f'(u)|$ possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on $f(u)$ is artificial.

Exact Controllability of Scalar Conservation Laws with an Additional Control in the Context of Entropy Solutions

In this paper, we study the exact controllability problem for nonlinear scalar conservation laws on a compact interval with a regular convex flux and in the framework of entropy solutions. With the boundary data and a source term depending only on the time as controls, we provide sufficient conditions for a state to be reachable in arbitrary small time. To do so we introduce a slightly modified wave-front tracking algorithm.

Provably stable and time‐accurate extensions of Runge–Kutta schemes for CFD computations on moving grids

SUMMARYExtending fixed‐grid time integration schemes for unsteady CFD applications to moving grids, while formally preserving their numerical stability and time accuracy properties, is a nontrivial task. A general computational framework for constructing stability‐preserving ALE extensions of Eulerian multistep time integration schemes can be found in the literature. A complementary framework for designing accuracy‐preserving ALE extensions of such schemes is also available. However, the application of neither of these two computational frameworks to a multistage method such as a Runge–Kutta (RK) scheme is straightforward. Yet, the RK methods are an important family of explicit and implicit schemes for the approximation of solutions of ordinary differential equations in general and a popular one in CFD applications. This paper presents a methodology for filling this gap. It also applies it to the design of ALE extensions of fixed‐grid explicit and implicit second‐order time‐accurate RK (RK2) methods. To this end, it presents the discrete geometric conservation law associated with ALE RK2 schemes and a method for enforcing it. It also proves, in the context of the nonlinear scalar conservation law, that satisfying this discrete geometric conservation law is a necessary and sufficient condition for a proposed ALE extension of an RK2 scheme to preserve on moving grids the nonlinear stability properties of its fixed‐grid counterpart. All theoretical findings reported in this paper are illustrated with the ALE solution of inviscid and viscous unsteady, nonlinear flow problems associated with vibrations of the AGARD Wing 445.6. Copyright © 2011 John Wiley & Sons, Ltd.

Error estimate for the upwind finite volume method for the nonlinear scalar conservation law

In this paper we estimate the error of upwind first order finite volume schemes applied to scalar conservation laws. As a first step, we consider standard upwind and flux finite volume scheme discretization of a linear equation with space variable coefficients in conservation form. We prove that, in spite of their lack of consistency, both schemes lead to a first order error estimate. As a final step, we prove a similar estimate for the nonlinear case. Our proofs rely on the notion of geometric corrector, introduced in our previous paper by Bouche et al. (2005) [24] in the context of constant coefficient linear advection equations.

A remark on duality solutions for some weakly nonlinear scalar conservation laws

We investigate existence and uniqueness of duality solutions for a scalar conservation law with a nonlocal interaction kernel. Following Bouchut and James (1999) [3], a notion of duality solution for such a nonlinear system is proposed, for which we do not have uniqueness. However we prove that a natural definition of the flux allows to select a solution for which uniqueness holds.

Vanishing viscosity with short wave–long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave–long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α = o ( ε 1 / 2 ) . We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.

A class of hybrid DG/FV methods for conservation laws II: Two-dimensional cases

By comparing the discontinuous Galerkin (DG) methods, the k-exact finite volume (FV) methods and the lift collocation penalty (LCP) methods, a concept of ‘static reconstruction’ and ‘dynamic reconstruction’ was introduced for higher-order numerical methods in our previous work. Based on this concept, a class of hybrid DG/FV methods was presented for one-dimensional conservation law using a ‘hybrid reconstruction’ approach. In the hybrid DG/FV schemes, the lower-order derivatives of the piecewise polynomial are computed locally in a cell by the traditional DG method (called as ‘dynamic reconstruction’), while the higher-order derivatives are re-constructed by the ‘static reconstruction’ of the FV method, using the known lower-order derivatives in the cell itself and in its adjacent face neighboring cells. In this follow-up paper, the hybrid DG/FV schemes are extended onto two-dimensional unstructured and hybrid grids. The two-dimensional linear and non-linear scalar conservation law and Euler equations are considered. Some typical cases are tested to demonstrate the performance of the hybrid DG/FV method, and the numerical results show that they can reduce the CPU time and memory requirement greatly than the traditional DG method with the same order of accuracy in the same mesh.

On the propagation of statistical model parameter uncertainty in CFD calculations

This work considers a new class of finite-volume approximations for scalar and system nonlinear conservation laws with multiple sources of stochastic model parameter uncertainty. The deterministic propagation of model parameter uncertainty is achieved through the introduction of additional stochastic coordinates. Particular attention is given to the construction of specialized piecewise polynomial approximation spaces well suited to the high-order accurate approximation of solution discontinuities in both physical and stochastic dimensions without exhibiting Gibbs-like oscillations characteristic of polynomial approximation. The proposed discretization easily retrofits existing finite-volume CFD codes in use today. Numerical results are presented for inviscid Burgers equation with uncertain initial data as well as the compressible Reynolds-averaged Navier–Stokes equations with uncertain boundary data and turbulence model parameters.

Entropy-TVD Scheme for Nonlinear Scalar Conservation Laws

In this paper, we develop a so-called Entropy-TVD scheme for the non-linear scalar conservation laws. The scheme simultaneously simulates the solution and one of its entropy, and in doing so the numerical dissipation is reduced by carefully computing the entropy decrease. We prove that the scheme is feasible and TVD and satisfies the entropy condition. We also prove that the local truncation error of the scheme is of first-order. However, numerical tests show that the scheme has a second-order convergence rate, an order higher than its truncation error, in computing smooth solution, and in many cases is better than a second-order ENO scheme in resolving shocks and corners of rarefaction waves.

A Minimality Property for Entropic Solutions to Scalar Conservation Laws in 1 + 1 Dimensions

The Second Law of Thermodynamics asserts that the physical entropy of an adiabatic system is an increasing function in time. In this paper we will study a more stringent version of this law, according to which entropy should not only increase in time, but the rate of increase is optimal in absolute value among all possible evolutions. We will establish this property in the framework of non-linear scalar hyperbolic conservation law with strictly convex fluxes.

SCALAR CONSERVATION LAWS ON A HALF-LINE: A PARABOLIC APPROACH

The initial-boundary value problem for the (viscous) nonlinear scalar conservation law is considered, [Formula: see text] The flux f(ξ) ∈ C2(ℝ) is assumed to be convex (but not strictly convex, i.e. f″(ξ)≥ 0). It is shown that a unique limit u = lim ∊ → 0 u∊ exists. The classical duality method is used to prove uniqueness. To this end parabolic estimates for both the direct and dual solutions are obtained. In particular, no use is made of the Kružkov entropy considerations.

Stability Analysis and A Priori Error Estimates of the Third Order Explicit Runge–Kutta Discontinuous Galerkin Method for Scalar Conservation Laws

In this paper we present an analysis of the Runge–Kutta discontinuous Galerkin method for solving scalar conservation laws, where the time discretization is the third order explicit total variation diminishing Runge–Kutta method. We use an energy technique to prove the $\mathrm{L}^2$-norm stability for scalar linear conservation laws and to obtain a priori error estimates for smooth solutions of scalar nonlinear conservation laws. Quasi-optimal order is obtained for general numerical fluxes, and optimal order is given for upwind fluxes. The theoretical results are obtained for piecewise polynomials with any degree $k\geq1$ under the standard temporal-spatial CFL condition $\tau\leq\gamma h$, where h and $\tau$ are the element length and time step, respectively, and the positive constant $\gamma$ is independent of h and $\tau$.

Shock formation and breaking in granular avalanches

In this paper, we explore properties of shock wave solutions of the Gray-Thornton model for particle size segregation in granular avalanches.The model equation is a nonlinear scalar conservation law expressing conservation of mass under shear for the concentration of small particles in a bidisperse mixture. Shock waves are weak solutions of the partial differential equation across which the concentration jumps. We give precise criteria on smooth initial conditions under which a shock wave forms in the interior of the avalanche in finite time. Shocks typically lose stability as they are sheared by the flow, giving way to a complex structure in which a two-dimensional rarefaction wave interacts dynamically with a pair of shocks. The rarefaction represents a mixing zone, in which small and large particles are mixed as they are transported up and down (respectively) through the zone. The mixing zone expands and twice changes its detailed structure before reaching the boundary.

On the upstream mobility scheme for two-phase flow in porous media

When neglecting capillarity, two-phase incompressible flow in porous media is modelled as a scalar nonlinear hyperbolic conservation law. A change in the rock type results in a change of the flux function. Discretizing in one-dimensional with a finite volume method, we investigate two numerical fluxes, an extension of the Godunov flux and the upstream mobility flux, the latter being widely used in hydrogeology and petroleum engineering. Then, in the case of a changing rock type, one can give examples when the upstream mobility flux does not give the right answer.

Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere’s tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here “equatorial periodic solutions”, analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct “confined solutions”, which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test cases are presented.