A new nonlinear functional for general scalar conservation laws

The approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well-posedness theories of hyperbolic conservation laws, especially in one space dimensional case. In particular, a generalized entropy functional was constructed in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427-1442] for the L¹ stability of weak solutions. However, this generalized functional is so far only defined for scalar equations with convex flux function. In this paper, we introduce a new nonlinear functional which is motivated by the new Glimm functional introduced in [J.-L. Hua, Z.-H. Jiang, T. Yang, A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, preprint] for general scalar conservation laws. This functional improves the one given in [H.-X. Liu, T. Yang, A nonlinear functional for general scalar hyperbolic conservation laws, J. Differential Equations 235 (2) (2007) 658-667] and it can be viewed as a better attempt for the generalized entropy functional for general equations.     

Potential comparison and asymptotics in scalar conservation laws without convexity

Two kinds of optimal convergence orders in L¹-norm to a self-similar solution are proved or conjectured for various evolutionary problems so far. The first convergence order is of the magnitude of the similarity solution itself and the second one is of order 1/t. Employing a potential comparison technique to scalar conservation laws we may easily see that these asymptotic convergence orders are related to space and time translation of potentials. We present the technique clearly in the simple setting of scalar conservation laws in one space dimension.     

A Time-Fractional Step Method for Conservation Law Related Obstacle Problems

We are interested in approximating the solution of a first-order quasi-linear equation associated with a forced unilateral obstacle condition. With this view, we make use of the time-splitting method developed classically to compute discontinuous solutions of nonhomogeneous scalar conservation laws. Here, one proves that this fractional step method converges in L¹ to the weak entropy solution of the considered obstacle problem. In the case of the Cauchy problem, an L¹-error bound in is established.     

A nonlinear functional for general scalar hyperbolic conservation laws

A generalized entropy functional was introduced in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 1427-1442] for the scalar hyperbolic conservation laws with convex flux function. This functional was crucially used in the functional approach to the L¹ stability study on the system of hyperbolic conservation laws when each characteristic field is either genuinely nonlinear or linearly degenerate. However, how to construct the generalized entropy functional for scalar conservation laws with general flux, and then how to apply the functional approach to the L¹ study on general systems are still open. In this paper, we construct a new nonlinear functional which gives some partial answer to this question and we expect the analysis will shed some light on the future investigation in this direction.     

New energy identities and super convergence analysis of the energy conserved splitting FDTD methods for 3D Maxwell's equations

The energy‐conserved splitting finite‐difference time‐domain (EC‐S‐FDTD) method has recently been proposed to solve the Maxwell equations with second order accuracy while numerically keep the L² energy conservation laws of the equations. In this paper, the EC‐S‐FDTD scheme for the 3D Maxwell equations is proved to be energy‐conserved and unconditionally stable in the discrete H¹ norm. The EC‐S‐FDTD scheme is of second‐order accuracy both in time step and spatial steps, which suggests the super‐convergence of this scheme in the discrete H¹ norm. And the divergence of the electric field of the EC‐S‐FDTD scheme in the discrete L² norm is second‐order accurate. Numerical experiments confirm our theoretical analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

A least-squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L² convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H⁻¹-type norm, but not L²-coercive. Nevertheless, the L² convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L² norm, or by properly choosing the finite element spaces, providing implicit control of the L² norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L² convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. We believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).     

Lower compactness estimates for scalar balance laws

In this paper, we study the compactness in L of the semigroup (St)_t≥0 of entropy weak solutions to strictly convex scalar conservation laws in one space dimension. The compactness of St for each t > 0 was established by P. D. Lax. Upper estimates for the Kolmogorov e‐entropy of the image of bounded sets in L¹ n L^∞ through S_t were given by C. De Lellis and F. Golse. Here we provide lower estimates on this e‐entropy of the same order as the one established by De Lellis and Golse, thus showing that such an e‐entropy is of size ≈ 1/ε. Moreover, we extend these estimates of compactness to the case of convex balance laws. © 2012 Wiley Periodicals, Inc.     

Entropy formulation for fractal conservation laws

Using an integral formula of Droniou and Imbert (2005) for the fractional Laplacian, we define an entropy formulation for fractal conservation laws with pure fractional diffusion of order λ ∈]0, 1]. This allows to show the existence and the uniqueness of a solution in the L^∞ framework. We also establish a result of controled speed of propagation that generalizes the finite propagation speed result of scalar conservation laws. We finally let the non-local term vanish to approximate solutions of scalar conservation laws, with optimal error estimates for BV initial conditions as Kuznecov (1976) for λ = 2 and Droniou (2003) for λ ∈]1, 2].     

A second order explicit finite element scheme to multidimensional conservation laws and its convergence

A second order explicit finite element scheme is given for the numerical computation to multi-dimensional scalar conservation laws.L ^p convergence to entropy solutions is proved under some usual conditions. For two-dimensional problems, uniform mesh, and sufficiently smooth solutions a second order error estimate inL ² is proved under a stronger condition, Δt≤Ch ^2/4     

Computational inversion of electron micrographs using L2‐gradient flows—convergence analysis

A gradient flow‐based explicit finite element method (L2GF) for reconstructing the 3D density function from a set of 2D electron micrographs has been proposed in recently published papers. The experimental results showed that the proposed method was superior to the other classical algorithms, especially for the highly noisy data. However, convergence analysis of the L2GF method has not been conducted. In this paper, we present a complete analysis on the convergence of L2GF method for the case of using a more general form regularization term, which includes the Tikhonov‐type regularizer and modified or smoothed total variation regularizer as two special cases. We further prove that the L²‐gradient flow method is stable and robust. These results demonstrate that the iterative variational reconstruction method derived from the L²‐gradient flow approach is mathematically sound and effective and has desirable properties. Copyright © 2013 John Wiley & Sons, Ltd.     

Splitting method applied to hyperbolic problem with source term

This work deals with the study of an operator splitting applied to approximate scalar conservation laws with source term by use of homogeneous laws. A convergence result towards the entropy weak solution is given. The reasoning is based on fine properties of BV ∩ L^∞ functions and Kruskov techniques.     

Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds

Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to non-linear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vector-field on the manifold and depends on the unknown function as a parameter. We introduce notions of entropy solutions in the class of bounded measurable functions and in the class of measure-valued mappings. We establish the well-posedness theory for conservation laws on a manifold, by generalizing both Kruzkov's and DiPerna's theories originally developed in the Euclidian setting. The class of geometry-compatible (as we call it) conservation laws is singled out as an important case of interest, which leads to robust L^p estimates independent of the geometry of the manifold. On the other hand, general conservation laws solely enjoy the L¹ contraction property and leads to a unique contractive semi-group of entropy solutions. Our framework allows us to construct entropy solutions on a manifold via the vanishing diffusion method or the finite volume method.     

Local error analysis for approximate solutions of hyperbolic conservation laws

We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of estimating the local truncation error, through the use of localized test-functions. In the convex scalar case, they can be converted intoL _loc estimates, following theLip convergence theory developed by Tadmor et al. Comparisons between the local truncation error and theL _loc -error show remarkably similar behavior. Numerical results are presented for the convex scalar case, where the theory is valid, as well as for nonconvex scalar examples and the Euler equations of gas dynamics. The local truncation error has proved a reliable smoothness indicator and has been implemented in adaptive algorithms in [Karni, Kurganov and Petrova, J. Comput. Phys. 178 (2002) 323–341].     

Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws

Nonlinear geometric optics with various frequencies for entropy solutions only in L^∞ of multidimensional scalar conservation laws is analyzed. A new approach to validate nonlinear geometric optics is developed via entropy dissipation through scaling, compactness, homogenization, and L¹-stability. New multidimensional features are recognized, especially including nonlinear propagations of oscillations with high frequencies. The validity of nonlinear geometric optics for entropy solutions in L^∞ of multidimensional scalar conservation laws is justified.     

N-waves in elasticity

We study the large time behavior of solutions of the one-dimensional equations of elasticity, typically a nonconvex system of conservation laws. We show that solutions with compact initial support converge in L₁ to a superposition of N-waves, at algebraic rate. Likewise, we show that the perturbation expansion of weakly nonlinear geometric optics is uniformly valid to second order. Our analysis uses approximate scalar laws, together with L₁ stability of scalar solutions and decay in total variation of the wave speed.     

Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

For the scalar conservation laws with discontinuous flux, an infinite family of (A, B)‐interface entropies are introduced and each one of them is shown to form an L¹‐contraction semigroup (see [2]). One of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution. Away from the interface, boundedness of total variation of the solution has been proved in a recent paper [6]. In this paper, we discuss this particular issue in detail and produce a counterexample to show that the solution, in general, has unbounded total variation near the interface. In fact, this example illustrates that smallness of the BV norm of the initial data is immaterial. We hereby settle the question of determining for which of the aforementioned (A, B) pairs the solution will have bounded total variation in the case of strictly convex fluxes. © 2010 Wiley Periodicals, Inc.     

Convergence analysis of the LDG method applied to singularly perturbed problems

Considering a two‐dimensional singularly perturbed convection–diffusion problem with exponential boundary layers, we analyze the local discontinuous Galerkin (DG) method that uses piecewise bilinear polynomials on Shishkin mesh. A convergence rate O(N^‐1 lnN) in a DG‐norm is established under the regularity assumptions, while the total number of mesh points is O(N²). The rate of convergence is uniformly valid with respect to the singular perturbation parameter ε. Numerical experiments indicate that the theoretical error estimate is sharp. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Robust residual‐ and recovery‐based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems

We consider a posteriori error estimation for a multipoint flux mixed finite element method for two‐dimensional elliptic interface problems. Within the class of modified quasi‐monotonically distributed coefficients, we derive a residual‐type a posteriori error estimator of the weighted sum of the scalar and flux errors which is robust with respect to the jumps of the coefficients. Moreover, we develop robust implicit and explicit recovery‐type estimators through gradient recovery in an H(curl)‐conforming finite element space. In particular, we apply a modified L² projection in the implicit recovery procedure so as to reduce the computational cost of the recovered gradient. Numerical experiments confirm the theoretical results.     

General phase transition models for vehicular traffic with point constraints on the flow

In this paper, we present a general phase transition model that describes the evolution of vehicular traffic along a one‐lane road. Two different phases are taken into account, according to whether the traffic is low or heavy. The model is given by a scalar conservation law in the free‐flow phase and by a system of 2 conservation laws in the congested phase. The free‐flow phase is described by a one‐dimensional fundamental diagram corresponding to a Newell‐Daganzo type flux. The congestion phase is described by a two‐dimensional fundamental diagram obtained by perturbing a general fundamental flux. In particular, we study the resulting Riemann problems in the case a local point constraint on the flow of the solutions is enforced.     

A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations

The local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L^∞(L²) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L²(L²) norm for the first penalty method with a penalty parameter of order one (h^?1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001