Systems Of Nonlinear Conservation Laws Research Articles

Crowd dynamics through non-local conservation laws

We present a Lax-Friedrichs type scheme to compute the solutions of a class of non-local and non-linear systems of conservation laws in several space dimensions. The convergence of the approximate solutions is proved by providing suitable L 1 , L ∞ and BV uniform bounds. To illustrate the performances of the scheme, we consider an application to crowd dynamics. Numerical integrations show the formation of lanes in groups moving in opposite directions. This is joint work with R.M. Colombo (INDAM Unit, University of Brescia).

On the Convergence of a Shock Capturing Discontinuous Galerkin Method for Nonlinear Hyperbolic Systems of Conservation Laws

In this paper, we present a shock capturing discontinuous Galerkin (SC-DG) method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [14], our approach is new in that we do not use streamline diffusion (SD) stabilization. It is proved that an artificial-viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued (emv) solution. The result is valid for general systems and arbitrary order discontinuous Galerkin method.

Error Estimate for Time-Explicit Finite Volume Approximation of Strong Solutions to Systems of Conservation Laws

We study the finite volume approximation of strong solutions to nonlinear systems of conservation laws. We focus on time-explicit schemes on unstructured meshes, with entropy satisfying numerical fluxes. The numerical entropy dissipation is quantified at each interface of the mesh, which enables to prove a weak–BV estimate for the numerical approximation under a strengthen CFL condition. Then we derive error estimates in the multidimensional case, using the relative entropy between the strong solution and its finite volume approximation. The error terms are carefully studied, leading to a classical $h^1/4$ estimate in $L^2$ under this strengthen CFL condition.

Macroscopic modeling approach to estimate traffic-related emissions in urban areas

This paper integrates a macroscopic dynamic traffic assignment (DTA) model for urban traffic flow with an instantaneous emission model to investigate traffic-related emissions in urban areas of arbitrary shape. It is assumed that homogeneous travelers continuously distributed over the urban areas tend to choose a path to minimize their total travel cost based on instantaneous traffic information. The macroscopic DTA model consists of a two-dimensional hyperbolic system of nonlinear conservation laws with source terms and an Eikonal-type equation used to describe the path-choice behavior of travelers. A solution algorithm for the model is designed as a Runge–Kutta Discontinuous Galerkin method for the hyperbolic system coupled with a fast sweeping method for the Eikonal-type equation on unstructured meshes. A case study investigating macroscopic characteristics of urban traffic flow and predicting exhaust emissions emitted by various types of vehicles in urban areas is conducted to illustrate the applicability of the model and the effectiveness of the algorithm.

On the Accuracy of WENO and CWENO Reconstructions of Third Order on Nonuniform Meshes

Third order WENO and CWENO reconstruction are widespread high order reconstruction techniques for numerical schemes for hyperbolic conservation and balance laws. In their definition, there appears a small positive parameter, usually called $\epsilon$, that was originally introduced in order to avoid a division by zero on constant states, but whose value was later shown to affect the convergence properties of the schemes. Recently, two detailed studies of the role of this parameter, in the case of uniform meshes, were published. In this paper we extend their results to the case of finite volume schemes on non-uniform meshes, which is very important for h-adaptive schemes, showing the benefits of choosing $\epsilon$ as a function of the local mesh size $h_j$. In particular we show that choosing $\epsilon=h_j^2$ or $\epsilon=h_j$ is beneficial for the error and convergence order, studying on several non-uniform grids the effect of this choice on the reconstruction error, on fully discrete schemes for the linear transport equation, on the stability of the numerical schemes. Finally we compare the different choices for $\epsilon$ in the case of a well-balanced scheme for the Saint-Venant system for shallow water flows and in the case of an h-adaptive scheme for nonlinear systems of conservation laws and show numerical tests for a two-dimensional generalisation of the CWENO reconstruction on locally adapted meshes.

The Brio system with initial conditions involving Dirac masses: A result afforded by a distributional product

The Brio system is a 2 × 2 fully nonlinear system of conservation laws which arises as a simplified model in the study of plasmas. The present paper offers explicit solutions to this system subjected to initial conditions containing Dirac masses. The concept of a solution emerges within the framework of a distributional product and represents a consistent extension of the concept of a classical solution. Among other features, the result shows that the space of measures is not sufficient to contain all solutions of this problem.

Nonlocal Systems of Conservation Laws in Several Space Dimensions

We present a Lax--Friedrichs-type algorithm to numerically integrate a class of nonlocal and nonlinear systems of conservation laws in several space dimensions. The convergence of the approximate solutions is proved, also providing the existence of a solution in a slightly more general setting than in other results in the current literature. An application to a crowd dynamics model is considered.

Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

We propose a simple modification of standard WENO finite volume methods for Cartesian grids, which retains the full spatial order of accuracy of the one-dimensional discretization when applied to nonlinear multidimensional systems of conservation laws. We derive formulas, which allow us to compute high-order accurate point values of the conserved quantities at grid cell interfaces. Using those point values, we can compute a high-order flux at the center of a grid cell interface. Finally, we use those point values to compute high-order accurate averaged fluxes at cell interfaces as needed by a finite volume method. The method is described in detail for the two-dimensional Euler equations of gas dynamics. An extension to the three-dimensional case as well as to other nonlinear systems of conservation laws in divergence form is straightforward. Furthermore, similar ideas can be used to improve the accuracy of WENO type methods for hyperbolic systems which are not in divergence form. Several test computations confirm the high-order accuracy for smooth nonlinear problems.

Shock capturing with entropy-based artificial viscosity for staggered grid discontinuous spectral element method

This paper presents a shock capturing technique for a staggered grid discontinuous spectral element method (DSEM), which adds localized and smooth artificial viscosity to systems of nonlinear conservation laws. The artificial diffusivity model used in this work is a modified form of the entropy viscosity (EV) presented by Guermond et al. (2011). We extend the application of this method to high-order discontinuous schemes for the simulation of high speed flows with discontinuities on staggered grids. Direct implementation of the entropy viscosity method in DSEM leads to a non-smooth artificial viscosity, which in turn leads to oscillations and instability of the solution. To smoothen the artificial viscosity, the EV method is coupled with a spectral filter and an interface treatment technique. The resulting artificial viscosity is locally large near discontinuities and transitions smoothly to zero in smooth flow regions. The method enables using elements with orders higher than unity while avoiding adaptive mesh refinement and preserving the locality and compactness of the discontinuous Galerkin (DG) scheme. The method is implemented for the inviscid compressible Euler equations in two space dimensions and its effectiveness is demonstrated through its application to a series of benchmark problems.

Dynamic Correlators of Fermi-Pasta-Ulam Chains and Nonlinear Fluctuating Hydrodynamics

We study the equilibrium time correlations for the conserved fields of classical anharmonic chains and argue that their dynamic correlator can be predicted on the basis of nonlinear fluctuating hydrodynamics. In fact, our scheme is more general and would also cover other one-dimensional Hamiltonian systems, for example, classical and quantum fluids. Fluctuating hydrodynamics is a nonlinear system of conservation laws with noise. For a single mode, it is equivalent to the noisy Burgers equation, for which explicit solutions are available. Our focus is the case of several modes. No exact solution has been found so far, and we rely on a one-loop approximation. The resulting mode-coupling equations have a quadratic memory kernel and describe the time evolving 3×3 correlator matrix of the locally conserved fields. Long time asymptotics is computed analytically, and finite time properties are obtained through a numerical simulation of the mode-coupling equations.

Multi-dimensional hybrid Fourier continuation–WENO solvers for conservation laws

We introduce a multi-dimensional point-wise multi-domain hybrid Fourier-Continuation/WENO technique (FC–WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and essentially dispersionless, spectral, solution away from discontinuities, as well as mild CFL constraints for explicit time stepping schemes. The hybrid scheme conjugates the expensive, shock-capturing WENO method in small regions containing discontinuities with the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [A. Harten, Adaptive multiresolution schemes for shock computations, J. Comput. Phys. 115 (1994) 319–338]. We consider a WENO scheme of formal order nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws are investigated for problems with both smooth and non-smooth solutions. The Euler equations for gas dynamics are solved for the Mach 3 and Mach 1.25 shock wave interaction with a small, plain, oblique entropy wave using the hybrid FC–WENO, the pure WENO and the hybrid central difference–WENO (CD–WENO) schemes. We demonstrate considerable computational advantages of the new FC-based method over the two alternatives. Moreover, in solving a challenging two-dimensional Richtmyer–Meshkov instability (RMI), the hybrid solver results in seven-fold speedup over the pure WENO scheme. Thanks to the multi-domain formulation of the solver, the scheme is straightforwardly implemented on parallel processors using message passing interface as well as on Graphics Processing Units (GPUs) using CUDA programming language. The performance of the solver on parallel CPUs yields almost perfect scaling, illustrating the minimal communication requirements of the multi-domain strategy. For the same RMI test, the hybrid computations on a single GPU, in double precision arithmetics, displays five- to six-fold speedup over the hybrid computations on a single CPU. The relative speedup of the hybrid computation over the WENO computations on GPUs is similar to that on CPUs, demonstrating the advantage of hybrid schemes technique on both CPUs and GPUs.

Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate method for a general system of conservation laws. Furthermore, we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws. Convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown. Numerical experiments are presented to illustrate the robustness of the proposed schemes.

Multidimensional nonlinear systems of conservation laws

In this paper, we review the history and progress of the study in the field of multidimensional nonlinear systems of conservation laws. The course of the study can be described as local classical solutions, solutions with fan-shaped wave structure, solutions with flower-shaped wave structure, global solutions and mixed type equations. According to this line, we introduce main results obtained in recent decades, and present some open problems and difficulties.

Two-dimensional Riemann problem involving three J’s for a hyperbolic system of nonlinear conservation laws

In this paper, we study the Riemann problem for a two-dimensional hyperbolic system of conservation laws with initial data projecting exactly three contact discontinuities. We use the generalized characteristic analysis method to analyze interactions of contact discontinuities, and obtain six analytical solutions and corresponding criteria. Delta shock waves and spiral structures appear in some solutions. We also conduct numerical experiments, and numerical results confirm the geometric structures of the constructed solutions.

A space–time smooth artificial viscosity method for nonlinear conservation laws

We introduce a new methodology for adding localized, space–time smooth, artificial viscosity to nonlinear systems of conservation laws which propagate shock waves, rarefactions, and contact discontinuities, which we call the C-method. We shall focus our attention on the compressible Euler equations in one space dimension. The novel feature of our approach involves the coupling of a linear scalar reaction–diffusion equation to our system of conservation laws, whose solution C(x,t) is the coefficient to an additional (and artificial) term added to the flux, which determines the location, localization, and strength of the artificial viscosity. Near shock discontinuities, C(x,t) is large and localized, and transitions smoothly in space–time to zero away from discontinuities. Our approach is a provably convergent, spacetime-regularized variant of the original idea of Richtmeyer and Von Neumann, and is provided at the level of the PDE, thus allowing a host of numerical discretization schemes to be employed.We demonstrate the effectiveness of the C-method with three different numerical implementations and apply these to a collection of classical problems: the Sod shock-tube, the Osher–Shu shock-tube, the Woodward–Colella blast wave and the Leblanc shock-tube. First, we use a classical continuous finite-element implementation using second-order discretization in both space and time, FEM-C. Second, we use a simplified WENO scheme within our C-method framework, WENO-C. Third, we use WENO with the Lax–Friedrichs flux together with the C-equation, and call this WENO-LF-C. All three schemes yield higher-order discretization strategies, which provide sharp shock resolution with minimal overshoot and noise, and compare well with higher-order WENO schemes that employ approximate Riemann solvers, outperforming them for the difficult Leblanc shock tube experiment.

Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions

We extend the multi-level Monte Carlo (MLMC) in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data. The algorithm is presented and several issues arising in the massively parallel numerical implementation are addressed. In particular, we present a novel load balancing procedure that ensures scalability of the MLMC algorithm on massively parallel hardware. A new code is described and applied to simulate uncertain solutions of the Euler equations and ideal magnetohydrodynamics (MHD) equations. Numerical experiments showing the robustness, efficiency and scalability of the proposed algorithm are presented.

Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

We introduce a multi-domain Fourier-continuation/WENO hybrid method (FC–WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys essentially dispersionless, spectral character away from discontinuities, as well as mild CFL constraints (comparable to those of finite difference methods). The hybrid scheme employs the expensive, shock-capturing WENO method in small regions containing discontinuities and the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [J. Comput. Phys. 115 (1994) 319–338]. We consider WENO schemes of formal orders five and nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws is investigated for problems with both smooth and non-smooth solutions. In the latter case, we solve the Euler equations for gas dynamics for the standard test case of a Mach three shock wave interacting with an entropy wave, as well as a shock wave (with Mach 1.25, three or six) interacting with a very small entropy wave and evaluate the efficiency of the hybrid FC–WENO method as compared to a purely WENO-based approach as well as alternative hybrid based techniques. We demonstrate considerable computational advantages of the new FC-based method, suggesting a potential of an order of magnitude acceleration over alternatives when extended to fully three-dimensional problems.

A dispersive regularization of the modulational instability of stratified gravity waves

We derive high-order corrections to a modulation theory for the propagation of internal gravity waves in a density-stratified fluid with coupling to the mean flow. The methodology we use allows for strong modulations of wavenumber and mean flow, extending previous approaches developed for the quasi-monochromatic regime. The wave mean flow modulation equations consist of a system of nonlinear conservation laws that may be hyperbolic, elliptic or of mixed type. We investigate the regularizing properties of the asymptotic correction terms in the case when the system becomes unstable and ill-posed due to a change of type (loss of hyperbolicity). A linear analysis reveals that the regularization by the added correction terms does so by introducing a short-wave cut-off of the unstable wavenumbers. We perform various numerical experiments that confirm the regularizing properties of the correction terms, and show that the growth of unstable modes is tempered by nonlinearity. We also find an excellent agreement between the solution of the corrected modulation system and the modulation variables extracted from the numerical solution of the nonlinear Boussinesq equations.

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89–112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge–Kutta methods, Appl. Numer. Math. 58 (2008) 1675–1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed. A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.

Relaxation approaches to the optimal control of the Euler equations

The treatment of control problems governed by systems of conservation laws poses serious challenges for analysis and numerical simulations. This is due mainly to shock waves that occur in the solution of nonlinear systems of conservation laws. In this article, the problem of the control of Euler flows in gas dynamics is considered. Numerically, two semi-linear approximations of the Euler equations are compared for the purpose of a gradient-based algorithm for optimization. One is the Lattice-Boltzmann method in one spatial dimension and five velocities (D1Q5 model) and the other is the relaxation method. An adjoint method is used. Good results are obtained even in the case where the solution contains discontinuities such as shock waves or contact discontinuities.