General Systems Of Conservation Laws Research Articles

Wave Interactions of the Aw-Rascle Model for Generalized Chaplygin Gas

In this paper, we investigate the elementary wave interactions of the Aw-Rascle model for the generalized Chaplygin gas. We construct the unique solution by the characteristic analysis method and obtain the stability of the corresponding Riemann solutions under such small perturbations on the initial values. We find that the elementary wave interactions have a much more simple structure for Temple class than general systems of conservation laws. It is important to study the elementary waves interactions of the traffic flow system for the generalized Chaplygin gas not only because of their significance in practical applications in the traffic flow system, but also because of their basic role for the general mathematical theory.

Wave Interactions for the Suliciu Relaxation System

In this paper, we investigate the elementary wave interactions for the Suliciu relaxation system and construct uniquely the solution by the characteristic analysis method in the phase plane. We find that the elementary wave interactions have a much simpler structure for the Temple class than the general systems of conservation laws. It is observed that the Riemann solutions of the Suliciu relaxation system are stable under the small perturbation on the Riemann initial data.

Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods

We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (J Sci Comput 77:579–596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well known that high-order methods generate spurious oscillations near discontinuities which can develop in the solution for nonlinear problems, even when the initial data is smooth. We propose a novel multi-element SIAC filtering technique applied to the DGSEM as a shock capturing method. We design the SIAC filtering such that the numerical scheme remains high-order accurate and that the shock capturing is applied adaptively throughout the domain. The shock capturing method is derived for general systems of conservation laws. We apply the novel SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics equations to several standard test problems with a variety of boundary conditions.

Nonlinear wave analysis of geochemical injection for multicomponent two phase flow in porous media

We study nonlinear waves in two phase geochemical flow in one spatial dimension in porous media. We assume that each chemical species may flow in one or both phases with concentrations obeying thermodynamical equilibrium. We present a new methodology for reducing a number of equations applicable to injection problems for general systems of conservation laws in Geochemistry. This reduction is achieved by solving a nonlinear inverse problem. Nevertheless, we are able to perform a complete and explicit characteristic analysis, obtaining rarefaction and shock waves that are used to solve the representative Riemann problems, besides the main bifurcations structures appearing in the phase space are derived with explicit expressions. We illustrate the methodology by means of an example with four equations.

A Note on Weak Solutions of Conservation Laws and Energy/Entropy Conservation

A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such cases most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws; they are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions. Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake the role of physical admissibility conditions for weak solutions. We want to answer the question: what is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality? An archetypal example of such a result was derived for the incompressible Euler system in the context of Onsager’s conjecture in the early nineties. This general result can serve as a simple criterion to numerous systems of mathematical physics to prescribe the regularity of solutions needed for an appropriate companion law to be satisfied.

Scalar Green function bounds for instantaneous shock location and one-dimensional stability of viscous shock waves

In this paper, we investigate and prove the nonlinear stability of viscous shock wave solutions of a scalar viscous conservation law, using the methods developed for general systems of conservation laws by Howard, Mascia, Zumbrun and others, based on instantaneous tracking of the location of the perturbed viscous shock wave. In some sense, this paper extends the treatment in a previous expository work of Zumbrun [Instantaneous shock location ...] on Burgers equation to the general case, giving an exposition of these methods in the simplest setting of scalar equations. In particular we give by a rescaling argument a simple treatment of nonlinear stability in the small-amplitude case.

A quadratic interaction estimate for conservation laws: motivations, techniques and open problems

In a series of joint works with S. Bianchini [3, 4, 5], we proved a quadratic interaction estimate for general systems of conservation laws. Aim of this paper is to present the results obtained in the three cited articles [3, 4, 5], discussing how they are related with the general theory of hyperbolic conservation laws. To this purpose, first we explain why this quadratic estimate is interesting, then we give a brief overview of the techniques we used to prove it and finally we present some related open problems.

Quadratic Interaction Functional for General Systems of Conservation Laws

For the Glimm scheme approximation \({u_\varepsilon}\) to the solution of the system of conservation laws in one space dimension $$u_t + f(u)_x = 0, \qquad u(0, x) = u_0(x) \in \mathbb{R}^n,$$with initial data u 0 with small total variation, we prove a quadratic (w.r.t. Tot. Var. (u 0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux f are made (apart from smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems.More precisely, we obtain the following results: a new analysis of the interaction estimates of simple waves; a Lagrangian representation of the derivative of the solution, i.e., a map \({\mathtt{x}(t, w)}\) which follows the trajectory of each wave w from its creation to its cancellation; the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves; a new functional \({\mathfrak{Q}}\) controlling the variation in speed of the waves w.r.t. time. This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems.The main result is that the distribution \({D_{t} \hat \sigma_k(t,w)}\) is a measure with total mass \({\leq \mathcal{O}(1) {\rm Tot. Var.} (u_0)^2}\) , where \({\hat{\sigma}_k(t, w)}\) is the speed given to the wave w by the Riemann problem at the grid point \({(i\varepsilon, \mathtt{x}(i\varepsilon, w)), t \in [i\varepsilon, (i + 1)\varepsilon)}\).

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.

The Inviscid Limits to Piecewise Smooth Solutions for a General Parabolic System

We study the viscous limit problem for a general system of conservation laws. We prove that if the solution of the underlying inviscid problem is piecewise smooth with finitely many noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding viscous system which converge to the inviscid solutions away from shock discontinuities at a rate of ε1 as the viscosity coefficient ε vanishes.

Upwind-biased FORCE schemes with applications to free-surface shallow flows

In this paper we develop numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple-space dimensions on structured meshes. The proposed method is an extension of the multidimensional FORCE flux developed by Toro et al. (2009) [14]. Here we introduce upwind bias by modifying the shape of the staggered mesh of the original FORCE method. The upwind bias is evaluated using an estimate of the largest eigenvalue, which in any case is needed for selecting a time step. The resulting basic flux is first-order accurate and monotone. For the linear advection equation, the proposed UFORCE method reproduces exactly the upwind Godunov method. Extension to non-linear systems has been done empirically via the two-dimensional inviscid shallow water equations. Second order of accuracy in space and time on structured meshes is obtained in the framework of finite volume methods. The proposed method improves the accuracy of the solution for small Courant numbers and intermediate waves associated with linearly degenerate fields (contact discontinuities, shear waves and material interfaces). It achieves comparable accuracy to that of upwind methods with approximate Riemann solvers, though retaining the simplicity and efficiency of centred methods. The performance of the schemes is assessed on a suite of test problems for the two-dimensional shallow water equations.

Linearization of the Riemann problem for a triangular system of conservation laws and delta shock wave formation process

AbstractUsing the weak asymptotic method, we approximate a triangular system of conservation laws arising from the so‐called generalized pressureless gas dynamics by a diagonal linear system. Then, we apply the usual method of characteristics to find approximate solution to the original system. As a consequence, we shall see how the delta shock wave naturally arises along the characteristics.Also, we propose a procedure that could be applied to more general systems of conservation laws. Copyright © 2009 John Wiley & Sons, Ltd.

Adaptive mesh fluid simulations on GPU

We describe an implementation of compressible inviscid fluid solvers with block-structured adaptive mesh refinement on Graphics Processing Units using NVIDIA's CUDA. We show that a class of high resolution shock capturing schemes can be mapped naturally on this architecture. Using the method of lines approach with the second order total variation diminishing Runge-Kutta time integration scheme, piecewise linear reconstruction, and a Harten-Lax-van Leer Riemann solver, we achieve an overall speedup of approximately 10 times faster execution on one graphics card as compared to a single core on the host computer. We attain this speedup in uniform grid runs as well as in problems with deep AMR hierarchies. Our framework can readily be applied to more general systems of conservation laws and extended to higher order shock capturing schemes. This is shown directly by an implementation of a magneto-hydrodynamic solver and comparing its performance to the pure hydrodynamic case. Finally, we also combined our CUDA parallel scheme with MPI to make the code run on GPU clusters. Close to ideal speedup is observed on up to four GPUs.

Decay estimate for a new measure of rarefaction waves in general systems of conservation laws

The sharp decay estimate of rarefaction waves in terms of a partial ordering among positive measures is not only interesting in itself but also crucial in the study of convergence rate of vanishing viscosity approximations, cf. Bressan and Yang (2004) [10]. Such an estimate is well established for genuinely nonlinear system of conservation laws, cf. Bressan and Yang (2004) [9]. But similar result is not available for non-genuinely nonlinear system. In this paper, we give a new measure about the rarefaction waves. In addition, a sharp decay estimate of the new measure is given for the cubic nonlinear system of conservation laws.

FORCE schemes on unstructured meshes I: Conservative hyperbolic systems

This paper is about the construction of numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple space dimensions on structured and unstructured meshes. The work is a multi-dimensional extension of the one-dimensional FORCE flux and is closely related to the work of Nessyahu–Tadmor and Arminjon. The resulting basic flux is first-order accurate and monotone; it is then extended to arbitrary order of accuracy in space and time on unstructured meshes in the framework of finite volume and discontinuous Galerkin methods. The performance of the schemes is assessed on a suite of test problems for the multi-dimensional Euler and Magnetohydrodynamics equations on unstructured meshes.

$L^1$ continuous dependence for the Euler equations of compressible fluids dynamics

We prove the $L^1$ continuous dependence of entropy solutions for the $2 \times 2$ (isentropic) and the $3\times 3$ (non-isentropic) systems of inviscid fluid dynamics in one-space dimension. We follow the approach developed by the second author for solutions with small total variation to general systems of conservation laws in [11, 14]. For the systems of fluid dynamics under consideration here, our estimates are more precise and we cover entropy solutions with large total variation.

Entropy satisfying flux vector splittings and kinetic BGK models

We establish forward and backward relations between entropy satisfying BGK relaxation models such as those introduced previously by the author and the first order flux vector splitting numerical methods for general systems of conservation laws. Classically, to a kinetic BGK model that is compatible with some family of entropies we can associate an entropy flux vector splitting. We prove that the converse is true: any entropy flux vector splitting can be interpreted by a kinetic model, and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes.We deduce a new proof of discrete entropy inequalities under a sharp CFL condition that generalizes the monotonicity criterion in the scalar case. In particular, this gives a stability condition for numerical kinetic methods with noncompact velocity support.A new interpretation of general kinetic schemes is also provided via approximate Riemann solvers.We deduce the construction of finite velocity relaxation systems for gas dynamics, and obtain a HLLC scheme for which we are able to prove positiveness of density and internal energy, and discrete entropy inequalities.

Nonclassical Riemann solvers and kinetic relations I. A nonconvex hyperbolic model of phase transitions

We consider a nonlinear hyperbolic model describing phase transitions in nonlinear elastodynamics. The Riemann problem is solved uniquely, provided we supplement the fundamental conservation laws (mass, momentum) with a kinetic relation. The latter takes into account small-scale mechanisms, such as the viscosity and capillary effects in the material under consideration. Our construction generalizes, to solutions of arbitrary large amplitude of the model under study, an approach proposed by Hayes and LeFloch for general systems of conservation laws. The Riemann solutions may contain rarefaction waves, (compressive) classical shock waves, as well as (undercompressive) nonclassical shocks.

Nonlinear dynamics of Kelvin–Helmholtz unstable magnetized jets: Three-dimensional effects

A numerical study of the Kelvin–Helmholtz instability in compressible magnetohydrodynamics is presented. The three-dimensional simulations consider shear flow in a cylindrical jet configuration, embedded in a uniform magnetic field directed along the jet axis. The growth of linear perturbations at specified poloidal and axial mode numbers demonstrate intricate nonlinear coupling effects. The physical mechanisms leading to induced secondary Kelvin–Helmholtz instabilities at higher mode numbers are identified. The initially weak magnetic field becomes locally dominant in the nonlinear dynamics before and during saturation. Thereby, it controls the jet deformation and eventual breakup. The results are obtained using the Versatile Advection Code [G. Tóth, Astrophys. Lett. Commun. 34, 245 (1996)], a software package designed to solve general systems of conservation laws. An independent calculation of the same Kelvin–Helmholtz unstable jet configuration using a three-dimensional pseudospectral code gives important insights into the coupling and excitation events of the various linear mode numbers.

Existence de chocs faibles pour des systèmes quasi-linéaires hyperboliques multidimensionnels

For general systems of conservation laws, in any space dimension, we prove the existence of weak shocks corresponding to a simple real genuinely nonlinear eigenvalue of the system and for a time of existence T independent of the strength of the shock. For instance, the results apply to Euler's equations of gas dynamics.