Systems Of Conservation Laws Research Articles

A triangle-based positive semi-discrete Lagrangian–Eulerian scheme via the weak asymptotic method for scalar equations and systems of hyperbolic conservation laws

In this paper, we show how to construct a positive semi-discrete Lagrangian–Eulerian scheme on triangular grids that asymptotically satisfies multidimensional scalar hyperbolic problems and first-order nonlinear systems of conservation laws. The construction of such numerical scheme is based on the improved concept of space–time no-flow curves. For scalar equations, through a sequence of computational grid functions that tend to satisfy the underlying hyperbolic conservation law(s) in a weak sense in the space variable and in a strong sense in the time variable, we prove that the proposed scheme converges to the unique weak solution satisfying a type of Kruzhkov entropy condition. For nonlinear systems of first-order hyperbolic conservation laws, we show that the proposed scheme also satisfies a (weak) positivity principle. This provides an effective numerical analysis tool (based on weak asymptotic methods) to construct a family of continuous functions that asymptotically satisfies scalar conservation laws with discontinuous nonlinearity and systems that have irregular solutions. In order to assess the robustness of the proposed approach, we solve a series of numerical tests on several hyperbolic problems (scalar and systems) using the positive semi-discrete Lagrangian–Eulerian scheme on triangular grids. The results confirm that the proposed method is efficient (since it is Riemann-solver-free) and provides good resolution for scalar models and for the solution of systems of conservation laws.

Experimental convergence rate study for three shock‐capturing schemes and development of highly accurate combined schemes

AbstractWe study experimental convergence rates of three shock‐capturing schemes for hyperbolic systems of conservation laws: the second‐order central‐upwind (CU) scheme, the third‐order Rusanov‐Burstein‐Mirin (RBM), and the fifth‐order alternative weighted essentially non‐oscillatory (A‐WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and convergence rates. We apply the studied schemes to the shallow water equations and conduct their comprehensive numerical convergence study. We verify that while the studied schemes achieve their formal orders of accuracy on smooth solutions, after the shock formation, a part of the computed solutions is affected by shock propagation and both the pointwise and integral convergence rates reduce there. Moreover, while the convergence rates for the CU and A‐WENO schemes, which rely on nonlinear stabilization mechanisms, reduce to the first order, the RBM scheme, which utilizes a linear stabilization, is clearly second‐order accurate. Finally, relying on the conducted experimental convergence rate study, we develop two new combined schemes based on the RBM and either the CU or A‐WENO scheme. The obtained combined schemes can achieve the same high order of accuracy as the RBM scheme in the smooth areas while being non‐oscillatory near the shocks.

A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws

The hyperbolic problem has a unique entropy solution, which maintains the entropy inequality. As such, people hope that the numerical results should maintain the discrete entropy inequalities accordingly. In view of this, people tend to construct entropy stable (ES) schemes. However, traditional numerical schemes cannot directly maintain discrete entropy inequalities. To address this, we here construct an ES finite difference scheme for the nonlinear hyperbolic systems of conservation laws. The proposed scheme can not only maintain the discrete entropy inequality, but also enjoy high-order accuracy. Firstly, we construct the second-order accurate semi-discrete entropy conservative (EC) schemes and ensure that the schemes meet the entropy identity when an entropy pair is given. Then, the second-order EC schemes are used as a building block to achieve the high-order accurate semi-discrete EC schemes. Thirdly, we add a dissipation term to the above schemes to obtain the high-order ES schemes. The term is based on the Weighted Essentially Non-Oscillatory (WENO) reconstruction. Finally, we integrate the scheme using the third-order Runge–Kutta (RK) approach in time. In the end, plentiful one- and two-dimensional examples are implemented to validate the capability of the scheme. In summary, the current scheme has sharp discontinuity transitions and keeps the genuine high-order accuracy for smooth solutions. Compared to the standard WENO schemes, the current scheme can achieve higher resolution.

Tightly coupled hyperbolic treatment of buoyant two-phase flow and transport in porous media

Buoyant incompressible multiphase flow and transport in porous media typically is simulated by coupling an elliptic flow equation with a hyperbolic transport description. The tight coupling between flow and transport either requires a fully implicit solution algorithm or very small time steps, if solved sequentially. In any case, however, a large linear system has to be solved each time step.Here a new solution approach is devised, which relies on solving a coupled hyperbolic system of conservation laws with an explicit finite volume method. Consequently, only local operations have to be performed, which is a great advantage in case of massive parallel simulations and if GPUs are employed.The devised method is based on the isothermal Euler equations with momentum source terms accounting for resistance due to the porous medium and buoyancy. In this system the pressure is proportional to the density and in case of very small Mach numbers and relative density variation, the computed velocity field is divergence free and converges towards the mean total volumetric flow in the porous domain. To account for saturation transport, the system was augmented by an additional hyperbolic equation. In order to obtain the numerical fluxes, a characteristic based approximative Riemann solver was developed and 2nd order accuracy in space and time is achieved by piecewise linear reconstruction.Two-phase flow test cases with buoyant plumes and a lock exchange problem demonstrate that the devised hyperbolic approach recovers the correct incompressible solutions. Numerical experiments also demonstrate that the method is accurate and efficient.Thus the devised method is very promising, and conceptually it also can be applied for flow and transport in heterogeneous media; but it remains to be investigated how well it performs for such cases.

On the Consistency of Arnoldi‐Based Krylov Methods for Conservation Laws

AbstractConservation and consistency are fundamental properties of discretizations of systems of hyperbolic conservation laws. Re‐ cently, these concepts have been extended to the realm of iterative methods by defining locally conservative and flux consistent iterations. In this note, the current status of such iterative methods is summarized. In particular, it has been shown that Krylov subspace methods are locally conservative, but that they are not flux consistent. Here, we approach the problem of quantifying the flux inconsistency of Krylov subspace methods. Krylov methods introduce a time retardation factor into discretizations of linear conservation laws. It has thusfar been unknown how to compute the precise value of this factor. This issue is resolved herein for Arnoldi‐based Krylov subspace methods.

The initial boundary value problem for second order traffic flow models with vacuum: Existence of entropy weak solutions

We study the IBVP for a non-strictly hyperbolic 2 × 2 system of conservation laws on an interval with characteristic boundaries, modelling traffic dynamics including vacuum states on a road stretch. After giving a detailed characterization of the admissible states at the boundary in terms of Riemann solver and entropy conditions, we prove existence of entropy weak solutions for data of bounded variation in the Riemann invariant coordinates by convergence of wave-front tracking approximations.

Global entropy stability for a class of unlimited second-order schemes for 2D hyperbolic systems of conservation laws on unstructured meshes

We introduce a class of second-order accurate finite volume schemes defined on two dimensional unstructured meshes. They satisfy a global discrete entropy stability for an arbitrary strictly convex entropy provided the second-order discretization verifies a dissipative inequality. These schemes do not require usual limitation techniques. Numerical experiments are performed to confirm the accuracy and the shock capturing efficiency of the schemes.

An entropy-stable updated reference Lagrangian smoothed particle hydrodynamics algorithm for thermo-elasticity and thermo-visco-plasticity

This paper introduces a novel upwind Updated Reference Lagrangian Smoothed Particle Hydrodynamics (SPH) algorithm for the numerical simulation of large strain thermo-elasticity and thermo-visco-plasticity. The deformation process is described via a system of first-order hyperbolic conservation laws expressed in referential description, chosen to be an intermediate configuration of the deformation. The linear momentum, the three incremental geometric strains measures (between referential and spatial domains), and the entropy density of the system are treated as conservation variables of this mixed coupled approach, thus extending the previous work of the authors in the context of isothermal elasticity and elasto-plasticity. To guarantee stability from the SPH discretisation standpoint, appropriate entropy-stable upwinding stabilisation is suitably designed and presented. This is demonstrated via the use of the Ballistic free energy of the coupled system (also known as Lyapunov function), to ensure the satisfaction of numerical entropy production. An extensive set of numerical examples is examined in order to assess the applicability and performance of the algorithm. It is shown that the overall algorithm eliminates the appearance of spurious modes (such as hour-glassing and non-physical pressure fluctuations) in the solution, typical limitations observed in the classical Updated Lagrangian SPH framework.

Some remarks on relativistic fluids of divergence type

Relativistic fluids of divergence type constitute a special class of fluids whose states are determined from the knowledge of a scalar generating function χ and a dissipation tensor i.e. a second rank, symmetric and traceless tensor. Both depend upon fourteen variables, known in the literature as Lagrange multipliers, and these multipliers satisfy a symmetric, quasilinear, first order system determined by . For particular choices of the generating function χ, this system can be symmetric-hyperbolic and causal. We show in this work that this characteristic property of fluids of divergence type originates in the overdetermined nature of their dynamical equations. Combining their overdetermined nature with the work of Friedrichs on overdetermined system of conservation laws with more recent work by Boillat, Ruggeri and coworkers, we prove that fluids of divergence type are locally determined by a vector potential depending upon the Lagrange multipliers. For the case where the dynamical variables describing fluid states contain a symmetric energy momentum tensor, this vector potential is the gradient of a scalar field and this field is precisely the generating function χ introduced by Pennisi and independently by Geroch and Lindblom. Examples of scalar generating functions χ are discussed where this system is a symmetric hyperbolic and causal system in an open vicinity of equilibrium states.

Existence result for a two-dimensional system of conservation laws with unilateral constraints

The aim of this paper is to study a pressureless model with unilateral constraint in two dimensions. The corresponding one-dimensional case is a traffic flow model and was studied in Berthelin et al. (2008). The two-dimensional extension is linked to a pedestrian flow. Several challenges, geometric and analytical, appear with respect to the one-dimensional case. We get the existence of weak solutions for general initial data and the constraints taken in the strong sense.

Stability of Riemann solutions to a class of non-strictly hyperbolic systems of conservation laws

This paper is concerned with Riemann problem for a class of non-strictly hyperbolic systems of conservation laws. The two kinds of Riemann solutions including vacuum and delta shock wave are constructed. Under the generalized Rankine-Hugoniot relation and entropy condition, we establish the existence and uniqueness of delta shock wave solutions. Furthermore, by studying the interactions among of the delta shock wave and vacuum as well as contact discontinuity, the Riemann solutions with four kinds of different structures are obtained. Additionally, the stability of the Riemann solutions is obtained under certain perturbation of the initial data.

Nonlinear wave interactions in geochemical modeling

This paper is concerned with the study of the main wave interactions in a system of conservation laws in geochemical modeling. We study the modeling of the chemical complexes on the rock surface. The presence of stable surface complexes affects the relative permeability. We add terms representing surface complexes to the accumulation function in the model presented in [14]. This addition allows to take into account the interaction of ions with the rock surface in the modeling of the oil recovery by the injection of carbonated water. Compatibility hypotheses with the modeling are made on the coefficients of the system to obtain meaningful solutions. We developed a Riemann solver taking into account the complexity of the interactions and bifurcations of nonlinear waves. Such bifurcations occur at the inflection and resonance surfaces. We present the solution of a generalized eigenvalue problem in a (n+1)-dimensional case, which allows the construction of rarefaction curves. A method to find the discontinuous solutions is also presented. We find the solution path for some examples.

Port-Hamiltonian Formulations of Some Elastodynamics Theories of Isotropic and Linearly Elastic Shells: Naghdi–Reissner’s Moderately Thick Shells

The port-Hamiltonian system approach is intended to be an innovative and unifying way of modeling multiphysics systems, by expressing all of them as systems of conservation laws. Indeed, the increasing developments in recent years allow finding better control and coupling strategies. This work aimed to apply such an approach to Naghdi–Reissner’s five-kinematic-field shell model in linear elasticity, while including often-neglected higher-order intrinsic geometric coupling effects, therefore preparing the theoretical background required for the coupling (or interconnection) with an acoustic fluid model and the different types of interactions that can arise among them. The model derived thusly can be used for controller design in a wide variety of applications such as inflatable space structures, launcher tank vibration damping, payload vibration protection using smart materials, and many other related applications.

Well-balanced adaptive compact approximate Taylor methods for systems of balance laws

Compact Approximate Taylor (CAT) methods for systems of conservation laws were introduced by Carrillo and Parés in 2019. These methods, based on a strategy that allows one to extend high-order Lax-Wendroff methods to nonlinear systems without using the Cauchy-Kovalevskaya procedure, have arbitrary even order of accuracy 2p and use (2p+1)-point stencils, where p is an arbitrary positive integer. More recently in 2021 Carrillo, Macca, Parés, Russo and Zorío introduced a strategy to get rid of the spurious oscillations close to discontinuities produced by CAT methods. This strategy led to the so-called Adaptive CAT (ACAT) methods, in which the order of accuracy – and thus the width of the stencils – is adapted to the local smoothness of the solution. The goal of this paper is to extend CAT and ACAT methods to systems of balance laws. To do this, the source term is written as the derivative of its indefinite integral that is formally treated as a flux function. The well-balanced property of the methods is discussed and a variant that allows in principle to preserve any stationary solution is presented. The resulting methods are then applied to a number of systems going from a linear scalar conservation law to the 2D Euler equations with gravity, passing by the Burgers equations with source term and the 1D shallow water equations: the order and well-balanced properties are checked in several numerical tests.

Divergence-Preserving MHD Solutions Using a Noniterative Hyperbolic Conservation Law Scheme

The divergence constraint, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\nabla$</tex-math> </inline-formula> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\cdot$</tex-math> </inline-formula> B <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$=$</tex-math> </inline-formula> 0, presents a central challenging numerical feature of electromagnetic and magnetohydrodynamic solvers. In general, this constraint is not strictly enforced in hyperbolic solvers and tends to accumulate error over the duration of a simulation, polluting the solution with unphysical behavior. In this article, we present a numerical scheme that constructs a system of hyperbolic conservation laws that already satisfy the divergence constraint, starting from the potential form of Maxwell’s equations and basic derivative relationships. The new system consists only of the first-order spatial differentiation, in contrast to the potential equations which are second order; this allows them to be directly integrated into the existing hyperbolic solvers with no additional implementation changes needed. Furthermore, since the system of conservation laws constructed automatically satisfies the divergence criteria, no subiteration is required to resolve the divergence-free solution. We present the results of integrating this new conservation law approach in an existing finite volume plasmadynamic solver that discretizes the coupled Maxwell and Navier–Stokes equations. Numerical tests indicate that the new approach can strongly resolve physical solutions. The divergence error is also measured and quantified for these problems, showing that the new system suppresses divergence error by many orders of magnitude.

A family of second-order dissipative finite volume schemes for hyperbolic systems of conservation laws

We propose and study a family of formally second-order accurate schemes to approximate weak solutions of hyperbolic systems of conservation laws. Theses schemes are based on a dissipative property satisfied by the second-order discretization in space. They are proven to satisfy a global entropy inequality for a generic strictly convex entropy. These schemes do not involve limitation techniques. Numerical results are provided to illustrate their accuracy and stability.

Development of singularities in the relativistic Euler equations

The purpose of this paper is to study the phenomenon of singularity formation in large data problems for C 1 C^1 solutions to the Cauchy problem of the relativistic Euler equations. The classical theory established by [P. D. Lax [J. Math. Phys. 5 (1964), pp. 611–613] shows that, for 2 × 2 2\times 2 hyperbolic systems, the break-down of C 1 C^1 solutions occurs in finite time if initial data contain any compression in some truly non-linear characteristic field under some additional conditions, which include genuine non-linearity and the strict positivity of the difference between two corresponding eigenvalues. These harsh structural assumptions mean that it is highly non-trivial to apply this theory to archetypal systems of conservation laws, such as the (1+1)-dimensional relativistic Euler equations. Actually, in the (1+1)-dimensional spacetime setting, if the mass-energy density ρ \rho does not vanish initially at any finite point, the essential difficulty in considering the possible break-down is coming up with a way to obtain sharp enough control on the lower bound of ρ \rho . To this end, based on introducing several key artificial quantities and some elaborate analysis on the difference of the two Riemann invariants, we characterized the decay of mass-energy density lower bound in time, and ultimately made some concrete progress. On the one hand, for the C 1 C^1 solutions with large data and possible far field vacuum to the isentropic flow, we verified the theory obtained by P. D. Lax in 1964. On the other hand, for the C 1 C^1 solutions with large data and strictly positive initial mass-energy density to the non-isentropic flow, we exhibit a numerical value N N , thought of as representing the strength of an initial compression, above which all initial data lead to a finite-time singularity formation. These singularities manifest as a blow-up in the gradient of certain Riemann invariants associated with corresponding systems.

T5 configurations and hyperbolic systems

In this paper, we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in [B. Kirchheim, S. Müller and V. Šverák, Studying Nonlinear PDE by Geometry in Matrix Space (Springer, 2003), Sec. 7], and many of its properties have already been shown in [A. Lorent and G. Peng, Null Lagrangian measures in subspaces, compensated compactness and conservation laws, Arch. Ration. Mech. Anal. 234(2) (2019) 857–910; A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156]. In particular, in [A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156], it is shown that the differential inclusion does not contain any [Formula: see text] configurations. Here, we continue that study by showing that the differential inclusion does not contain [Formula: see text] configurations.

“A posteriori” limited high order and robust schemes for transient simulations of fluid flows in gas dynamics

In this paper, we propose a novel approximation strategy for time-dependent hyperbolic systems of conservation laws for the Euler system of gas dynamics that aims to represent the dynamics of strong interacting discontinuities. The goal of our method is to allow an approximation with a high-order of accuracy in smooth regions of the flow, while ensuring robustness and a non-oscillatory behaviour in the regions of steep gradients, in particular across shocks.Following the Multidimensional Optimal Order Detection (MOOD) ([15,17]) approach, a candidate solution is computed at a next time level via a high-order accurate explicit scheme ([3,5]). A so-called detector determines if the candidate solution reveals any spurious oscillations or numerical issue and, if so, only the troubled cells are locally recomputed via a more dissipative scheme. This allows to design a family of “a posteriori” limited, robust and positivity preserving, as well as high accurate, non-oscillatory and effective scheme. Among the detecting criteria of the novel MOOD strategy, two different approaches from literature, based on the work of [15,17] and on [35], are investigated. Numerical examples in 1D and 2D, on structured and unstructured meshes, are proposed to assess the effective order of accuracy for smooth flows, the non-oscillatory behaviour on shocked flows, the robustness and positivity preservation on more extreme flows.

On the time-dependent solution of fluid models driven by an &lt;i&gt;M&lt;/i&gt;/&lt;i&gt;M&lt;/i&gt;/1 queue using a probabilistic approach

More study has been performed on the steady state analysis of fluid models. However, the transient analysis of fluid models has not been extensively carried due to complexity of the problem. This is because the system of conservation laws for which no closed form solution is available. The motivation behind this paper is to provide a new methodology to find transient distribution of buffer content for fluid models. In this paper, the time-dependent solution of a fluid model driven by an M/M/1 queue is derived using a simple probability approach. Finally, numerical results are illustration for the proposed approach.