Hyperbolic Systems Of Balance Laws Research Articles

Dynamical behaviour of a logarithmically sensitive chemotaxis model under time-dependent boundary conditions

Abstract This article studies the dynamical behaviour of classical solutions of a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, with time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in the $H^2$ -space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero as time goes to infinity. There is no smallness restriction on the magnitude of the initial perturbations. Moreover, numerical simulations show that the assumptions on the boundary data are necessary for the above-mentioned results to hold true. In addition, numerical results indicate that the solutions converge asymptotically to time-periodic states if the boundary data are time-periodic.

Semi-implicit-Type Order-Adaptive CAT2 Schemes for Systems of Balance Laws with Relaxed Source Term

AbstractIn this paper, we present two semi-implicit-type second-order compact approximate Taylor (CAT2) numerical schemes and blend them with a local a posteriori multi-dimensional optimal order detection (MOOD) paradigm to solve hyperbolic systems of balance laws with relaxed source terms. The resulting scheme presents the high accuracy when applied to smooth solutions, essentially non-oscillatory behavior for irregular ones, and offers a nearly fail-safe property in terms of ensuring the positivity. The numerical results obtained from a variety of test cases, including smooth and non-smooth well-prepared and unprepared initial conditions, assessing the appropriate behavior of the semi-implicit-type second order CATMOOD schemes. These results have been compared in the accuracy and the efficiency with a second-order semi-implicit Runge-Kutta (RK) method.

A Multiscale Method for Two-Component, Two-Phase Flow with a Neural Network Surrogate

Understanding the dynamics of phase boundaries in fluids requires quantitative knowledge about the microscale processes at the interface. We consider the sharp-interface motion of the compressible two-component flow and propose a heterogeneous multiscale method (HMM) to describe the flow fields accurately. The multiscale approach combines a hyperbolic system of balance laws on the continuum scale with molecular-dynamics (MD) simulations on the microscale level. Notably, the multiscale approach is necessary to compute the interface dynamics because there is—at present—no closed continuum-scale model. The basic HMM relies on a moving-mesh finite-volume method and has been introduced recently for the compressible one-component flow with phase transitions by Magiera and Rohde in (J Comput Phys 469: 111551, 2022). To overcome the numerical complexity of the MD microscale model, a deep neural network is employed as an efficient surrogate model. The entire approach is finally applied to simulate droplet dynamics for argon-methane mixtures in several space dimensions. To our knowledge, such compressible two-phase dynamics accounting for microscale phase-change transfer rates have not yet been computed.

Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior – an approximation of the steady solution – which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.

Stabilization of a multi-dimensional system of hyperbolic balance laws

Stabilization of a multi-dimensional system of hyperbolic balance laws

Exponential Asymptotic Stability of Riemann Shocks of Hyperbolic Systems of Balance Laws

Exponential Asymptotic Stability of Riemann Shocks of Hyperbolic Systems of Balance Laws

A flux-differencing formula for split-form summation by parts discretizations of non-conservative systems: Applications to subcell limiting for magneto-hydrodynamics

In this paper, we show that diagonal-norm summation by parts (SBP) discretizations of general non-conservative systems of hyperbolic balance laws can be rewritten as a finite-volume-type formula, also known as flux-differencing formula, if the non-conservative terms can be written as the product of a local and a symmetric contribution. Furthermore, we show that the existence of a flux-differencing formula enables the use of recent subcell limiting strategies to improve the robustness of the high-order discretizations. The methods are valid on unstructured curvilinear grids using tensor-product basis functions.To demonstrate the utility of the novel flux-differencing formula, we construct hybrid schemes that combine high-order SBP methods (the discontinuous Galerkin spectral element method and a high-order SBP finite difference method) with a compatible low-order finite volume (FV) scheme at the subcell level. We apply the hybrid schemes to solve challenging magnetohydrodynamics (MHD) problems featuring strong shocks.

Hyperbolic systems of balance laws with stiff source

The method of redistribution of damping is employed for solving the Cauchy problem, in the BV setting, for hyperbolic systems of balance laws with partially dissipative source that becomes stiff as the relaxation time shrinks to zero. One may then pass to the zero relaxation limit.

A new two-dimensional blood flow model with arbitrary cross sections

A new two-dimensional model for blood flows in arteries with arbitrary cross sections is derived. The domain consists of a narrow, large vessel that extends along an axial direction, with cross sections described by radial and angular coordinates. The model consists of a system of balance laws for conservation of mass and balance of momentum in the axial and angular directions. The equations are derived by applying asymptotic analysis to the incompressible Navier-Stokes equations in a moving domain with an elastic membrane, and integrating in the radial direction in each cross section. The resulting model is a system of hyperbolic balance laws with source terms. The main properties of the system are discussed and a positivity-preserving well-balanced central-upwind scheme is presented. The merits of the scheme will be tested in a variety of scenarios. In particular, simulations using an idealized aorta model are shown. We analyze the time evolution of the blood flow under different initial conditions such as perturbations to steady states, which parametrizes a bulging in a vessel’s wall. We consider different situations given by distinct variations in the vessel’s elasticity.

Global existence of classical solutions for the gas flows near vacuum through ducts expanding with space and time

In this study, we examine the global existence of classical solutions for the gas flows near vacuum through ducts expanding with space and time. It is described by the initial-boundary value problem of compressible Euler equations together with a sufficiently small variable parameter, which can be viewed as a hyperbolic system of balance laws whose source is a non-dissipative term when Riemann invariants are applied. We prove a couple of global existence theorems of classical solutions under the suitable conditions of expanding ducts and the initial-boundary data whose C0 norms can be large. The analysis depends primarily on the local existence theorem and on uniform a priori estimates, which are obtained by giving the maximum principle and introducing new generalized Lax transformations. Furthermore, the limit behavior of expanding ducts at infinity and the long-time behavior of global classical solutions along all characteristic curves and vertical lines are also determined. Lastly, we explore the feasibility of the initial value problem for such expanding ducts.

A Well-Balanced Active Flux Method for the Shallow Water Equations with Wetting and Drying

Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this work, the Active Flux method is extended for the first time to a nonlinear hyperbolic system of balance laws, namely, to the shallow water equations with bottom topography. We demonstrate how to achieve an Active Flux method that is well-balanced, positivity preserving, and allows for dry states in one spatial dimension. Because of the continuous reconstruction all these properties are achieved using new approaches. To maintain third order accuracy, we also propose a novel high-order approximate evolution operator for the update of the point values. A variety of test problems demonstrates the good performance of the method even in presence of shocks.

Switched hyperbolic balance laws and differential algebraic equations

Motivated by several applications, we investigate the well-posedness of a switched system composed by a system of linear hyperbolic balance laws and by a system of linear algebraic differential equations. This setting includes networks and looped systems of hyperbolic balance laws. The obtained results are globally in time, provided that the inputs have finite (but not necessarily small) total variation.

Partial dissipation and sub-shock

To study the dissipation property of a physical system one first considers infinitesimal waves for the analysis of weakly nonlinear phenomena. For some physical systems, the dissipation is partial and there is the appearance of sub-shocks in a strong traveling trajectory. The phenomenon of partial dissipation can occur for systems of hyperbolic balance laws and also for viscous conservation laws in continuum physics. We illustrate the phenomenon for a simple relaxation model and for the Navier-Stokes equations for compressible media. The admissibility criteria and the formation of sub-shocks are studied through the zero viscosity limit.

Convergence to a diffusive contact wave for solutions to a system of hyperbolic balance laws

We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.

Saturated boundary feedback control of quasi-linear hyperbolic balance laws with application to LWR traffic flow stabilization

The saturated boundary stabilization problem for quasi-linear hyperbolic systems of balance laws is considered under H2-norm in this paper, where the boundary conditions of the system are subject to actuator saturations. The resulting closed-loop system is proven to be locally exponentially stable with respect to the steady states in the presence of saturations. To this end, the sector nonlinearity model is introduced to deal with the saturation term, and then sufficient conditions for ensuring the locally exponential stability are established in terms of a set of matrix inequalities by employing the Lyapunov function method along with a sector condition. Furthermore, these results are applied to the stabilization of the two-lane traffic flow dynamic represented by Lighthill–Whitham–Richards (LWR) model. By utilizing variable speed limit (VSL) devices, a saturated boundary feedback controller is designed to stabilize the two-lane traffic flow, and the exponential convergence of the quasi-linear traffic flow system in H2 sense is validated by numerical simulations.

Nonresonant bilinear forms for partially dissipative hyperbolic systems violating the Shizuta–Kawashima condition

In the context of hyperbolic systems of balance laws, the Shizuta–Kawashima coupling condition guarantees that all the variables of the system are dissipative even though the system is not totally dissipative. Hence it plays a crucial role in terms of sufficient conditions for the global in time existence of classical solutions. However, it is easy to find physically based models that do not satisfy this condition, especially in several space dimensions. In this paper, we consider two simple examples of partially dissipative hyperbolic systems violating the Shizuta–Kawashima condition (SK) in 3D, such that some eigendirections do not exhibit dissipation at all. We prove that if the source term is nonresonant (in a suitable sense) in the direction where dissipation does not play any role, then the formation of singularities is prevented, despite the lack of dissipation, and the smooth solutions exist globally in time. The main idea of the proof is to couple Green function estimates for weakly dissipative hyperbolic systems with the space–time resonance analysis for dispersive equations introduced by Germain, Masmoudi and Shatah. More precisely, the partially dissipative hyperbolic systems violating (SK) are endowed, in the nondissipative directions, with a special structure of the nonlinearity, the so-called nonresonant bilinear form for the wave equation (see Pusateri and Shatah, CPAM 2013).

Well-balanced positivity preserving adaptive moving mesh central-upwind schemes for the Saint-Venant system

We extend the adaptive moving mesh (AMM) central-upwind schemes recently proposed in Kurganov et al. [Commun. Appl. Math. Comput. 3 (2021) 445–479] in the context of one- (1-D) and two-dimensional (2-D) Euler equations of gas dynamics and granular hydrodynamics, to the 1-D and 2-D Saint-Venant system of shallow water equations. When the bottom topography is nonflat, these equations form hyperbolic systems of balance laws, for which a good numerical method should be capable of preserving a delicate balance between the flux and source terms as well as preserving the nonnegativity of water depth even in the presence of dry or almost dry regions. Therefore, in order to extend the AMM central-upwind schemes to the Saint-Venant systems, we develop special positivity preserving reconstruction and evolution steps of the AMM algorithms as well as special corrections of the solution projection step in (almost) dry areas. At the same time, we enforce the moving mesh to be structured even in the case of complicated 2-D computational domains. We test the designed method on a number of 1-D and 2-D examples that demonstrate robustness and high resolution of the proposed numerical approach.

Boundary Feedback Stabilization of Freeway Traffic Networks: ISS Control and Experiments

The boundary feedback control of networks of freeway traffic is considered in this article by means of the partial differential equation (PDE)-based techniques. The control and measurements are all located at the boundaries of each link. We have established the boundary control model for the system, a linear hyperbolic system of balance laws, which includes not only the traffic flow dynamics of the network described by combining the linearized Aw&#x2013;Rascle&#x2013;Zhang (ARZ) traffic flow model of each link but also the integrated on-ramping metering and variable speed limit control modeled as the boundary condition. As the traffic demand of the network is fluctuated, the boundary input-to-state stability (ISS) controller is designed to suppress the disturbance and regulate the traffic flow into the boundedness regions of the desired states. Based on a novel Lyapunov function, some sufficient conditions in terms of the matrix inequalities are derived for the ISS boundary stabilization in the <inline-formula> <tex-math notation="LaTeX">${L}^{2}$ </tex-math></inline-formula>-norm. The numerical simulation is given to illustrate the effectiveness of the developed boundary feedback control. Moreover, an interesting traffic experiment is carried out by using the traffic simulation software, AIMSUN, and some specific traffic data are collected and analyzed to verify the feasibility and effectiveness of the boundary control.

Time-asymptotic stability for first-order symmetric hyperbolic systems of balance laws in dissipative compressible fluid dynamics

This paper identifies a non-(or /iso-)thermal variant of Ruggeri’s 1983 formulation of viscous heat-conductive fluid dynamics as a hyperbolic system of balance laws and shows that both the original model and this variant have (a) time-asymptotically stable equilibria and (b) principal parts deriving from a protopotential: a single scalar function that induces the temporospatial flux as an appropriate part of its Hessian.

Unconditionally stable and parallel Discontinuous Galerkin solver

We describe a parallel and quasi-explicit Discontinuous Galerkin (DG) kinetic scheme for solving systems of balance laws. The solver is unconditionally stable (i.e., the CFL number can be arbitrary) and has the complexity of an explicit scheme. It can be applied to any hyperbolic system of balance laws, see [2,21]. In this work, we assess the performance of the scheme in the particular cases of the three-dimensional wave equation and of Maxwell's equations. We measure the benefit of the unconditional stability by performing experiments with very large CFL numbers. In addition, the parallel possibilities of the method are investigated.