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Understanding the dynamics of hyperbolic balance laws is of paramount interest in the realm of fluid mechanics. Nevertheless, fundamental questions on the analysis and the numerics for distinctive hyperbolic features related to turbulent flow motion remain vastly open. Recent progress on the mathematical side reveals novel routes to face these concerns. This includes findings about the failure of the entropy principle to ensure uniqueness, the use of structure-preserving concepts in high-order numerical methods, and the advent of tailored probabilistic approaches. Whereas each of these three directions on hyperbolic modelling are of completely different origin they are all linked to small- or subscale features in the solutions which are either enhanced or depleted by the hyperbolic nonlinearity. Thus, any progress in the field might contribute to a deeper understanding of turbulent flow motion on the basis of the continuum-scale mathematical models. We present an overview on the mathematical state-of-the-art in the field and relate it to the scientific work in the DFG Priority Research Programme 2410. As such, the survey is not necessarily targeting at readers with comprehensive knowledge on hyperbolic balance laws but instead aims at a general audience of reseachers which are interested to gain an overview on the field and associated challenges in fluid mechanics. [ABSTRACT FROM AUTHOR]

We study BV solutions for a 2 × 2 system of hyperbolic balance laws. We show that when initial data have small total variation on (− ∞ , ∞) and small amplitude, and decay sufficiently fast to a constant equilibrium state as | x | → ∞ , a Cauchy problem (with generic data) has a unique admissible BV solution defined globally in time. Here the solution is admissible in the sense that its shock waves satisfy the Lax entropy condition. We also study asymptotic behavior of solutions. In particular, we obtain a time decay rate for the total variation of the solution, and a convergence rate of the solution to its time asymptotic solution. Our system is a modification of a Keller-Segel type chemotaxis model. Its flux function possesses new features when comparing to the well-known model of Euler equations with damping. This may help to shed light on how to extend the study to a general system of hyperbolic balance laws in the future. [ABSTRACT FROM AUTHOR]

We establish global well-posedness and asymptotic behavior of solutions to a system of balance laws modeling traffic flow with nonconcave fundamental diagram. We prove the results by finding a convex entropy-entropy flux pair and verifying the Kawashima condition, the sub-characteristic condition, and the partial dissipative inequality in the framework of Dafermos. This problem is of specific interest since nonconcave fundamental diagrams arise naturally in traffic flow. [ABSTRACT FROM AUTHOR]

The classical paper [40] by S. K. Godunov had a revolutionary effect on the field of numerical simulations of compressible fluid flows. Yet, for some twenty years after its publication, it was mostly used by the mathematical community in the Soviet Union. The seminal paper of van Leer [85] has inaugurated the period of universal interest in high-resolution extensions of Godunov's scheme. The first step to take was to modify the (locally) self-similar solution to the Riemann Problem (at discontinuities) by allowing piecewise-polynomial (rather than piecewise-constant) initial data. The GRP (Generalized Riemann Problem) analysis [6] provided analytical solutions (for piecewise-linear data) that could be readily implemented in a high-resolution robust code. This review paper focuses on the evolution of some mathematical aspects of this method. It addresses the three concepts of consistency, stability and convergence in the context of compact finite difference (or finite volume) schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of "balance laws", a common viewpoint in relevant physical conservation laws. The first significant observation is that under very mild conditions a weak solution is indeed a solution to the balance law (obtained by a formal application of the Gauss-Green formula). Since high-resolution schemes require the computation of several quantities per mesh cell (e.g., slopes), the notion of "consistency" must be extended to this framework. A combination of consistency hypothesis with stability of the scheme leads to a suitable convergence theorem, generalizing the classical convergence theorem of Lax and Wendroff [52]. Finally, the limit functions are shown to be entropy solutions by using a notion of "Godunov compatibility", which serves as a substitute to the entropy condition. • The concept of "consistency" is explored in detail, for general high resolution schemes. • Consistency (and its order) of a finite difference scheme depends on the underlying space of approximation. • A generalization of the Lax-Wendroff convergence theorem is given. • The "entropy condition" is replaced by the concept of "Godunov compatibility". [ABSTRACT FROM AUTHOR]

In this paper, we develop an oscillation-free discontinuous Galerkin (OFDG) method for solving the shallow water equations with a non-flat bottom topography. Due to the nonlinear hyperbolic nature of the shallow water equation, the exact solution may contain shock discontinuities; thus, the numerical solution may generate spurious oscillations if no special treatment is taken near the discontinuities. Besides, it is desirable to construct a numerical scheme that possesses the well-balanced property, which preserves exactly the hydrostatic equilibrium solutions up to the machine error. Based on the existing well-balanced DG schemes proposed in Xing (J Comput Phys 257:536–553, 2014), Xing and Shu (Commun Comput Phys 1: 100–134, 2006), we add an extra damping term into these schemes to control the spurious oscillations. One of the advantages of the damping term in suppressing the oscillations is that the damping technique seems to be more convenient for the theoretical analysis, at least in the semi-discrete analysis. With the careful construction of the damping term, the proposed method not only achieves non-oscillatory property without compromising any order of accuracy, but also inherits the the well-balanced property from original schemes. We also present a detailed procedure for constructing the damping term and a theoretical analysis for the preservation of the well-balanced property. Extensive numerical experiments, including one- and two-dimensional space, demonstrate that the proposed method has desired properties without sacrificing any order of accuracy. [ABSTRACT FROM AUTHOR]

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. • Enrichment of classical modal Discontinuous Galerkin bases for to hyperbolic balance laws. • Construction of an approximately well-balanced scheme using a prior approximating the steady solution. • Convergence results and error estimates for the enriched scheme. • Construction of the prior using parametric Physics-Informed Neural Networks. • Thorough validation with numerical experiments in one and two dimensions. [ABSTRACT FROM AUTHOR]

We propose and study a one-dimensional 2 × 2 hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic critical thresholds for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic. [ABSTRACT FROM AUTHOR]

In this work, we develop a method for learning interpretable, thermodynamically stable and Galilean invariant partial differential equations (PDEs) based on the conservation-dissipation formalism of irreversible thermodynamics. As governing equations for non-equilibrium flows in one dimension, the learned PDEs are parameterized by fully connected neural networks and satisfy the conservation-dissipation principle automatically. In particular, they are hyperbolic balance laws and Galilean invariant. The training data are generated from a kinetic model with smooth initial data. Numerical results indicate that the learned PDEs can achieve good accuracy in a wide range of Knudsen numbers. Remarkably, the learned dynamics can give satisfactory results with randomly sampled discontinuous initial data and Sod's shock tube problem although it is trained only with smooth initial data. [ABSTRACT FROM AUTHOR]

Abstract With increasing engineering demands, there need high order accurate schemes embedded with precise physical information in order to capture delicate small scale structures and strong waves with correct “physics”. There are two families of high order methods: One is the method of line, relying on the Runge-Kutta (R-K) time-stepping. The building block is the Riemann solution labeled as the solution element “1”. Each step in R-K just has first order accuracy. In order to derive a fourth order accuracy scheme in time, one needs four stages labeled as “ 1⊙1⊙1⊙1=4”. The other is the one-stage Lax-Wendroff (LW) type method, which is more compact but is complicated to design numerical fluxes and hard to use when applied to highly nonlinear problems. In recent years, the pair of solution element and dynamics element, labeled as “2”, are taken as the building block. The direct adoption of the dynamics implies the inherent temporal-spatial coupling. With this type of building blocks, a family of two-stage fourth order accurate schemes, labeled as “ 2⊙2=4”, are designed for the computation of compressible fluid flows. The resulting schemes are compact, robust and efficient. This paper contributes to elucidate how and why high order accurate schemes should be so designed. To some extent, the “ 2⊙2=4” algorithm extracts the advantages of the method of line and one-stage LW method. As a core part, the pair “2” is expounded and LW solver is revisited. The generalized Riemann problem (GRP) solver, as the discontinuous and nonlinear version of LW flow solver, and the gas kinetic scheme (GKS) solver, the microscopic LW solver, are all reviewed. The compact Hermite-type data reconstruction and high order approximation of boundary conditions are proposed. Besides, the computational performance and prospective discussions are presented.

• A sequence of auxiliary problems is constructed from scaling the balance laws. • Source terms of the auxiliary problems are decoupled from the states. • For a class of source terms the auxiliary problems generate a convergent sequence. • Existence of a convergent sequence ensures the existence of a balance law solution. • An iterative algorithm is obtained where any conservative scheme can be used. The scaling of the exact solution of a hyperbolic balance law generates a family of scaled problems in which the source term does not depend on the current solution. These problems are used to construct a sequence of solutions whose limiting function solves the original hyperbolic problem. Thus this gives rise to an iterative procedure. Its convergence is demonstrated both theoretically and analytically. The analytical demonstration is in terms of a local in time convergence and existence theorem in the L 2 framework for the class of problems in which the source term s (q) is bounded, with s (0) = 0 , is locally Lipschitz and belongs to C 2 (R) ∩ H 1 (R). A convex flux function, which is usual for existence and uniqueness for conservation laws, is also needed. For the numerical demonstration, a set of model equations is solved, where a conservative finite volume method using a low-dissipation flux is implemented in the iteration stages. The error against reference solutions is computed and compared with the accuracy of two conventional first order schemes in order to assess the gaining in accuracy of the present procedure. Regarding the accuracy only a first order scheme is explored because the development of a useful procedure is of interest in this work, high-order accurate methods should increase the computational cost of the global procedure. Numerical tests show that the present approach is a feasible method of solution. [ABSTRACT FROM AUTHOR]

Summary: In this work, we consider a hyperbolic one‐dimensional (1D) model for blood flow through compliant axisymmetric tilted vessels. The pressure is a function of the cross‐sectional area and other model parameters. Important features of the model are inherited at the discrete level by the numerical scheme. For instance, the existence of steady states may provide important information about the flow properties at low computational cost. Here, we characterize a large class of smooth equilibrium solutions by means of quantities that remain invariant. At the discrete level, the well‐balanced property in the numerical scheme leads to accurate results when steady states are perturbed. On the other hand, the model is equipped with an entropy function and an entropy inequality that can help us choose the physically relevant weak solutions. A large class of semidiscrete entropy‐satisfying numerical schemes is described. In addition, preservation of positivity for the cross‐sectional area is achieved. Numerical results show the scheme is robust, stable, and accurate. The ultimate goal of this article is the numerical application to cases that are more relevant from the medical viewpoint. In particular, a numerical simulation of cardiac cycles with appropriate parameters shows that increasing the rigidity of the artery walls delays the formation of shock waves. Gravity effects are also measured in tilted vessels, and a simulation using an idealized aorta model was conducted. [ABSTRACT FROM AUTHOR]

This paper presents novel high-order accurate discontinuous Galerkin (DG) schemes for the compressible Euler equations under gravitational fields. A notable feature of these schemes is that they are well-balanced for a general known hydrostatic equilibrium state and, at the same time, provably preserve the positivity of density and pressure. In order to achieve the well-balanced and positivity-preserving properties simultaneously, a novel DG spatial discretization is carefully designed with suitable source term reformulation and a properly modified Harten--Lax--van Leercontact (HLLC) flux. Based on some technical decompositions as well as several key properties of the admissible states and HLLC flux, rigorous positivity-preserving analyses are carried out. It is proven that the resulting well-balanced DG schemes, coupled with strong-stability-preserving time discretizations, satisfy a weak positivity property, which implies that one can apply a simple existing limiter to effectively enforce the positivity-preserving property, without losing high-order accuracy and conservation. The proposed methods and analyses are illustrated with the ideal equation of state (EOS) for notational convenience only, while the extensions to general EOS are straightforward and are discussed in the supplementary material. Extensive one- and two-dimensional numerical tests demonstrate the desired properties of these schemes, including the exact preservation of the equilibrium state, the ability to capture small perturbation of such state, the robustness for solving problems involving low density and/or low pressure, and good resolution for smooth and discontinuous solutions. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

Instabilities of electron plasma waves in high-mobility semiconductor devices have recently attracted a lot of attention as a possible candidate for closing the THz gap. Conventional moments-based transport models usually neglect time derivatives in the constitutive equations for vectorial quantities, resulting in parabolic systems of partial differential equations (PDE). To describe plasma waves however, such time derivatives need to be included, resulting in hyperbolic rather than parabolic systems of PDEs; thus the fundamental nature of these equation systems is changed completely. Additional nonlinear terms render the existing numerical stabilization methods for semiconductor simulation practically useless. On the other hand there are plenty of numerical methods for hyperbolic systems of PDEs in the form of conservation laws. Standard numerical schemes for conservation laws, however, are often either incapable of correctly handling the large source terms present in semiconductor devices due to built-in electric fields, or rely heavily on variable transformations which are specific to the equation system at hand (e.g. the shallow water equations), and can not be generalized easily to different equations. In this paper we develop a novel well-balanced numerical scheme for hyperbolic systems of PDEs with source terms and apply it to a simple yet non-linear electron transport model. [ABSTRACT FROM AUTHOR]

We consider the development of high-order space and time numerical methods based on implicit-explicit multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence, the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system. From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several drawbacks and typically lose efficiency in describing the parabolic limit regime. In this work, in the context of implicit-explicit linear multistep methods we construct high-order space-time discretizations which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Several numerical examples confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

The numerical modeling of one-dimensional (1D) domains joined by symmetric or asymmetric bifurcations or arbitrary junctions is still a challenge in the context of hyperbolic balance laws with application to flow in pipes, open channels or blood vessels, among others. The formulation of the Junction Riemann Problem (JRP) under subsonic conditions in 1D flow is clearly defined and solved by current methods, but they fail when sonic or supersonic conditions appear. Formulations coupling the 1D model for the vessels or pipes with other container-like formulations for junctions have been presented, requiring extra information such as assumed bulk mechanical properties and geometrical properties or the extension to more dimensions. To the best of our knowledge, in this work, the JRP is solved for the first time allowing solutions for all types of transitions and for any number of vessels, without requiring the definition of any extra information. The resulting JRP solver is theoretically well-founded, robust and simple, and returns the evolving state for the conserved variables in all vessels, allowing the use of any numerical method in the resolution of the inner cells used for the space-discretization of the vessels. The methodology of the proposed solver is presented in detail. The JRP solver is directly applicable if energy losses at the junctions are defined. Straightforward extension to other 1D hyperbolic flows can be performed.

The classical Saint–Venant shallow water equations on complex geometries have wide applications in many areas including coastal engineering and atmospheric modeling. The main numerical challenge in simulating Saint–Venant equations is to maintain the high order of accuracy and well-balanced property simultaneously. In this paper, we propose a high-order accurate and well-balanced discontinuous Galerkin (DG) method on two dimensional (2D) unstructured meshes for the Saint–Venant shallow water equations. The technique used to maintain well-balanced property is called constant subtraction and proposed in Yang et al. (J Sci Comput 63:678–698, 2015). Hierarchical reconstruction limiter with a remainder correction technique is introduced to control numerical oscillations. Numerical examples with smooth and discontinuous solutions are provided to demonstrate the performance of our proposed DG methods. [ABSTRACT FROM AUTHOR]

Gas flow through pipeline networks can be described using 2 × 2 hyperbolic balance laws along with coupling conditions at nodes. The numerical solution at steady state is highly sensitive to these coupling conditions and also to the balance between flux and source terms within the pipes. To avoid spurious oscillations for near equilibrium flows, it is essential to design well-balanced schemes. Recently Chertock, Herty & Özcan[11] introduced a well-balanced method for general 2 × 2 systems of balance laws. In this paper, we simplify and extend this approach to a network of pipes. We prove well-balancing for different coupling conditions and for compressors stations, and demonstrate the advantage of the scheme by numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is concerned with the rarefaction waves for a model system of hyperbolic balance laws in the whole space and in the half space. We prove the asymptotic stability of rarefaction waves under smallness assumptions on the initial perturbation and on the amplitude of the waves. The proof is based on the L2 energy method. [ABSTRACT FROM AUTHOR]

The paper discusses the long time behavior of BV solutions to the Cauchy problem for hyperbolic systems of balance laws with partial dissipation, when the relaxed system is adiabatic. [ABSTRACT FROM AUTHOR]

Gas flow through pipeline networks can be described using 2 × 2 hyperbolic balance laws along with coupling conditions at nodes. The numerical solution at steady state is highly sensitive to these coupling conditions and also to the balance between flux and source terms within the pipes. To avoid spurious oscillations for near equilibrium flows, it is essential to design well-balanced schemes. Recently Chertock, Herty & Özcan[11] introduced a well-balanced method for general 2 × 2 systems of balance laws. In this paper, we simplify and extend this approach to a network of pipes. We prove well-balancing for different coupling conditions and for compressors stations, and demonstrate the advantage of the scheme by numerical experiments. [ABSTRACT FROM AUTHOR]

Implicit–Explicit (IMEX) schemes are a powerful tool in the development of numerical methods for hyperbolic systems with stiff sources. Here we focus our attention on the asymptotic properties of such schemes, like the preservation of steady-states (well-balanced property) and the behavior in presence of small space–time scales (asymptotic preservation property). We analyze conditions under which the standard additive approach based on taking the fluxes explicitly and the sources implicitly yields a well-balanced behavior. In addition, we consider a partitioned strategy which possesses better well-balanced properties. The behavior of the additive and partitioned approaches under classical scaling limits is then studied in the context of asymptotic-preserving schemes. Additional order conditions that guarantee the correct behavior of the schemes in the Navier–Stokes regime are derived. Several examples illustrate these asymptotic behaviors and the performance of the new methods. [ABSTRACT FROM AUTHOR]

We propose a generalized polynomial chaos based stochastic Galerkin methods for scalar hyperbolic balance laws with random geometric source terms or random initial data. This method is well-balanced (WB), in the sense that it captures the stochastic steady state solution with high order accuracy. The framework of the stochastic WB schemes is presented in details, along with several numerical examples to illustrate their accuracy and effectiveness. The goal of this paper is to show that the stochastic WB scheme yields a more accurate numerical solution at steady state than the non-WB ones. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of the mathematical entropy for hyperbolic systems of balance laws with (not necessarily symmetric) relaxation. As applications, we deal with the Timoshenko system, the Euler-Maxwell system and the Euler-Cattaneo-Maxwell system. Especially, for the Euler-Cattaneo-Maxwell system, we observe that its dissipative structure is of the regularity-loss type and investigate the corresponding decay property. Furthermore, we prove the global existence and asymptotic stability of solutions to the Euler-Cattaneo-Maxwell system for small initial data. [ABSTRACT FROM AUTHOR]

We present a semi-analytical, implicit solution to the generalized Riemann problem (GRP) for non-linear systems of hyperbolic balance laws with stiff source terms. The solution method is based on an implicit, time Taylor series expansion and the Cauchy–Kowalewskaya procedure, along with the solution of a sequence of classical Riemann problems. Our new GRP solver is then used to construct locally implicit ADER methods of arbitrary accuracy in space and time for solving the general initial–boundary value problem for non-linear systems of hyperbolic balance laws with stiff source terms. Analysis of the method for model problems is carried out and empirical convergence rate studies for suitable tests problems are performed, confirming the theoretically expected high order of accuracy. [ABSTRACT FROM AUTHOR]

For hyperbolic systems of balance laws governing relaxation processes, in one space dimension, with source incurring nonnegative entropy production and satisfying a Kawashima-type condition, it is shown that when the initial data have small total variation on (-∞, ∞) and decay rapidly to zero, as |x| → ∞, then the Cauchy problem possesses a unique admissible BV solution, in the large, with total variation decaying to zero, as t → ∞. [ABSTRACT FROM AUTHOR]

This paper deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [Hyperbolic systems of balance laws with weak dissipation, J. Hyp. Diff. Equations 3 (2006) 505-527.], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented by [Ch. Arvanitis, Ch. Makridakis and A. E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic conservation laws, SIAM J. Numer. Anal. 42(4) (2004) 1357-1393]; [S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235-277] for systems of hyperbolic conservation laws without source terms. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analytical study of initial–boundary value problems for a system of hyperbolic balance laws derived from a repulsive chemotaxis model with logarithmic sensitivity. In the first part of the paper we show that, subject to the Dirichlet boundary conditions, classical solutions exist globally in time for large initial data. Asymptotically in time, the solutions are shown to converge to their boundary data at an exponential rate as time goes to infinity. Numerical simulations are supplied to corroborate the analytical results. The analytic approach developed herein can be utilized to handle a family of initial and initial–boundary value problems of the model and related models with similar mathematical structure. As a demonstration of the effectiveness of our approach, in the second part of the paper we show that, subject to the Neumann–Dirichlet boundary conditions, classical solutions exist and converge to constant equilibrium states for large initial data and for arbitrary values of the chemical diffusion coefficient. This improves a previous result obtained in [30] where the smallness of the chemical diffusion coefficient was required. [ABSTRACT FROM AUTHOR]

The paper establishes the existence and long time behavior of BV solutions to a hyperbolic system of balance laws with partially dissipative source, manifesting relaxation. The component of the state vector that is actively damped decays to zero, while the complementary component, whose total mass is not necessarily conserved, behaves asymptotically as a heat kernel. [ABSTRACT FROM AUTHOR]

By extending the analysis in [C. M. Dafermos, Hyperbolic systems of balance laws with weak dissipation, J. Hyperbolic Differ. Equ. 3 (2006) 505-527], this note constructs global BV solutions to the Cauchy problem for strictly hyperbolic systems of balance laws endowed with a convex entropy, under the assumption that the entropy production is positive definite. [ABSTRACT FROM AUTHOR]

The rate of convergence for vanishing viscosity approximations to hyperbolic balance laws is established. The result applies to systems that are strictly hyperbolic and genuinely nonlinear with a source term satisfying a special mechanism that induces dissipation. The proof relies on error estimates that measure the interaction of waves. Shock waves are treated by monitoring the evolution of suitable Lyapunov functionals, whereas interactions involving rarefaction waves are accommodated by employing a sharp decay estimate. [ABSTRACT FROM AUTHOR]

This paper is about the numerical solutions for a computational fluid dynamics-population balance model (CFD-PBM) coupled model of two-phase flow in a liquid-liquid extraction column. The model accounts for a complete description between both the dispersed and continuous phases, and constitutes a hyperbolic system of equations having a linearly degenerate nature. A numerical algorithm based on operator splitting approach for the numerical solution of the model is used. The homogeneous part is solved using the TVD MUSCL-Hancock scheme. Numerical results are presented, demonstrating the accuracy of the proposed methods and in particular, the accurate numerical description of the flow in the vicinity of the contact discontinuities. [ABSTRACT FROM AUTHOR]

Abstract: We consider strictly hyperbolic and genuinely nonlinear systems of hyperbolic balance laws in one-space dimension. Sharp decay estimates are derived for the positive waves in an entropy weak solution. The result is obtained by introducing a partial ordering within the family of positive Radon measures, using symmetric rearrangements and a comparison with a solution of Burgers''s equation with impulsive sources as well as lower semicontinuity properties of continuous Glimm-type functionals. [Copyright &y& Elsevier]

We consider the Cauchy problem for an n x n strictly hyperbolic system of balance laws ut + f(u)x = g(x, u), x ∈ ℝ, t > 0, ||g(x, ·)||C² ≤ M̃ (x) ∈ L¹, endowed with the initial data u(0, .) = uo ∈ L¹ ∩ BV(ℝ; ℝn). Each characteristic field is assumed to be genuinely nonlinear or linearly degenerate and nonresonant with the source, i.e., |λi(u)| ≥ c > 0 for all i ∈ {1, . . . ,n}. Assuming that the L¹ norms of ||g(x, ·)||C¹ and ||uo||BV(ℝ) are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [D. Amadori, L. Gosse, and G. Guerra, Arch. Ration. Mech. Anal., 162 (2002), pp. 327-366] to unbounded (in L∞) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup. [ABSTRACT FROM AUTHOR]

• We develop a method for finding models of heat conduction via ML, which can simulate complex process based on database. • The learned models satisfy the CD principle, and thereby are hyperbolic balance laws, which are interpretable and stable. • The learned models can be solved by any traditional numerical method. • The learned models achieve very high accuracy, maintain long-time stability and perform well with discontinuous initial data. • "Warm-up" technique is used, and some neural networks are trained by other well trained neural networks. In this paper, we develop a method for finding models of heat conduction via machine learning. Integrating machine learning and the conservation-dissipation formulism (CDF) of irreversible thermodynamics, we obtain a system of PDEs for the heat conduction. The learned PDEs satisfy the conservation-dissipation principle and thereby are hyperbolic balance laws, which can be solved by conventional numerical methods. In the training process, we use a "warm-up" technique and train a neural network by another trained neural network. Numerical tests show that the learned models can achieve very high accuracy, perform well in a long time for a wide range of Knudsen numbers, and have an excellent generalization ability. The code is available at https://github.com/JinSZhao/Finding-Models-of-Heat-Conduction-via-Machine-Learning. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First a high-order WENO reconstruction procedure is applied to the cell averages at the current time level. Second, the temporal evolution of the reconstruction polynomials is computed locally inside each cell using the governing equations. In the original ENO scheme of Harten et al. and in the ADER schemes of Titarev and Toro, this time evolution is achieved via a Taylor series expansion where the time derivatives are computed by repeated differentiation of the governing PDE with respect to space and time, i.e. by applying the so-called Cauchy–Kovalewski procedure. However, this approach is not able to handle stiff source terms. Therefore, we present a new strategy that only replaces the Cauchy–Kovalewski procedure compared to the previously mentioned schemes. For the time-evolution part of the algorithm, we introduce a local space–time discontinuous Galerkin (DG) finite element scheme that is able to handle also stiff source terms. This step is the only part of the algorithm which is locally implicit. The third and last step of the proposed ADER finite volume schemes consists of the standard explicit space–time integration over each control volume, using the local space–time DG solutions at the Gaussian integration points for the intercell fluxes and for the space–time integral over the source term. We will show numerical convergence studies for nonlinear systems in one space dimension with both non-stiff and with very stiff source terms up to sixth order of accuracy in space and time. The application of the new method to a large set of different test cases is shown, in particular the stiff scalar model problem of LeVeque and Yee [R.J. LeVeque, H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, Journal of Computational Physics 86 (1) (1990) 187–210], the relaxation system of Jin and Xin [S. Jin, Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics 48 (1995) 235–277] and the full compressible Euler equations with stiff friction source terms. [Copyright &y& Elsevier]

Global BV solutions are constructed to the Cauchy problem for strictly hyperbolic systems of balance laws endowed with a rich family of entropies and source that is merely weakly dissipative, of the type induced by relaxation mechanisms. [ABSTRACT FROM AUTHOR]