Hyperbolic Equations Research Articles

Asymptotic behavior of Riemann solutions for the one-dimensional mean-field games in conservative form with the logarithmic coupling term

Construction of Riemann solutions for a hyperbolic system in conservative form arising from the one-dimensional mean-field games with the quadratic Hamiltonian and the logarithmic coupling term are provided in detail. Moreover, the delta shock formation is concretely analyzed from the limit of double-shock-solution as well as the emergence of vacuum state is also specifically discussed from the limit of double-rarefaction-solution when the coupling coefficient drops to zero. Accordingly, the remarkable cavitation and concentration phenomena can be closely observed and explored. Additionally, the numerical experiments are also presented in correspondence to authenticate the theoretical analysis results.

Gradient-Based Monte Carlo Methods for Relaxation Approximations of Hyperbolic Conservation Laws

Particle methods based on evolving the spatial derivatives of the solution were originally introduced to simulate reaction-diffusion processes, inspired by vortex methods for the Navier–Stokes equations. Such methods, referred to as gradient random walk methods, were extensively studied in the ’90s and have several interesting features, such as being grid-free, automatically adapting to the solution by concentrating elements where the gradient is large, and significantly reducing the variance of the standard random walk approach. In this work, we revive these ideas by showing how to generalize the approach to a larger class of partial differential equations, including hyperbolic systems of conservation laws. To achieve this goal, we first extend the classical Monte Carlo method to relaxation approximation of systems of conservation laws, and subsequently consider a novel particle dynamics based on the spatial derivatives of the solution. The methodology, combined with asymptotic-preserving splitting discretization, yields a way to construct a new class of gradient-based Monte Carlo methods for hyperbolic systems of conservation laws. Several results in one spatial dimension for scalar equations and systems of conservation laws show that the new methods are very promising and yield remarkable improvements compared to standard Monte Carlo approaches, either in terms of variance reduction as well as in describing the shock structure.

Maximum principle preserving time implicit DGSEM for linear scalar hyperbolic conservation laws

The properties of the high-order discontinuous Galerkin spectral element method (DGSEM) with implicit backward Euler time stepping are investigated for the approximation of hyperbolic linear scalar conservation equation in multiple space dimensions. We first prove that the DGSEM scheme in one space dimension preserves a maximum principle for the cell-averaged solution when the time step is large enough. This property however no longer holds in multiple space dimensions and we propose to use the flux-corrected transport (FCT) limiting [5] based on a low-order approximation using graph viscosity to impose a maximum principle on the cell-averaged solution. These results allow us to use a linear scaling limiter [58] in order to impose a maximum principle at nodal values within elements, while limiting the cell average with the FCT limiter improves the accuracy of the limited solution. Then, we investigate the inversion of the linear systems resulting from the time implicit discretization at each time step. We prove that the diagonal blocks are invertible and provide efficient algorithms for their inversion. Numerical experiments in one and two space dimensions are presented to illustrate the conclusions of the present analyses.

Well-posedness for a class of pseudo-differential hyperbolic equations on the torus

In this paper we establish the well-posedness of the Cauchy problem for a class of pseudo-differential hyperbolic equations on the torus. The class considered here includes a space-like fractional order Laplacians. By applying the toroidal pseudo-differential calculus we establish regularity estimates, existence and uniqueness in the scale of the standard Sobolev spaces on the torus.

Multidimensional Hyperbolic Equations with Nonlocal Potentials: Families of Global Solutions

In spaces of arbitrary dimensions, hyperbolic differential–difference equations with potentials undergoing translations along arbitrary spatial directions are investigated. The fundamental novelty of this study is that, unlike previous investigations, no restrictions are imposed on the symbol of the differential–difference operator contained in the equation. For the investigated equation, we succeed to explicitly construct a multiparametric family of classical global solutions.

Regularity of the affine hyperbolic sphere and related equations

Regularity of the affine hyperbolic sphere and related equations

Averaging a High-Frequency Hyperbolic System of Quasilinear Equations with Large Summands

Averaging a High-Frequency Hyperbolic System of Quasilinear Equations with Large Summands

The conduit equation: Hyperbolic approximation and generalized Riemann problem

The conduit equation is a dispersive non-integrable scalar equation modeling the flow of a low-viscous buoyant fluid embedded in a highly viscous fluid matrix. This equation can be written in a particular form reminiscent of the famous Godunov form proposed in 1961 for the Euler equations of compressible fluids. We propose a hyperbolic approximation of the conduit equation by retaining the Godunov-type structure. The comparison of solutions to the conduit equation and those to the approximate hyperbolic system is performed: the wave fission of a large initial perturbation of a rectangular or Gaussian form. The results are in good agreement. New generalized solutions to the conduit equation composed of a finite set of waves of the same period and linked with a constant solution by generalized Rankine-Hugoniot relations are discovered. Such multi-hump structures interact with each other almost as solitary waves: they collide, merge, and reconstruct after the interaction. This partly indicates the stability of such multi-hump solutions under small perturbations. The exact and approximate hyperbolic system describes such an interaction with good accuracy.

Discrete inverse problem for a hyperbolic equation, properties of the solution to a discrete direct and auxiliary discrete problems

This paper considers the formulation of a discrete inverse problem for a hyperbolic equation. First, the continuous inverse problem is reduced to a convenient form for research. In the inverse problem, the required function is considered even. Since the Dirac delta function is present in the problem data, the structure of a generalized solution to the Cauchy problem for a hyperbolic equation is determined. The solution to the Cauchy problem for a hyperbolic equation is determined only for positive values in time, therefore the solution to the Cauchy problem for negative values in time is determined using odd continuation. After some transformations, the formulation of the continuous inverse problem is reduced to a form convenient for research. A grid domain is introduced, and for all functions in the problem statement the corresponding grid functions and a discrete analogue of the Dirac delta function are determined. Differential operators, initial conditions and additional data of the inverse problem are approximated by finite differences. Assuming that a solution to the discrete inverse problem exists, we prove the data lemma of the discrete inverse problem. In order to study the discrete inverse problem for a hyperbolic equation, a theorem on the existence and uniqueness of the discrete direct problem is proved, as well as on the properties of the solution to this discrete problem. In the course of proving the theorem, a discrete analogue of d'Alembert's formula for solving the Cauchy problem for a hyperbolic equation was obtained. The theorem on the existence of a unique solution to the auxiliary discrete problem and its properties is proved.

A comparison of hyperbolic sine creep life equations and data correlation methods for these equations

ABSTRACT Hyperbolic sine (HS) minimum creep rate equations combined with the Monkman-Grant equation are attractive for correlating time to rupture data because of their physical basis in dislocation creep theory, numerical stability, and conformance to physical boundary conditions at zero and infinite stress. In this study, uniaxial creep life data of Sanicro 25 and HR6W alloys were correlated to two recently published HS creep life equations with three commonly used objective functions. For each alloy, the correlation results (fit to the data and predicted strength at 100,000 h time to rupture) were essentially the same using either HS equation. Multiple equivalent solutions (same goodness of fit but different values of the fitting constants) were obtained with the HR6W data set. A discussion on obtaining initial estimates of fitting constants is provided as an Appendix.

The problem with the missing Goursat condition at the boundary of the domain for a degenerate hyperbolic equation with a singular coefficient

The problem with the missing Goursat condition at the boundary of the domain for a degenerate hyperbolic equation with a singular coefficient

Problem of Linear Conjugation with Multipoint Conditions in the Case of Multiple Nodes for Higher-Order Strictly Hyperbolic Homogeneous Equations

Problem of Linear Conjugation with Multipoint Conditions in the Case of Multiple Nodes for Higher-Order Strictly Hyperbolic Homogeneous Equations

Observers for hyperbolic systems with multiple delays in the nonlocal boundary condition and its application to secure image encryption

In this article, we propose a step-by-step observer design for a first-order linear hyperbolic system with multiple delays in the nonlocal boundary condition. In the proposed approach inspired by the backstepping methodology, the observer gains are obtained following seven successive steps. Next, we present the application of this observer design in the context of image encryption. We consider a system where a local nonlinear term causing a chaotic vibration is added to the nonlocal boundary condition with time lags. A synchronization system for this chaotic system can be constructed using the same observer design method. The chaotic states generated by both systems can be used to encrypt and decrypt image data. The time lags, and weight functions included in the nonlocal boundary condition play the role of common encryption keys. By considering a system that includes arbitrarily many delays, this approach allows for a significant increase in the number of common encryption keys compared to conventional methods, resulting in greatly improved communication security.

An absolutely convergent fixed-point fast sweeping WENO method on triangular meshes for steady state of hyperbolic conservation laws

High order accuracy fast sweeping methods have been developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs) on rectangular meshes, while they were not available yet on unstructured meshes. In this paper, we extend high order accuracy fast sweeping methods to unstructured triangular meshes by applying fixed-point iterative sweeping techniques to a fifth-order finite volume unstructured WENO scheme, for solving steady state solutions of hyperbolic conservation laws. Similar as other fast sweeping methods, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady state solutions. An advantage of fixed-point fast sweeping methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. This also provides certain convenience in designing high order fast sweeping methods on unstructured meshes. As in the first order fast sweeping methods on triangular meshes, we introduce multiple reference points to determine alternating sweeping directions on unstructured meshes. All the cells on the mesh are ordered according to their centroids' distances to those reference points, and the resulted orderings provide sweeping directions for iterations. To make the residue of the fast sweeping iterations converge to machine zero / round off errors, we follow the approach in our early work of developing the absolutely convergent fixed-point fast sweeping WENO methods on rectangular meshes, and adopt high order WENO scheme with unequal-sized sub-stencils, specifically here a fifth-order finite volume unstructured WENO scheme for spatial discretization. Extensive numerical experiments, including problems with complex domain geometries, are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth-order fast sweeping scheme on triangular meshes. Furthermore, the proposed method is compared with the forward Euler time marching method and the popular third order total variation diminishing Rung-Kutta (TVD-RK3) time-marching method for steady state computations. Numerical examples show that the developed fixed-point fast sweeping WENO method is the most efficient scheme among them, and especially it can save up to 70% CPU time costs than TVD-RK3 in order to converge to steady state solutions.

On Lattice Boltzmann Methods based on vector-kinetic models for hyperbolic partial differential equations

In this paper, we are concerned about the lattice Boltzmann methods (LBMs) based on vector-kinetic models for hyperbolic partial differential equations. In addition to usual lattice Boltzmann equation (LBE) derived by explicit discretisation of vector-kinetic equation (VKE), we also consider LBE derived by semi-implicit discretisation of VKE and compare the relaxation factors of both. We study the properties such as H-inequality, total variation boundedness and positivity of both the LBEs, and infer that the LBE due to semi-implicit discretisation naturally satisfies all the properties while the LBE due to explicit discretisation requires more restrictive condition on relaxation factor compared to the usual condition obtained from Chapman-Enskog expansion. We also derive the macroscopic finite difference form of the LBEs, and utilise it to establish the consistency of LBEs with the hyperbolic system. Further, we extend this LBM framework to hyperbolic conservation laws with source terms, such that there is no spurious numerical convection due to imbalance between convection and source terms. We also present a D2Q9 model that allows upwinding even along diagonal directions in addition to the usual upwinding along coordinate directions. The different aspects of the results are validated numerically on standard benchmark problems.

Quantum algorithms for nonlinear partial differential equations

Computations of nonlinear partial differential equations (PDEs) are some of the most important problems in science and engineering, but come with tremendous costs. Quantum algorithms have the potential to speed up these computations, but current methods are bound to weak or polynomial nonlinearity, short time of validity, or display no quantum advantage otherwise. We construct new quantum algorithms–based on level set methods – for Hamilton-Jacobi and scalar hyperbolic PDEs that generate algorithms with advantages with respect to critical numerical parameters, even for computing the physical observables. These algorithms hold for arbitrary nonlinearity for the equations under study and are valid globally in time. In addition, we also introduce a new quantum encoding, called level set encoding, that allows the efficient computation of physical observables. These PDEs are important for many applications like geometric optics, semi-classical limit of the Schrödinger equations, and high-frequency limit of symmetric linear hyperbolic systems (for example the elastic wave and Maxwell equations). It can display up to exponential advantage –depending on the initial conditions and domain of the relevant observables– in both the dimension of the PDE and the error in computing its observables with respect to classical finite difference methods for these PDEs. Compared to classical finite difference methods, our algorithm also has an advantage with respect to the number of initial data, which has potential important applications in Monte Carlo simulations and uncertainty quantification. For more general PDEs, while an advantage with respect to a very large number of initial data is possible, this is mostly unrealistic for problems of practical interest, and new methods are required.

An Experimental Study Focusing on the Filling Process and Consolidation Characteristics of Geotextile Tubes Filled with Fine-Grained Tungsten Tailings

With advancements in mineral processing technology, the disposal of fine-grained tailings has increasingly become a significant challenge. The geotextile tube method, characterized by its use of a permeable fabric and its cost-effectiveness, has gradually been applied in dam construction and other engineering projects involving tailings. This method offers a novel approach to addressing the storage issues of fine-grained tailings and promotes sustainable utilization. In this paper, the fine tailings that remained after the cyclone classification of Ganzhou tungsten ore were taken as the research object. Specifically, this research endeavored to evaluate the effects of various filling heights and concentrations on the geotextile tube-filling and consolidation process. The results revealed that the filling concentration had a significant impact on the filling benefit of the geotextile tubes, while the filling height had a minimal effect. During the consolidation drainage stage, the dry density, internal friction angle, cohesion, and compression modulus of the tailings in the bags increased with an increasing consolidation time and filling concentration. However, the physical and mechanical properties of the tailings in the geotextile tubes decreased with an increased filling height. Ultimately, this research developed a hyperbolic equation that makes it possible to forecast the ultimate settlement value at various filling heights and concentrations, better representing how the settlement of geotextile tubes changes over the consolidation time.

Global solvability of a mixed problem for a singular semilinear hyperbolic 1d system

Using the method of characteristics and the Banach fixed point theorem (for the Bielecki metric), in the paper it is established the existence and uniqueness of a global (continuous) solution of the mixed problem in the rectangle $\Pi=\{(x,t)\colon 0<x<l<\infty,\ 0<t<T<\infty\}$ for the first order hyperbolic system of semi-linear equations of the form $$ \dfrac{\partial u}{\partial t}+\Lambda(x,t) \dfrac{\partial u}{\partial x}=f(x,t,u,v,w), \dfrac{\partial v}{\partial x}=g(x,t,u,v,w), \dfrac{\partial w}{\partial t }=h(x,t,u,v,w), $$ for a singular with orthogonal (degenerate) and non-orthogonal to the coordinate axes characte\-ristics and with nonlinear boundary conditions, where $\Lambda(x,t)=\mathrm{diag}(\lambda_1(x,t),\ldots,\lambda_k(x,t)),$ $u=(u_1,\ldots,u_k),$ $v=(v_1,\ldots,v_m),$ $w=(w_1,\ldots,w_n),$ $f=(f_1,\ldots,f_k),$ $g=(g_1,\ldots,g_m),$ $h=(h_1,\ldots,h_n)$ and besides $\textrm{sign } \lambda_i(0,t)=\mathrm{const}\neq 0$, $\textrm{sign } \lambda_i(l,t)=\mathrm{const}\neq 0 $ $\text{for all} \ t \in [0, T] $ and for all $i \in \{1,\ldots,k\}$. The presence of non-orthogonal and degenerate characteristics of the hyperbolic system for physical reasons indicates that part of the oscillatory disturbances in the medium propagates with a finite speed, and part with an unlimited one. Such a singularity (degeneracy of characteristics) of the hyperbolic system allows mathematical interpretation of many physical processes, or act as auxiliary equations in the analysis of multidimensional problems.

On Hyperbolic Equations with a Translation Operator in Lowest Derivatives

In the half-plane, a solution to a two-dimensional hyperbolic equation with a translation operator in the lowest derivative with respect to a spatial variable varying along the entire real axis is constructed in an explicit form. It is proven that the solutions obtained are classical if the real part of the symbol of a differential-difference operator in the equation is positive.

Adaptive observer design for coupled ODE–hyperbolic PDE systems with application to traffic flow estimation

This paper studies the state estimation problem for a class of coupled linear ODE–hyperbolic PDE systems with unknown in-domain parameters. Based on the swapping transformation and the least squares parameter estimation method, a Luenberger-type adaptive boundary observer is designed to estimate the ODE–PDE states under the unknown parameters. The exponential convergence conditions w.r.t. the observer feedback gains are given by employing the Lyapunov technique. Finally, an application to traffic state estimation of the Aw–Rascle–Zhang (ARZ) model involving the unknown relaxation time and the boundary input speed deviation is provided to validate the theoretical result.