Hyperbolic Equations Research Articles

The inverse problem of identifying complex hyperbolic equation source terms in electromagnetic propagation

Abstract This article investigates the inverse problem of determining the source term of the hyperbolic equation for electromagnetic propagation using terminal data. This study is an important method for identifying propagation sources in electromagnetics. Unlike wave equations, the complexity of the underlying equations can make theoretical analysis quite difficult. Firstly, the uniqueness of the inverse problem was proved using the energy method. Then, based on the optimal control framework, the inverse problem was transformed into an optimal control problem, and the existence of the optimal solution and its necessary conditions were established. Secondly, the global uniqueness and stability of the optimal solution have been proven, which is a completely new conclusion. This has laid a solid theoretical foundation for numerical algorithms. Finally, it is proposed to apply the Landweber iteration method and conjugate gradient method to this problem, and some numerical examples are provided to demonstrate the effectiveness and convergence speed of these two algorithms.

Reducing polynomial degree by one for inner-stage operators affects neither stability type nor accuracy order of the Runge–Kutta discontinuous Galerkin method

The Runge–Kutta (RK) discontinuous Galerkin (DG) method is a mainstream numerical algorithm for solving hyperbolic equations. In this paper, we use the linear advection equation in one and two dimensions as a model problem to prove the following results: For an arbitrarily high-order RKDG scheme in Butcher form, as long as we use the P k P^k approximation in the final stage, even if we drop the k k th-order polynomial modes and use the P k − 1 P^{k-1} approximation for the DG operators at all inner RK stages, the resulting numerical method still maintains the same type of stability and convergence rate as those of the original RKDG method. Numerical examples are provided to validate the analysis. The numerical method analyzed in this paper is a special case of the Class A RKDG method with stage-dependent polynomial spaces proposed by Chen, Sun, and Xing [arXiv preprint, arXiv:2402.15150, 2024]. Our analysis provides theoretical justifications for employing cost-effective and low-order spatial discretization at specific RK stages for developing more efficient DG schemes without affecting stability type and accuracy order of the original method.

Controllability of some nonlocal‐impulsive Volterra evolution systems via measures of noncompactness

This study investigates the controllability of a Volterra evolution equation with impulsive terms and nonlocal initial conditions. With the aid of the resolvent operator generated by the linear part of the equation, mild solutions can be defined. Notably, the resolvent operator lacks compactness and equicontinuity. Additionally, the compactness of the impulsive and nonlocal functions is not required. Sufficient conditions for controllability are obtained through measures of noncompactness in Banach spaces. Functional differential equations and hyperbolic partial differential equations can be solved with these results. An example is given to illustrate the validity of the presented results.

Stable Difference Scheme for the Numerical Solution of the Source Identification Problem for Hyperbolic Equations

Stable Difference Scheme for the Numerical Solution of the Source Identification Problem for Hyperbolic Equations

Simultaneous identification of the initial data in a degenerate and singular hyperbolic wave equation

Simultaneous identification of the initial data in a degenerate and singular hyperbolic wave equation

Random Walks and Lorentz Processes

Random walks and Lorentz processes serve as fundamental models for Brownian motion. The study of random walks is a favorite object of probability theory, whereas that of Lorentz processes belongs to the theory of hyperbolic dynamical systems. Here we first present an example where the method based on the probabilistic approach led to new results for the Lorentz process: concretely, the recurrence of the planar periodic Lorentz process with a finite horizon. Afterwards, an unsolved problem—related to a 1981 question of Sinai on locally perturbed periodic Lorentz processes—is formulated as an analogous problem in the language of random walks.

Decoupling of modes for low regularity hyperbolic systems

Decoupling of modes for low regularity hyperbolic systems

From kinetic flocking model of Cucker–Smale type to self-organized hydrodynamic model

We investigate the hydrodynamic limit problem for a kinetic flocking model. We develop a GCI-based Hilbert expansion method, and establish rigorously the asymptotic regime from the kinetic Cucker–Smale model with a confining potential in a mesoscopic scale to the macroscopic limit system for self-propelled individuals, which is derived formally by Aceves-Sánchez et al. in 2019. In the traditional kinetic equation with collisions, for example, Boltzmann-type equations, the key properties that connect the kinetic and fluid regimes are: the linearized collision operator (linearized collision operator around the equilibrium), denoted by [Formula: see text], is symmetric, and has a nontrivial null space (its elements are called collision invariants) which include all the fluid information, i.e. the dimension of Ker([Formula: see text]) is equal to the number of fluid variables. Furthermore, the moments of the collision invariants with the kinetic equations give the macroscopic equations. The new feature and difficulty of the corresponding problem considered in this paper is: the linearized operator [Formula: see text] is not symmetric, i.e. [Formula: see text], where [Formula: see text] is the dual of [Formula: see text]. Moreover, the collision invariants lies in Ker([Formula: see text]), which is called generalized collision invariants (GCI). This is fundamentally different with classical Boltzmann-type equations. This is a common feature of many collective motions of self-propelled particles with alignment in living systems, or many active particle system. Another difficulty (also common for active system) is involved by the normalization of the direction vector, which is highly nonlinear. In this paper, using Cucker–Smale model as an example, we develop systematically a GCI-based expansion method, and micro–macro decomposition on the dual space, to justify the limits to the macroscopic system, a non-Euler-type hyperbolic system. We believe our method has wide applications in the collective motions and active particle systems.

A lower bound estimate of solutions to the Cauchy problems for a hyperbolic Monge–Ampère equation

This paper investigates the Cauchy problems for a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system via Riemannian invariants. Based on the first a priori estimate of the solutions, the second a priori estimate of the derivation of the solutions, and the third a priori estimate of the continuous mould of the first‐order partial derivatives of the solutions to quasilinear hyperbolic system, a lower bound estimate of classical solutions to the Cauchy problems for a hyperbolic Monge–Ampère equation is derived.

Multigrid Reduction‐In‐Time Convergence for Advection Problems: A Fourier Analysis Perspective

ABSTRACTA long‐standing issue in the parallel‐in‐time community is the poor convergence of standard iterative parallel‐in‐time methods for hyperbolic partial differential equations (PDEs), and for advection‐dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two‐level variant of the iterative parallel‐in‐time method of multigrid reduction‐in‐time (MGRIT). This closed‐form theory allows for new insights into the poor convergence of MGRIT for advection‐dominated PDEs when using the standard approach of rediscretizing the fine‐grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse‐grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady‐state advection‐dominated PDEs. We apply this convergence theory to show that, for certain semi‐Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse‐grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re‐purposed in the MGRIT context to develop more robust parallel‐in‐time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach.

Energy bounds for discontinuous Galerkin spectral element approximations of well-posed overset grid problems for hyperbolic systems

We show that even though the Discontinuous Galerkin Spectral Element Method is stable for hyperbolic boundary-value problems, and the overset domain problem is well-posed in an appropriate norm, the energy of the approximation of the latter is bounded by data only for fixed polynomial order, mesh, and time. In the absence of dissipation, coupling of the overlapping domains is destabilizing by allowing positive eigenvalues in the system to be integrated in time. This coupling can be stabilized in one space dimension by using the upwind numerical flux. To help provide additional dissipation, we introduce a novel penalty method that applies dissipation at arbitrary points within the overlap region and depends only on the difference between the solutions. We present numerical experiments in one space dimension to illustrate the implementation of the well-posed penalty formulation, and show spectral convergence of the approximations when sufficient dissipation is applied.

Cauchy problem for a loaded hyperbolic equation with the Bessel operator

Abstract This work is devoted to the study of the Cauchy problem for a loaded differential equation with the Bessel operator. When studying problems for loaded equations, the properties of Erdélyi-Kober operators are used as transformation operators concerning a relation. We obtain an explicit form of the solution to the Cauchy problem for a loaded one-dimensional differential equation. At the end of the work, we will show several examples on graphs.

Rigidity of Lyapunov exponents for derived from Anosov diffeomorphisms

Abstract For a class of volume-preserving partially hyperbolic diffeomorphisms (or non-uniformly Anosov) $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ homotopic to linear Anosov automorphism, we show that the sum of the positive (negative) Lyapunov exponents of f is bounded above (respectively below) by the sum of the positive (respectively negative) Lyapunov exponents of its linearization. We show this for some classes of derived from Anosov (DA) and non-uniformly hyperbolic systems with dominated splitting, in particular for examples described by Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math.115(1) (2000), 157–193]. The results in this paper address a flexibility program by Bochi, Katok and Rodriguez Hertz [Flexibility of Lyapunov exponents. Ergod. Th. & Dynam. Sys.42(2) (2022), 554–591].

Exponential Ergodicity for the Stochastic Hyperbolic Sine-Gordon Equation on the Circle

Exponential Ergodicity for the Stochastic Hyperbolic Sine-Gordon Equation on the Circle

The hysteretic Aw–Rascle–Zhang model

AbstractA novel hyperbolic system of partial differential equations is introduced to model traffic flows. This system comprises three equations, with two being linearly degenerate; its main feature is the inclusion of a hysteretic term in a generalized Aw–Rascle–Zhang (ARZ) model. First, a maximum principle for the diffusive version of the model is proven. Then, it is demonstrated that a solution to the Riemann problem exists, which is unique among solutions that are monotone in velocity; all waves exploited in the construction have suitable viscous profiles. Through several examples it is shown how, as a consequence of different driving habits, the system can model the decay, emergence, or persistence of stop‐and‐go waves (a feature that is missing in the ARZ model), and such behavior is characterized by a simple geometric condition. Furthermore, the model allows the study of traffic flows with a mixture of drivers whose hysteresis loops are either clockwise or counterclockwise. In particular, the presence of sufficiently many of the former dampens speed oscillations.

ON A BOUNDARY VALUE PROBLEM FOR HIGH-ORDER HYPERBOLIC EQUATION WITH IMPULSE DISCRETE MEMORY

The investigation focuses on a boundary value problem for a high-order hyperbolic equation with impulse discrete memory in a rectangular domain. By introducing new functions, the problem is transformed into a set of boundary value problems for a first-order differential equation with impulse discrete memory, which depends on unknown functions and integral relations. D.S. Dzhumabaev's parametrization method is applied to this equivalent problem. The domain is divided according to the time variable, and functional parameters representing discrete memory values are introduced within the interior domains. As a result, the family of boundary value problems for the first-order differential equation with impulse discrete memory and unknown functions is converted into an equivalent family of integral-multipoint boundary value problems involving functional parameters and unknown functions. These equivalent problems include initial value problems for first-order differential equations related to the new functions. The solutions to the initial value problems are expressed using Volterra integral equations. By substituting these solutions into the boundary and impulse conditions, a system of linear functional equations concerning the functional parameters is derived. An algorithm is developed to solve the equivalent problem, and sufficient conditions for the unique solvability of the family of integral-multipoint boundary value problems with functional parameters and unknown functions are provided. Additionally, sufficient conditions for the unique solvability of the original boundary value problem for the high-order hyperbolic equation with impulse discrete memory are established based on the initial data.

Symmetric form of the magnetohydrodynamics equations

Numerical methods of continuum mechanics. Academy of Sciences of the USSR, Siberian Branch, Computing Center. Volume 3, Issue 1, Novosibirsk, 1972. This paper had been originally written in Russian and translated into English by K. A. Ivanova. It is well known that many linear hyperbolic systems of mathematical physics are symmetric hyperbolic systems. In the case of nonlinear equations, the question of whether they can be written in symmetric form is far from simple.

Mixed Crank-Nicolson and Galerkin Methods for Solving Nonlinear Hyperbolic Partial Differential Equation

In this work,the approximation solution for nonlinear hyperbolic partial differential equation(NLHPDE) is obtained by using the mixed Crank-Nicolson (CN) schemeand the GalerkinMethod(GM) and it is symbolized by (MCNGM).At first theCNis utilized for the variable of time to obtain the discrete weak form for the NLHPDE, and then the GM is utilized which reduces the DWF into theGalerkin nonlinear algebraic system (GNLAS) at each step of time. Through utilizing the predictor-corrector techniques which are symbolized by the obtained GNLAS is transformed into Galerkin linear algebraic system (GLAS) which is solved by applying the Cholesky method. The convergence of the method is studied. Some examples are given to illustrate the efficiency and the accuracy for the proposed method.

Global existence and blow-up of solutions for mixed local and nonlocal hyperbolic equations

In this paper, we consider the following mixed local and nonlocal hyperbolic equation: u t t − Δ u + μ ( − Δ ) s u = | u | p − 2 u , in Ω × R + , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , in Ω , u ( x , t ) = 0 , in ( R N ∖ Ω ) × R 0 + , where s ∈ ( 0 , 1 ), N > 2, p ∈ ( 2 , 2 s ∗ ], μ is a nonnegative real parameter, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω, Δ is the Laplace operator, ( − Δ ) s is the fractional Laplace operator. By combining the Galerkin approach with the modified potential well method, we obtain the global existence, vacuum isolating, and blow-up of solutions for the aforementioned problem, provided certain assumptions are fulfilled. Specifically, we study the existence of global solutions for the above problem in the cases of subcritical and critical initial energy levels, as well as the finite time blow-up of solutions. Then, we investigate the blow-up of solutions for the above problem in the case of supercritical initial energy level, as well as upper and lower bounds of blow-up time of solutions.

Regional Controllability of the Position and Fractional Speed of Linear Hyperbolic Systems

Regional Controllability of the Position and Fractional Speed of Linear Hyperbolic Systems