Second-order Hyperbolic Systems Research Articles

A Mixed Problem for a Class of Second-Order Nonlinear Hyperbolic Systems with Dirichlet and Poincaré Boundary Conditions

A Mixed Problem for a Class of Second-Order Nonlinear Hyperbolic Systems with Dirichlet and Poincaré Boundary Conditions

A Mixed Problem for a Class of Second-Order Nonlinear Hyperbolic Systems with Dirichlet and Poincaré Boundary Conditions

Для одного класса гиперболических систем второго порядка изучена смешанная задача с граничными условиями Дирихле и Пуанкаре. В линейном случае дается представление решения задачи в явном виде, исследованы также вопросы единственности и разрешимости поставленной задачи в зависимости от характера нелинейностей, присутствующей в системе. Библиография: 15 названий.

Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system

The Cauchy problems are studied for a first-order multidimensional symmetric linear hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter $\tau>0$ multiplying the second derivatives with respect to $x$ and $t$. The existence and uniqueness of weak solutions of all three systems and $\tau$-uniform estimates for solutions of systems with perturbations are established. Estimates for the difference of solutions of the original system and the systems with perturbations are given, including ones of order $O(\tau^{\alpha/2})$ in the norm of $C(0,T;L^2(\mathbb{R}^n))$, for an initial function $\mathbf w_0$ in the Sobolev space $H^\alpha(\mathbb{R}^n)$, $\alpha=1,2$, or the Nikolskii space $H_2^{\alpha}(\mathbb{R}^n)$, $0<\alpha<2$, $\alpha\neq 1$, and under appropriate assumptions on the free term of the first-order system. For ${\alpha=1/2}$ a wide class of discontinuous functions $\mathbf w_0$ is covered. Estimates for derivatives of any order with respect to $x$ for solutions and of order $O(\tau^{\alpha/2})$ for their differences are also deduced. Applications of the results to the first-order system of gas dynamic equations linearized at a constant solution and to its perturbations, namely, the linearized second-order parabolic and hyperbolic quasi-gasdynamic systems of equations, are presented. Bibliography: 34 titles.

Uniform dissipativity for mixed-order hyperbolic systems, with an application to relativistic fluid dynamics

This paper extends a characterization of uniform dissipativity given in a previous publication by two of the authors for second-order hyperbolic systems to an equally natural class of hyperbolic systems of mixed order. These systems are finite-speed-of-propagation counterparts of symmetric hyperbolic-parabolic systems, and the characterizing condition obtained plays a role analogous to the Kawashima condition, inducing decay rates for all Fourier modes in a way that gives – at least for linear constant-coefficients problems, to which this paper restricts attention – uniform decay of solutions in L2 based Sobolev spaces.

Decay and subluminality of modes of all wave numbers in the relativistic dynamics of viscous and heat conductive fluids

To further confirm the causality and stability of a second-order hyperbolic system of partial differential equations that models the relativistic dynamics of barotropic fluids with viscosity and heat conduction [H. Freistühler and B. Temple, J. Math. Phys. 59, 063101 (2018)], this paper studies the Fourier–Laplace modes of this system and shows that all such modes, relative to arbitrary Lorentz frames, (a) decay with increasing time and (b) travel at subluminal speeds. Stability is also shown for the related model of non-barotropic fluids [H. Freistühler and B. Temple, Proc. R. Soc. A 470, 20140055 (2014) and H. Freistühler and B. Temple, Proc. R. Soc. A 473, 20160729 (2017)]. Even though these properties had been known for a while in the sense of numerical evidence, the fully analytical proofs for the subluminality of modes of arbitrary wave numbers in arbitrary frames given here appear to be the first regarding any four-field formulation of dissipative relativistic fluid dynamics.

Split generalized-[formula omitted] method: A linear-cost solver for multi-dimensional second-order hyperbolic systems

We propose a variational splitting technique for the generalized-α method to solve hyperbolic partial differential equations. We use tensor-product meshes to develop the splitting method, which has a computational cost that grows linearly with the total number of degrees of freedom for multi-dimensional problems. We use the generalized-α method for the temporal discretization while standard C0 finite elements as well as isogeometric elements for spatial discretization. . We perform spectral analysis on the amplification matrix to establish the unconditional stability of the method and to show how splitting affects the overall behavior of the time marching scheme for finite time step sizes, including the standard stability analysis limits 0 and ∞ as particular cases. We use various examples to demonstrate the performance of the method and the optimal approximation accuracy. In these examples, we compute the L2 and H1 norms of the error to show the optimal convergence of the discrete method in space and second-order accuracy in time. Lastly, we also use these tests to demonstrate the linear cost of the solver as the number of degrees of freedom grows.

Asymptotic stability in a second-order symmetric hyperbolic system modeling the relativistic dynamics of viscous heat-conductive fluids with diffusion

This paper establishes nonlinear asymptotic stability of homogeneous reference states in dissipative relativistic fluid dynamics. The result is a counterpart for general non-barotropic fluids of one obtained by the author in a previous paper on barotropic fluids. Differently from that of this earlier finding, the proof here crucially relies on analyzing the corresponding linearized problem in Fourier space, with different scalings for small and large wave numbers.

A waveform relaxation Newmark method for structural dynamics problems

In the conventional Newmark family for time integration of hyperbolic problems, both explicit and implicit methods are inherently sequential in the time domain and not well suited for parallel implementations due to unavoidable processor communication at every time step. In this work we propose a Waveform Relaxation Newmark (WRN $$_\beta $$ ) algorithm for the solution of linear second-order hyperbolic systems of ODEs in time, which retains the unconditional stability of the implicit Newmark scheme with the advantage of the lower computational cost of explicit time integration schemes. This method is unstructured in the time domain and is well suited for parallel implementation. We consider a Jacobi and Gauss–Seidel type splitting and study their convergence and stability. The performance of the WRN $$_\beta $$ algorithm is compared to a standard implicit Newmark method and the obtained results confirm the effectiveness of the Waveform Relaxation Newmark algorithm as a new class of more efficient time integrators, which is applicable, as shown in the numerical examples, to both the finite element method and meshfree methods (e.g. the reproducing kernel particle method).

Single-step full-state feedback control design for nonlinear hyperbolic PDEs

ABSTRACTThe present work proposes an extension of single-step formulation of full-state feedback control design to the class of distributed parameter system described by nonlinear hyperbolic partial differential equations (PDEs). Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law, both feedback control and stabilisation design objectives given as target stable dynamics are accomplished in one step. In particular, the mathematical formulation of the problem is realised via a system of first-order quasi-linear singular PDEs. By using Lyapunov's auxiliary theorem for singular PDEs, the necessary and sufficient conditions for solvability are utilised. The solution to the singular PDEs is locally analytic, which enables development of a PDE series solution. Finally, the theory is successfully applied to an exothermic plug-flow reactor system and a damped second-order hyperbolic PDE system demonstrating ability of in-domain nonlinear control law to achieve stabilisation.

Control Properties for Second-Order Hyperbolic Systems in Anisotropic Cases with Applications in Inhomogeneous and Anisotropic Elastodynamic Systems

We consider control properties for second-order hyperbolic systems in anisotropic cases with variable coefficient matrices in a bounded domain in ${\mathbb{R}}^n$ with $C^{2}$-boundary. Assuming th...

Exact boundary observability for a kind of second-order quasilinear hyperbolic systems

Exact boundary observability for a kind of second-order quasilinear hyperbolic systems

Global solutions to quasi-linear hyperbolic systems of viscoelasticity

In the present paper, we study a large-time behavior of solutions to a quasi-linear second-order hyperbolic system which describes a motion of viscoelastic materials. The system has dissipative properties consisting of a memory term and a damping term. It is proved that the solution exists globally in time in the Sobolev space, provided that the initial data are sufficiently small. Moreover, we show that the solution converges to zero as time tends to infinity. The crucial point of the proof is to derive uniform a priori estimates of solutions by using an energy method.

A consistent Moving Particle System Simulation method: Applications to parabolic/hyperbolic heat conduction type problems

In this paper, a consistent MPS (Moving Particle System Semi-implicit/Moving Particle System Simulation) method is established emanating from Taylor Expansion and Gauss’s Theorem. The proposed MPS method preserves the consistency between the Gradient and Laplacian approximations, and also has the ability to recover the conventional MPS method which lacks consistency. In addition, the proposed method preserves the first order accuracy in space whereas the traditional MPS method loses the accuracy in the case with arbitrary and non-uniform particle discretization. The applicability of the proposed method for the analysis of first- and second-order transient hyperbolic and parabolic systems is illustrated via a recently developed C–F model (Cattaneo–Fourier model) of heat conduction that has been proven to govern thermal transport in different scales, and also capture the various heat conduction physical phenomena. The approaches of imposing different boundary conditions in the heat conduction problems are investigated comprehensively as well. In addition, a unified GS4 i-Integration framework is exploited in the numerical simulations involving first- and second-order systems in time.

Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part II: an inverse source problem

In this paper, we consider Carleman-type estimate and consider an inverse problem for second order hyperbolic systems in an anisotropic case. In the previous Part I paper, we established a Carleman-type estimate for hyperbolic systems in which the coefficient matrices satisfy suitable conditions. We apply a Carleman estimate in the previous Part I paper to an inverse source problem for second-order hyperbolic systems in an anisotropic case and prove an estimate of the Hölder type.

Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part I: Carleman estimates

In this paper, we consider Carleman-type estimates and an inverse source problem for second-order hyperbolic systems in an anisotropic case. In Part I, we establish a Carleman-type estimate for hyperbolic systems in an anisotropic case by assuming that the coefficient matrices in the systems satisfy suitable conditions and taking the weight function where and for in a bounded domain in with -boundary and .

Stable Parareal in Time Method for First- and Second-Order Hyperbolic Systems

The parareal in time algorithm allows one to perform parallel simulations of time-dependent problems. This algorithm has been implemented on many types of time-dependent problems with some success. Recent contributions have allowed one to extend the domain of application of the parareal in time algorithm so as to handle long-time simulations of Hamiltonian systems. This improvement has managed to avoid the fatally large lack of accuracy of the plain parareal in time algorithm, which does not conserve invariant quantities. A somewhat similar difficulty occurs for problems where the solution lacks regularity, either initially or during the evolution, as is the case for hyperbolic systems of conservation laws. In this paper we identify the reasons for instabilities of the parareal in time algorithm and propose a simple way to cure them. We use the new method to solve a linear wave equation and a nonlinear Burgers' equation. The results illustrate the stability of this variant of the parareal in time algorithm.

DECAY RATES FOR SEMILINEAR VISCOELASTIC SYSTEMS IN WEIGHTED SPACES

We consider a class of second-order hyperbolic systems which describe viscoelastic materials, and we extend the recent results by Dharmawardane and Conti et al. More precisely, for all initial data (u0, u1)∈(Hs+1(ℝN) ∩ L1, γ(ℝN))×(Hs(ℝN) ∩ L1, γ(ℝN)) with γ∈[0, 1], we derive faster decay estimates for both dissipative structure or regularity-loss type models. To this end, we first transform our problem into Fourier space and then, by using the pointwise estimate derived by Dharmawardane et al., combined with a device to treat the Fourier transform in the low frequency region, we derive optimal decay results for the solutions to our problem. Finally, we use these decay estimates for the linear problem, combined with the weighted energy method introduced by Todorova and Yordanov, and tackle a semilinear problem.

Exact boundary controllability for a kind of second-order quasilinear hyperbolic systems

Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the local exact boundary observability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.

Second-order hyperbolic Fuchsian systems: Asymptotic behavior of geodesics in Gowdy spacetimes

Recent work by the authors led to the development of a mathematical theory dealing with `second--order hyperbolic Fuchsian systems', as we call them. In the present paper, we adopt a physical standpoint and discuss the implications of this theory which provides one with a new tool to tackle the Einstein equations of general relativity (under certain symmetry assumptions). Specifically, we formulate the `Fuchsian singular initial value problem' and apply our general analysis to the broad class of vacuum Gowdy spacetimes with spatial toroidal topology. Our main focus is on providing a detailed description of the asymptotic geometry near the initial singularity of these inhomogeneous cosmological spacetimes and, especially, analyzing the asymptotic behavior of timelike geodesics ---which represent the trajectories of freely falling observers --- and null geodesics. In particular, we numerically construct Gowdy spacetimes which contain a black hole--like region together with a flat Minkowski--like region. By using the Fuchsian technique, we investigate the effect of the gravitational interaction between these two regions and we study the unexpected behavior of geodesic trajectories within the intermediate part of the spacetime limited by these two regions.

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.