First-order Hyperbolic Systems Research Articles

Reflection of conormal pulse solutions to large variable-coefficient semilinear hyperbolic systems

We provide a rigorous justication of nonlinear geometric optics expansions for reflecting pulses in space dimensions n>1. The pulses arise as solutions to variable coefficient semilinear first-order hyperbolic systems. The justification applies to N×N systems with N interacting pulses which depend on phases that may be nonlinear. The coherence assumption made in a number of earlier works is dropped. We consider problems in which incoming pulses are generated from pulse boundary data as well as problems in which a single outgoing pulse reflects off a possibly curved boundary to produce a number of incoming pulses. Although we focus here on boundary problems, it is clear that similar results hold by similar methods for the Cauchy problem for N×N systems in free space.

On k-wave solutions of quasilinear systems of partial differential equations

In this paper, we establish a relation between two seemingly unrelated concepts for solving first-order hyperbolic quasilinear systems of partial differential equations in many dimensions. These concepts are based on a variant of the conditional symmetry method and on the generalized method of characteristics. We present the outline of recent results on multiple Riemann wave solutions of these systems. An auxiliary result concerning a modification of the Frobenius theorem for integration is used. We apply this result in order to show that the conditional symmetry method can deliver larger classes of multiple Riemann wave solutions, through a simpler procedure, than the one obtained from the generalized method of characteristics. We demonstrate that solutions can be interpreted physically as a superposition of k single waves. These theoretical considerations are illustrated by examples of hydrodynamic-type systems in (n+1) dimensions.

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

A coupled acoustic/elastic discontinuous Galerkin finite element method: Application to ultrasonic imaging of 3D-printed synthetic materials

We present the derivation of upwind numerical fluxes for the space discontinuous Galerkin (dG) finite element method applied to the numerical modeling of wave propagation in multidimensional coupled acoustic/elastic media. The space dG method is formulated using the first-order velocity-pressure and velocity-stress systems for acoustic and elastic media, respectively. After eigenanalysis of the first-order hyperbolic systems highlighting the eigenmodes of wave propagation, the upwind numerical fluxes on the acoustic/acoustic and acoustic/elastic interface are obtained in terms of exact solutions of relevant Riemann problems. Thanks to the proposed approach, explicit closed-form expressions of the upwind numerical fluxes are obtained on the acoustic/elastic interface for the general case of multidimensional anisotropic heterogeneous solid media coupled with acoustic fluids. The developed numerical fluxes are validated by analytical/numerical comparison considering the example of an acoustic domain with a circular elastic inclusion. Finally, the coupled solver is used to perform a multiparametric study on the microstructure's echogenicity in a 3D-printed synthetic material under ultrasonic imaging.

Double-pole unsplit complex-frequency-shifted multiaxial perfectly matched layer combined with strong-stability-preserved Runge-Kutta time discretization for seismic wave equation based on the discontinuous Galerkin method

The classical complex-frequency-shifted perfectly matched layer (CFS-PML) technique has attracted widespread attention for seismic wave simulations. However, few studies have addressed the double-pole variant of the CFS-PML scheme. The double-pole CFS-PML has a stronger capacity to absorb near-grazing incident waves and evanescent waves than the classical CFS-PML. Using the discontinuous Galerkin (DG) method, we derive a double-pole unsplit auxiliary ordinary differential equation CFS-multiaxial PML (AODE CFS-MPML) formulation, which combines a fourth-order strong-stability-preserved Runge-Kutta time discretization for wavefield simulation on an unstructured grid. The double-pole unsplit CFS-MPML formulations are obtained by introducing auxiliary memory variables and AODEs. The original stress-velocity equations and the double-pole unsplit AODE CFS-MPML equations are all first-order hyperbolic systems and suitable for the DG method. The attenuative variables are added directly to the original seismic wave equations without changing their formats. In contrast to the split perfectly matched layer (PML), we avoid reformulating PML equations in the nonattenuative modeling region. The original seismic wave equation is solved in the nonattenuative modeling domain, whereas the double-pole unsplit AODE CFS-MPML equation is implemented in the PML absorbing region. Three numerical examples validate the performance of the double-pole unsplit AODE CFS-MPML technique. The isotropic and anisotropic experiments demonstrate that our developed double-pole unsplit AODE CFS-MPML is more stable and obtains more accurate solutions than the classical CFS-PML. The second example indicates the flexibility of the combination of the DG method with the double-pole CFS-MPML on undulating topography. The final example displays the applicability and effectiveness of our method in a 3D situation.

Wavefield simulation with the discontinuous Galerkin method for a poroelastic wave equation in triple-porosity media

In this paper, we derive a new poroelastic wave equation in triple-porosity media and develop a weighted Runge-Kutta (RK) discontinuous Galerkin method (DGM) for solving it. Based on Biot’s theory and Lagrangian formulas, we obtain 3D Biot’s equations in a heterogeneous anisotropic triple-porosity medium. We also summarize poroelastic wave equations in single- and double-porosity media. The traditional two-phase theory is a special case of the porous theory. Compared with single- or dual-porosity wave equations, our triple-porosity wave equation generates more accurate wavefield information. The isotropic and anisotropic cases are considered. Subsequently, we formulate the new poroelastic equation into a first-order hyperbolic conservation system, which is suitable to be solved by DGM. An optimized local Lax-Friedrichs flux and an implicit-weighted RK time discretization scheme are used for this computation. We use two types of mesh elements — quadrilateral and unstructured triangular elements. We find that there are two kinds of slow P waves, P1 and P2 waves, in double-porosity media, whereas three kinds of slow P waves, P1, P2, and P3 waves, exist in three-porosity media. We also study the analytical and numerical solutions of propagation velocities for different waves in isotropic media without dissipation using the Jacobian matrix of DGM and provide a comparison of field variables about three types of wave equations. Finally, we conduct a series of examples to quantitatively investigate the propagation properties of seismic waves in isotropic and anisotropic multiporosity media computed by DGM. The slow P wave in multiporosity media with dissipation decays rapidly, which also will lead to phase distortion. Numerical results verify the correctness and applicability of our proposed new equation and indicate that the weighted RK DGM is a stable and accurate algorithm to simulate wave propagation in poroelastic media.

Cauchy Problem for Incompressible Neo-Hookean Materials

In this paper we consider the Cauchy problem for neo-Hookean incompressible elasticity in spatial dimension $$d \geqq 2$$ . The Cauchy problem can be formulated in terms of maps $$x(t,\cdot )$$ with domain a reference space $${\mathbb {R}}^d_\xi $$ , and with values in space $${\mathbb {R}}^d_x$$ . Initial data consists of initial deformation $$\phi (\xi ) = x(0,\xi )$$ and velocity $$\psi (\xi ) = \partial x(t,\xi )/\partial t |_{t=0}$$ . We consider the initial deformations of the form $$x(0, \xi ) = A \xi + \varphi (\xi )$$ , where A is a constant $$SL(d, {\mathbb {R}})$$ matrix. We assume that $$\varphi $$ and $$\psi $$ are in Sobolev spaces $$(\varphi , \psi ) \in H^{s+1}({\mathbb {R}}^d)\times H^{s}({\mathbb {R}}^d)$$ . If $$s>s_{crit}= d/2+1$$ , well-posedness is well-known. We are here interested primarily in the low regularity case, $$s \le s_{crit}$$ . For $$d = 2, 3$$ , we prove existence and uniqueness for $$s_0 < s\le s_{crit}$$ , and we can prove the well-posedness, but for a smaller range, $$s_1 < s \le s_{crit}$$ , where, if $$d = 2$$ , $$s_0 = 7/4$$ and $$s_1 = 7/4 + (\sqrt{65}-7)/8$$ , and if $$d=3$$ , then $$s_0=2$$ and $$s_1 = 1 + \sqrt{3/2}$$ . For the full range (in s) results, as indicated above, we need additional Hölder regularity assumptions on certain combinations of second order derivatives of $$\varphi $$ . A key observation in the proof is that the equations of evolution for the vorticities decompose into a first-order hyperbolic system, for which a Strichartz estimate holds, and a coupled transport system. This allows one to set up a bootstrap argument to prove local existence and uniqueness. Continuous dependence on initial data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement $$s > s_{crit}$$ is necessary.

On Second-Order Parabolic and Hyperbolic Perturbations of a First-Order Hyperbolic System

On Second-Order Parabolic and Hyperbolic Perturbations of a First-Order Hyperbolic System

Variational optimality condition in control of hyperbolic systems with boundary delay parameters

An optimal control problem of a first-order hyperbolic system is studied, in which a boundary condition at one of the ends is determined from a controlled system of ordinary differential equations with constant state lag. Control functions are bounded and measurable functions. The system of ordinary differential equations at the boundary is linear in state. However the matrix of coefficients depends on control functions. Therefore, the optimality condition of Pontryagin’s maximum principle type in this problem is a necessary, but not a sufficient optimality condition. In this paper, the problem is reduced to an optimal control problem of a special system of ordinary differential equations. The proposed approach is based on the use of an exact formula of the cost functional increment. The reduced problem can be solved using a wide range of effective methods used for optimization problems in systems of ordinary differential equations. Problems of this kind arise when modeling thermal separation processes, suppression of mechanical vibrations in drilling, wave processes and population dynamics.

Leaderless Consensus of Semilinear Hyperbolic Multiagent Systems with Semipositive or Seminegative Definite Convection

This paper deals with a leaderless consensus of semilinear first-order hyperbolic partial differential equation-based multiagent systems (HPDEMASs). A consensus controller under an undirected graph is designed. Dealing with different convection assumptions, two different boundary conditions are presented, one right endpoint and the other left endpoint. Two sufficient conditions for leaderless consensus of HPDEMAS are presented by giving the gain range in the case of the symmetric seminegative definite convection coefficient and the semipositive definite convection coefficient, respectively. Two examples are presented to show the effectiveness of the control methods.

Generalized Fractional Algebraic Linear System Solvers

This paper is devoted to the numerical computation of algebraic linear systems involving several matrix power functions; that is finding x solution to \(\sum _{\alpha \in \mathbb {R}}A^{\alpha }x=b\). These systems will be referred to as Generalized Fractional Algebraic Linear Systems (GFALS). In this paper, we derive several gradient methods for solving these very computationally complex problems, which themselves require the solution to intermiediate standard Fractional Algebraic Linear Systems (FALS) \(A^{\alpha }x=b\), with \(\alpha \in \mathbb {R_+}\). The latter usually require the solution to many classical linear systems \(Ax=b\). We also show that in some cases, the solution to a GFALS problem can be obtained as the solution to a first-order hyperbolic system of conservation laws. We also discuss the connections between this PDE-approach and gradient-type methods. The convergence analysis is addressed and some numerical experiments are proposed to illustrate and compare the methods which are proposed in this paper.

Variational optimality condition in a control problem of a linear first-order hyperbolic system with boundary delay

Variational optimality condition in a control problem of a linear first-order hyperbolic system with boundary delay

Dynamic phase-field fracture with a first-order discontinuous Galerkin method for elastic waves

We present a new numerical approach for wave induced dynamic fracture. The method is based on a discontinuous Galerkin approximation of the first-order hyperbolic system for elastic waves and a phase-field approximation of brittle fracture driven by the maximum tension. The algorithm is staggered in time and combines an implicit midpoint rule for the wave propagation followed by an implicit Euler step for the phase-field evolution. At fracture, the material is degraded, and the waves are reflected at the diffusive interfaces. Two and three-dimensional examples demonstrate the advantages of the proposed method for the computation of crack growth and spalling initiated by reflected and superposed waves.

On well-posedness of a mathematical model for cerebrospinal fluid in the optic nerve sheath and the spinal subarachnoid space

More often than not it is assumed a-priori that first-order, time-dependent systems of non-linear partial differential equations will be hyperbolic; such assumption may be incorrect and could produce misleading results, or no results at all, due to instabilities of the system itself. The present paper is concerned with the detailed theoretical and numerical study of a 4×4 non-linear, time-dependent first-order system that has been proposed for modelling the dynamics of cerebrospinal fluid in the spinal subarachnoid space [14,15,53]. The system has also been proposed as a model for the dynamics of the cerebrospinal fluid surrounding the optic nerve [45]; see also [39]; in this case however, there are physiological aspects that require clarification, before a realistic application of the present model to optic nerve physiology is considered. Here we address mathematical aspects and prove a well-posedness theorem that identifies necessary and sufficient conditions for the system to be strictly hyperbolic and conditions under which the system is of mixed elliptic-hyperbolic type. This result was anticipated in [53] and full details are given in [45]. We then address more generally the numerical implications for such type of systems. Solving numerically systems that contain a mixed elliptic-hyperbolic region is challenging. First, the use of upwind methods, so popular for hyperbolic first-order systems, is ruled out automatically here. One is then left with having to choose centred-type methods with great care. Moreover, the non-conservative character of the present system adds another challenge. Results for various representative examples are shown and analysed and the potential of the proposed model for simulating CSF dynamics in physiological and pathophysiological settings is discussed.

Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions

In this work we study linear Maxwell equations with time- and space-dependent matrix-valued permittivity and permeability on domains with a perfectly conducting boundary. This leads to an initial boundary value problem for a first-order hyperbolic system with characteristic boundary. We prove a priori estimates for solutions in Hm. Moreover, we show the existence of a unique Hm-solution if the coefficients and the data are sufficiently regular and satisfy certain compatibility conditions. Since the boundary is characteristic for the Maxwell system, we have to exploit the divergence conditions in the Maxwell equations in order to derive the energy-type Hm-estimates. A combination of these estimates with several regularization techniques then yields the existence of solutions in Hm.

Finite-time stabilization for a first-order hyperbolic system with integral kernel by boundary control

This paper consider the finite-time boundary stabilization for a first-order hyperbolic system with integral kernel and lower order term. Via choosing a suitable integral transformation converting the original system into a finite-time stable object system, then, we obtain the control law and certify the finite-time stabilization of the closed-loop system combined with the finite-time stability of object system and invertibility of forward transformation.

Iterative learning control with high-order internal model for first-order hyperbolic systems

This paper studies the iterative learning control (ILC) algorithm for first-order hyperbolic systems. Unlike most of the ILC literature of distributed parameter systems, in the iteration domain, that require identical desired trajectories. Here the desired trajectories are iteratively varying and described by a high-order internal model (HOIM). The HOIM-based P-type ILC design is firstly introduced in this paper to the first-order hyperbolic systems, which enable the systems to achieve the perfect tracking for the iteration-varying desired trajectories on L2 space. Meanwhile, the convergence theorem of the proposed algorithm is established for first-order time-delay hyperbolic systems. Finally, simulation results testify the validity of the algorithm.

Propagation and interaction of weakly nonlinear plane waves in transversely isotropic elastic materials

The paper presents a study of the propagation and interaction of weakly nonlinear plane waves in isotropic and transversely isotropic media. It begins with a definition of stored energy functions of considered hyperelastic models. The equation of elastodynamics as well as the first-order quasilinear hyperbolic system for plane waves are provided. The eigensystem for this system is determined to study three-wave interaction coefficients. The main part of the paper concerns a discussion of these coefficients. Applying the weakly nonlinear asymptotics method, it is shown that in the case of transverse isotropy the inviscid Burgers’ equation describes an evolution of a single quasi-shear wave. The result contradicts the case of isotropy, where the equation with quadratic nonlinearity cannot describe any shear wave propagation. The paper ends with an example of numerical solutions for the obtained evolution equation.

An optimal control problem by a first-order hyperbolic system with delay and integral control constraints

A non-classic optimality condition and numerical algorithm for smooth boundary control in semi-linear first-order hyperbolic system with delay are presented. An additional integral control constraint is considered. The suggested approach is based on special variations of admissible continuously differentiable controls. These variations lead to a necessary optimality condition which is a base for a numerical method. An illustrative example demonstrates the effectiveness of the proposed approach.

Recovering Density and Speed of Sound Coefficients in the 2D Hyperbolic System of Acoustic Equations of the First Order by a Finite Number of Observations

We consider the coefficient inverse problem for the first-order hyperbolic system, which describes the propagation of the 2D acoustic waves in a heterogeneous medium. We recover both the denstity of the medium and the speed of sound by using a finite number of data measurements. We use the second-order MUSCL-Hancock scheme to solve the direct and adjoint problems, and apply optimization scheme to the coefficient inverse problem. The obtained functional is minimized by using the gradient-based approach. We consider different variations of the method in order to obtain the better accuracy and stability of the appoach and present the results of numerical experiments.