First-order Hyperbolic Systems Research Articles

Analysis of stresses in two-dimensional isostatic granular systems

We analyze a recently proposed continuous model for stress fields that develop in two-dimensional purely isostatic granular systems. We present a reformulation of the field equations, as a linear first-order hyperbolic system, and show that it is very convenient both for analysis and for numerical computations. Our analysis allows us to predict quantitatively the formation and directions of stress paths and, from these, trajectories and magnitudes of force chains, given the structure in terms of a particular fabric tensor. We further predict quantitatively changes of stresses along the paths, as well as leakage and branching of stress from the main paths into the cones that they make in terms of the fabric tensor. Numerical computations in both Cartesian and cylindrical coordinates verify the analytic results and illustrate the rich behavior discovered. All the phenomena predicted by our solutions have been observed experimentally, suggesting that stresses in isostatic systems can form a base model for a more developed stress theory in granular materials.

A highly accurate discontinuous Galerkin method for complex interfaces between solids and moving fluids

We have extended a new highly accurate numerical scheme for unstructured 2D and 3D meshes based on the discontinuous Galerkin approach to simulate seismic wave propagation in heterogeneous media containing fluid-solid interfaces. Because of the formulation of the wave equations as a unified first-order hyperbolic system in velocity stress, the fluid can be in movement along the interface. The governing equations within the moving fluid are derived from well-known first principles in fluid mechanics. The discontinuous Galerkin approach allows for jumps of the material parameters and the solution across element interfaces, which are handled by Riemann solvers or numerical fluxes. The use of Riemann solvers at the element interfaces makesthe treatment of the fluid particularly simple bysetting the shearmodulus in the fluid region to zero. No additional compatibility relations, such as vanishing shear stress or continuity of normal stresses, are necessary to couple the solid and fluid along an interface. The Riemann solver automatically recognizes the jump of the material coefficients at the interface and provides the correct numerical fluxes for fluid-solid contacts. Therefore, wave propagation in the entire computational domain containing heterogeneous media, namely moving fluids and elastic solids, can be described by a uniform set of acoustic and elastic wave equations. The accuracy of the proposed scheme is confirmed by comparing numerical results against analytic solutions. The potential of the new method was demonstrated in a 3D model problem typical for marine seismic exploration with a fluid-solid interface determined by a complicated bathymetry.

Exact Solutions of Hyperbolic Systems of Kinetic Equations. Application to Verhulst Model with Random Perturbation

For hyperbolic first-order systems of linear partial differential equations (master equations), appearing in description of kinetic processes in physics, biology and chemistry we propose new procedures to obtain their complete closed-form non-stationary solutions. The methods used include the classical Laplace cascade method as well as its recent generalizations for systems with more than 2 equations and more than 2 independent variables. As an example we present the complete non-stationary solution (probability distribution) for Verhulst model driven by Markovian coloured dichotomous noise.

Classical solvability of nonlinear initial-boundary problems for first-order hyperbolic systems

We prove the global classical solvability of initial-boundary problems for semilinear first-order hyperbolic systems subjected to local and nonlocal nonlinear boundary conditions. We also establish lower bounds for the order of nonlinearity demarkating a frontier between regular cases (classical solvability) and singular cases (blow-up of solutions).

A first-order system approach for diffusion equation. I: Second-order residual-distribution schemes

In this paper, we embark on a new strategy for computing the steady state solution of the diffusion equation. The new strategy is to solve an equivalent first-order hyperbolic system instead of the second-order diffusion equation, introducing solution gradients as additional unknowns. We show that schemes developed for the first-order system allow O( h) time step instead of O( h 2) and converge very rapidly toward the steady state. Moreover, this extremely fast convergence comes with the solution gradients (viscous stresses/heat fluxes for the Navier–Stokes equations) simultaneously computed with the same order of accuracy as the main variable. The proposed schemes are formulated as residual-distribution schemes (but can also be identified as finite-volume schemes), directly on unstructured grids. We present numerical results to demonstrate the tremendous gains offered by the new diffusion schemes, driving the rise of explicit schemes in the steady state computation for diffusion problems.

Rank-k solutions of hydrodynamic-type systems

We present a variant of the conditional symmetry method for obtaining rank-k solutions in terms of Riemann invariants for first-order quasilinear hyperbolic systems of PDEs in many dimensions and discuss examples of applying the proposed approach to fluid dynamics equations in n+1 dimensions in detail. We obtain several new types of algebraic, rational, and soliton-like solutions (including kinks, bumps, and multiple-wave solutions).

Remarks on hyperbolic systems of first order with constant coefficient characteristic polynomials

In this paper we shall deal with hyperbolic systems of first order with constant coefficient characteristic polynomials and give a necessary and sufficient condition for the Cauchy problem to be $C^{\infty}$ well-posed under the maximal rank condition (see the condition (R) below).

On real hyperbolic systems with characteristics of variable multiplicity

We consider the linearized hydrodynamic system for an ideal contractible fluid and a linearized system of Maxwell equations with nonlinear current J ( ρ , E ) . We suggest conditions on the symbol of the linear system that generalize the properties of symbols of the linearized systems under consideration. Under these conditions, we prove an existence and uniqueness theorem for linear nonstrictly hyperbolic systems. Linearization of nonlinear hyperbolic systems is used to study waves and their stability under the action of high-frequency perturbations of initial data in physical media. The linearization of a nonlinear real firstorder system for some smooth solution yields a real first-order hyperbolic system L of the form

Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations

Extending the general approach for first-order hyperbolic systems developed in [D. Appelö, T. Hagstrom, G. Kreiss, Perfectly matched layers for hyperbolic systems: general formulation, well-posedness and stability, SIAM J. Appl. Math., 2006, to appear], we construct PML equations for the mixed-type system governing propagation of optical wave packets in both 1D and 2D Bragg resonant photonic waveguides with a cubic nonlinearity, i.e. the coupled mode equations. We prove that in the linear case the layer equations are absorbing and perfectly matched. We also prove they are stable for constant parameters. A number of numerical experiments are performed to assess the layer’s performance in both the linear and nonlinear regimes.

Grid stabilization of high-order one-sided differencing I: First-order hyperbolic systems

We construct stable, maximal order boundary closures for high order central difference methods. The stability is achieved by adding a small number of additional subcell nodes near the boundaries at experimentally determined locations. We find that methods up through 8th order can be stabilized by the addition of a single node, up through 16th order by the addition of two nodes, and up through 22nd order with three extra nodes. We also consider the application of the technique to dispersion relation preserving methods, and we construct and test artificial dissipation operators.

Inverse scattering problem for hyperbolic systems on a semi-axis in the case of equal number of incident and scattered waves

The direct and inverse nonstationary scattering problems for the first-order hyperbolic systems on the semi-axis in the case of equal number of incident and scattered waves under consideration of two problems with the same given incident waves and different boundary conditions are studied. When there are some violations on the conditions, the non-uniqueness in the inverse scattering problem is discussed.

A Mixed Problem for a Hyperbolic System on the Plane with Delay in the Boundary Conditions

We examine well-posedness of the boundary value problem in a half-strip for a first-order linear hyperbolic system with delay (lumped and distributed) in the boundary conditions. In the case of the negative real parts of the eigenvalues of the corresponding spectral problem we prove a time uniform estimate for a solution to the homogeneous problem which enables us to justify the linearization principle for analysis of stability of stationary solutions to the nonlinear problem.

On the boundary controllability of first-order hyperbolic systems

We discuss some results concerning the boundary controllability and stabilizability of a hyperbolic system of conservation laws ∂ u ∂ t + ∂ f ( u ) ∂ x = 0 , t ⩾ 0 , x ∈ ] 0 , 1 [ , where we regard the boundary data (or a partial number of their components) as boundary input controls. In particular, we consider the problem of the global exact boundary controllability of a first order linear hyperbolic system with constant coefficients relative to the linear boundary conditions. Under generic orthogonality assumptions on the boundary and control matrices, and assuming a non-resonance condition of the characteristic speeds we show that one can steer in finite time the solution from any initial condition ϕ ∈ L 1 , to any terminal state ψ ∈ L 1 , even in the case where only a partial control of the boundary values is available.

Model Predictive Control of Cocurrent First-Order Hyperbolic PDE Systems

A continuum of repetition takes place in reaction−convection processes described by first-order hyperbolic partial differential equations (PDEs). Especially, in cocurrent cases, they have finite impulse response (FIR) characteristics. Employing recently developed FIR property preserving finite difference scheme for cocurrent systems, a cocurrent first-order hyperbolic PDE system is recast as a repetitive process. Then a model predictive control scheme for repetitive processes is used for efficient control of cocurrent first-order hyperbolic PDE systems.

Non-linear model reduction of multi-scale reaction-convection processes

This paper deals with reaction-convection processes modeled by firstorder hyperbolic PDE systems. Specifically, we consider processes with fast and slow reactions in the possible presence of fast transport mechanism. In order to derive non-stiff PDE or ODE approximations of the slow dynamics, we propose a model reduction method based on singular perturbation arguments. The efficacy of the approach is illustrated through a representative example.

Classical solutions for a multicomponent flow model in porous media

We consider an initial-boundary-value problem for a nonlinear differential system consisting of one equation of parabolic type coupled with an $n \times n$ semilinear hyperbolic system of first order. This system of equations describes the compressible, $ ( n +1 )$-component, miscible displacement in a porous medium, without including effects of molecular diffusion and dispersion. Assuming some regularity conditions on the data, we prove the existence (locally in time) and uniqueness of classical solutions.

Qualitative properties of solutions for linear and nonlinear hyperbolic PDE's

We present a number of results concerning large-time qualitative behavior of solutions for high-order hyperbolic equations and first-order hyperbolic systems. We discuss the properties of exponential stability and exponential dichotomy, construction of stable, unstable, and center manifolds, Grobman--Hartman type theorems on linearization of the phase portrait, and existence and uniqueness of time-bounded and almost periodic (AP) solutions.

First-order symmetrizable hyperbolic formulations of Einstein's equations including lapse and shift as dynamical fields

First-order hyperbolic systems are promising as a basis for numerical integration of Einstein's equations. In previous work, the lapse and shift have typically not been considered part of the hyperbolic system and have been prescribed independently. This can be expensive computationally, especially if the prescription involves solving elliptic equations. Therefore, including the lapse and shift in the hyperbolic system could be advantageous for numerical work. In this paper, two first-order symmetrizable hyperbolic systems are presented that include the lapse and shift as dynamical fields and have only physical characteristic speeds.

Basic structures of hyperbolic relaxation systems

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.

Basic structures of hyperbolic relaxation systems

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.