Hyperbolic Initial Value Problems Research Articles

Numerical investigation of the dynamics of linear spin s fields on a Kerr background: Late-time tails of spin s=±1,±2 fields

The time evolution of linear fields of spin $s = \pm 1$ and $s = \pm 2$ on Kerr black hole spacetimes are investigated by solving the homogeneous Teukolsky equation numerically. The applied numerical setup is based on a combination of conformal compactification and the hyperbolic initial value problem. The evolved basic variables are expanded in terms of spin-weighted spherical harmonics which allows us to evaluate all the angular derivatives analytically, whereas the evolution of the expansion coefficients, in the time-radial section, is determined by applying the method of lines implemented in a fourth order accurate finite differencing stencil. Concerning the initialization, in all of our investigations single mode excitations---either static or purely dynamical type initial data---are applied. Within this setup the late time tail behavior is investigated. Due to the applied conformal compactification the asymptotic decay rates are determined at three characteristic locations---in the domain of outer communication, at the event horizon and at future null infinity---simultaneously. Recently introduced new type of `energy' and `angular momentum' balance relations are also applied in order to demonstrate the feasibility and robustness of the developed numerical schema, and also to verify the proper implementation of the underlying mathematical model.

A Stability Criterion for High-Frequency Oscillations

— We show that a simple Levi compatibility condition determines stability of WKB solutions to semilinear hyperbolic initial-value problems issued from highly-oscillating initial data with large amplitudes. The compatibility condition involves the hyperbolic operator, the fundamental phase associated with the initial oscillation, and the semilinear source term; it states roughly that hyperbolicity is preserved around resonances. If the compatibility condition is satisfied, the solutions are defined over time intervals independent of the wavelength, and the associated WKB solutions are stable under a large class of initial perturbations. If the compatibility condition is not satisfied, resonances are exponentially amplified, and arbitrarily small initial perturbations can destabilize the WKB solutions in small time. In the unstable case, the key observation is that resonances correspond to weakly hyperbolic frequencies; the amplification proof then relies on a short-time Duhamel representation formula for solutions of zeroth-order pseudo-differential equations. Our examples include coupled Klein-Gordon systems, and systems describing Raman and Brillouin instabilities.

Resonant Leading Term Geometric Optics Expansions with Boundary Layers for Quasilinear Hyperbolic Boundary Problems

We construct and justify leading order weakly nonlinear geometric optics expansions for nonlinear hyperbolic initial value problems, including the compressible Euler equations. The technique of simultaneous Picard iteration is employed to show approximate solutions tend to the exact solutions in the small wavelength limit. Recent work [2] by Coulombel et al. studied the case of reflecting wave trains whose expansions involve only real phases. We treat generic boundary frequencies by incorporating into our expansions both real and nonreal phases. Nonreal phases introduce difficulties such as approximately solving complex transport equations and result in the addition of boundary layers with exponential decay. This also prevents us from doing an error analysis based on almost periodic profiles as in [2].

Existence of solutions for linear hyperbolic initial-boundary value problems in a rectangle

In a recent article, we achieved the well-posedness of linear hyperbolic initial and boundary value problems (IBVP) in a rectangle via semigroup method, and we found that there are only two elementary modes called hyperbolic and elliptic modes in the system. It seems that, there is only one set of boundary conditions for the hyperbolic mode, while there are infinitely many sets of boundary conditions for the elliptic mode, which can lead to well-posedness. In this article, we continue to consider linear hyperbolic IBVP in a rectangle in the constant coefficients case and we show that there are also infinitely many sets of boundary conditions for hyperbolic mode which will lead to the existence of a solution. We also have uniqueness in some special cases. The boundary conditions satisfy the reflection conditions introduced in Section 3, which turn out to be equivalent to the strictly dissipative conditions.

The linear hyperbolic initial and boundary value problems in a domain with corners

In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichs-symmetrizable.

Interval Analytic Method in Existence Result for Hyperbolic Partial Differential Equation

Without the usual assumption of monotonicity, we establish some results on the theory of hyperbolic differential inequalities which enable us to produce a majorising interval function for the solution of the hyperbolic initial value problem. Using this function, a variation of parameters formula and interval iterative technique, the existence of solution to the problem is established.

Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems

We study the spectral approximation properties of finite element and NURBS spaces from a global perspective. We focus on eigenfunction approximations and discover that the L2-norm errors for finite element eigenfunctions exhibit pronounced “spikes” about the transition points between branches of the eigenvalue spectrum. This pathology is absent in NURBS approximations. By way of the Pythagorean eigenvalue error theorem, we determine that the squares of the energy-norm errors of the eigenfunctions are the sums of the eigenvalue errors and the squares of the L2-norm eigenfunction errors. The spurious behavior of the higher eigenvalues for standard finite elements is well-known and therefore inherited by the energy-norm errors along with the spikes in the L2-norm of the eigenfunction errors. The eigenvalue pathology is absent for NURBS. The implications of these results to the corresponding elliptic boundary-value problem and parabolic and hyperbolic initial-value problems are discussed.

A matrix formulation to the wave equation with non-local boundary condition

Recently, various sequential numerical schemes have been proposed for the solution of non-classical hyperbolic initial value problems which involve non-local integral terms over the spatial domain. In this paper, we focus on the wave equation with the non-local boundary condition. Two matrix formulation techniques based on the shifted standard and shifted Chebyshev bases are proposed for the numerical solution. Several numerical examples and also some comparisons with another methods will be investigated to confirm the efficiency of this procedure.

Travelling Waves for a Certain First-Order Coupled PDE System

This paper concentrates on a particular first-order coupled PDE system. It provides both a detailed treatment of the existence and uniqueness of monotone travelling waves to various equilibria, by differential-equation theory and by probability theory and a treatment of the corresponding hyperbolic initial-value problem, by analytic methods. The initial-value problem is studied using characteristics to show existence and uniqueness of a bounded solution for bounded initial data (subject to certain smoothness conditions). The concept of weak solutions to partial differential equations is used to rigorously examine bounded initial data with jump discontinuities. For the travelling wave problem the differential-equation treatment makes use of a shooting argument and explicit calculations of the eigenvectors of stability matrices. The probabilistic treatment is careful in its proofs of martingale (as opposed to merely local-martingale) properties. A modern change-of-measure technique is used to obtain the best lower bound on the speed of the monotone travelling wave --- with Heaviside initial conditions the solution converges to an approximate travelling wave of that speed (the solution tends to one ahead of the wave-front and to zero behind it). Waves to different equilibria are shown to be related by Doob $h$-transforms. Large-deviation theory provides heuristic links between alternative descriptions of minimum wave speeds, rigorous algebraic proofs of which are provided.

Entropy solutions for weakly coupled hyperbolic systems in several space dimensions

Systems of nonlinear hyperbolic conservation laws in several space dimensions are considered which are characterized by the fact that the coupling of the equations is only due to source terms. These weakly coupled systems arise in a variety of applications like hydrological problems, the theory of reactive flows, relaxation schemes, or mathematical biology. We present an existence and uniqueness theorem for certain entropy solutions of a general class of weakly coupled hyperbolic initial value problems. The approach relies on the analysis of the associated parabolically regularized Cauchy problem.¶The result for the general problem is applied tophysically relevant examples. Existence and uniquenessof entropy solutions for problems from combustion theory,hydrology and other areas is verified.

Asymptotic expansions of the solutions to a class of quasilinear hyperbolic initial value problems

Asymptotic expansions of the solutions to a class of quasilinear hyperbolic initial value problems

Transport of Gyration-Dominated Space Plasmas of Thermal Origin 2: Numerical Solution

A numerical solution has been found for the hyperbolic initial-value problem, which follows from the 16-moment set of equations describing the transport of a plasma in a strong magnetic field. In addition to the time-dependent evolution of mass and momentum, these equations describe the variations in anisotropic temperatures and heat fluxes for both ion and electron species. The numerical solution employs a high-order Godunov method to achieve third-order accuracy in space and second-order accuracy in time. The method includes an approximate Riemann solver which is suitable for a system of equations that cannot entirely be expressed in conservation form. In addition, a new technique has been developed to overcome the stiffness that occurs in regions where plasma flows are strongly reactive. Simple test cases showing correct wave behavior are presented.

Spectral methods for periodic initial value problems with nonsmooth data

In this paper we consider hyperbolic initial value problems subject to periodic boundary conditions with nonsmooth data. We show that if we filter the data and solve the problem by the Galerkin-Collocation method, recently proposed by us, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth. For this we have to perform a local smoothing of the computed solution.

Almost global existence of solutions of quasi‐linear hyperbolic equations

AbstractWe consider the quasi‐linear hyperbolic initial value problem (1) of the Introduction, and prove that for any T>0 there is a bound such that if the norm of the initial data is smaller than that bound then the solution of (1) exists on all of [0, T].

Stabilization of the Lax-Wendroff method and a generalized one-step Runge-Kutta method for hyperbolic initial-value problems

In order ot integrate hyperbolic systems we distinguish explicit and implicit time integrators. Implicit methods allow large integration steps, but require more storage and are more difficult to implement than explicit methods. However explicit methods are subject to a restriction on the integration step. This restriction is a drawback if the variation of the solution in time is so small that accuracy considerations would allow a larger integration step. In this report we apply a smoothing technique in order to stabilize the Lax-Wendroff method and a generalized one-step Runge-Kutta method. Using this technique, the integration step is not limited by stability considerations.

Bounds for the solution of hyperbolic problems

This article describes a newly developed and implemented method for computing guaranteed errorbounds for the solution of hyperbolic initial value problems. The basic concepts—modified fixed point theorems and approximated operators—allow an a posteriori error-estimation automatically. Therefore, no a priori knowledge of Lipschitz constants, monotonicity properties or additional error analysis is necessary.

Monotonicity and finite element methods for hyperbolic initial value problems

AbstractA well known theórem about super‐ and subfunctions for the solution of hyperbolic initial value problems constructs differentiable functions as upper and lower bounds (see Walter [1], 21 XIII). The proof can be done by transforming the differential equation problem into a set of integral equations, using the monotonicity‐properties of the arising integral operators. This proof needs an integral representation for twice differentiable functions. It is shown that this proceeding can be generalized to get upper and lower bounds in terms of finite element functions. To do this, we give an integral representation for continuous, piecewise differentiable functions, including the discontinuities of their derivatives. Then the generalization of the classical proof yields interface conditions for the finite element functions. Finally, it is demonstrated how to realize numerically these conditions.

Qualitative theory for hyperbolic characteristic initial value problems

SynopsisCharacteristic initial value problems associated with hyperbolic equations of the form uxy + g(x, y)u = 0 are considered for (x, y)∈ℝ+× ℝ+. New criteria for the existence of a nodal line asymptotic to the axes are established, as are criteria for the existence of a zero beyond such a nodal line. Some numerical solutions are presented in graphical form and discussed relative to what is known about oscillation properties of such problems.

On the numerical radius and its applications

We give a brief account of the numerical radius of a linear bounded operator on a Hilbert space and some of its better-known properties. Both finite- and infinite- dimensional aspects are discussed, as well as applications to stability theory of finite-difference approximations for hyperbolic initial-value problems.

Oscillation criteria for third order hyperbolic characteristic initial value problems

SynopsisSufficient conditions for oscillation of solutions to third order hyperbolic characteristic initial value problems are established. The results generalize known oscillation criteria for second order hyperbolic problems.