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We consider a generalization of the projecting operators method for the case of Cauchy problem for systems of 1D evolution differential equations of first order with variable coefficients. It is supposed that the coefficients dependence on the only variable x is weak, that is described by a small parameter introduction. Such problem corresponds, for example, to the case of wave propagation in a weakly inhomogeneous medium. As an example, we specify the problem to adiabatic acoustics. For the Cauchy problem, to fix unidirectional modes, the projection operators are constructed. The method of successive approximations (perturbation theory) is developed and based on pseudodifferential operators theory. The application of these projection operators allows to obtain approximate evolution equations corresponding to the separated directed waves.

We consider linear and nonlinear hyperbolic SPDEs with mixed derivatives with additive space-time Gaussian white noise of the form $Y_{xt}=F(Y) + \sigma W_{xt}.$ Such equations, which transform to linear and nonlinear wave equations, including Klein-Gordon, Liouville's and the sine-Gordon equation, are related to what Zimmerman (1972) called a diffusion equation. An explicit numerical scheme is employed in both deterministic and stochastic examples. The scheme is checked for accuracy against known exact analytical solutions for deterministic equations. In the stochastic case with $F=0$, solutions yield sample paths for the Brownian sheet whose statistics match well exact values. Generally the boundary conditions are chosen to be initial values $Y(x,0)$ and boundary values $Y(0,t)$ on the quarter-plane or subsets thereof, which have been shown to lead to existence and uniqueness of solutions. For the linear case solutions are compared at various grid sizes and wave-like solutions were found, with and without noise, for non-zero initial and boundary conditions. Surprisingly, wave-like structures seemed to emerge with zero initial and boundary conditions and purely noise source terms with no signal. Equations considered with nonlinear $F$ included quadratic and cubic together with the sine-Gordon equation. For the latter, wave-like structures were apparent with $\sigma \le 0.25$ but they tended to be shattered at larger values of $\sigma$. Previous work on stochastic sine-Gordon equations is briefly reviewed.

This paper is concerned with linear hyperbolic systems of partial differential equations for which certain of the associated propagation speeds are a great deal larger than the other propagation speeds. In certain cases the fast modes allowed by such a system are not present in the true physical solution. Yet the fact that such modes are allowed means that when one tries to compute a numerical solution to an initial-boundary value problem, the errors generated can propagate quite rapidly. In particular, when the boundary data used for the computation are less accurate then the initial data, the fast modes can cause a rapid contamination of the calculation in the interior. To prevent this, one would like to have boundary conditions which prevent fast waves from entering the region. The goal of this paper is to find such conditions.

Hyperbolic grid generation procedures are described which have been used in external flow simulations about complex configurations. For many practical applications a single well-ordered (i.e., structured) grid can be used to mesh an entire configuration, in other problems, composite or unstructured grid procedures are needed. Although the hyperbolic partial differential equation grid generation procedure has mainly been utilized to generate structured grids, extension of the procedure to semiunstructured grids is briefly described. Extensions of the methodology are also described using two-dimensional equations.

Experiments were conducted on mesh moving and local mesh refinement algorithms that are used with a finite difference scheme to solve initial-boundary value problems for vector systems of hyperbolic partial differential equations in one dimension. The mesh moving algorithms move a coarse base mesh by a mesh movement function to follow and isolate spatially distinct phenomena. The local mesh refinement method recursively divides the time step and spatial cells in regions where error indicators are high until a prescribed error tolerance is satisfied. The adaptive mesh algorithms are implemented in a code with an initial mesh generator, a MacCormack finite difference scheme, and an error estimator. Experiments are conducted for several different problems to determine the efficiency of the adaptive methods and their combinations and to gauge their effectiveness in solving one-dimensional problems.

A novel explicit and implicit Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) scheme is presented for hyperbolic equations such as Burgers equation and compressible Euler equations. The proposed scheme performs better than the original SUPG stabilized method in multi-dimensions. To demonstrate the numerical accuracy of the scheme, various numerical experiments have been carried out for 1D and 2D Burgers equation as well as for 1D and 2D Euler equations using Q4 and T3 elements. Furthermore, spectral stability analysis is done for the explicit 2D formulation. Finally, a comparison is made between explicit and implicit versions of the KSUPG scheme.

Snapshot matrices built from solutions to hyperbolic partial differential equations exhibit slow decay in singular values, whereas fast decay is crucial for the success of projection- based model reduction methods. To overcome this problem, we build on previous work in symmetry reduction [Rowley and Marsden, Physica D (2000), pp. 1-19] and propose an iterative algorithm that decomposes the snapshot matrix into multiple shifting profiles, each with a corresponding speed. Its applicability to typical hyperbolic problems is demonstrated through numerical examples, and other natural extensions that modify the shift operator are considered. Finally, we give a geometric interpretation of the algorithm.

This description of results from the project covers research in two areas: (1) Wave propagation for equations describing elastoplastic deformations of granular materials. (2) Hyperbolic conservation laws.

This description of results from the project covers research in two areas: (1) Wave propagation for equations describing elastoplastic deformations of granular materials. (2) Hyperbolic conservation laws.

This description of results from the project covers research in two areas: (1) Wave propagation for equations describing elastoplastic deformations of granular materials. (2) Hyperbolic conservation laws.

This project concerns properties of wave propagation in partial differential equations that are nonlinear and of hyperbolic type. The research is driven largely by applications to granular materials (notably soils in the current grant period), and to the propagation of shock waves in nonlinear elastic materials. Results utilize a combination of mathematical analysis and numerical simulation.

Transient solutions of the hyperbolic type partial differential equations are needed for solving many engineering problems such as computing stress waves for gun dynamics or determining shock behaviors of penetration mechanics. Variational procedures using the bilinear formulations with adjoint variables can serve as the theoretical basis in the derivation of algorithms for the finite element methods, giving direct numerical solutions for partial derivatives of the functions to be found for these problems. The adjoint system can be arranged in a manner that it is a reflected mirror of the original system in time. Generalized boundary conditions employ many types of springs relating the various spatial partial derivatives. They are defined to satisfy the boundaries of the concomitant for the bilinear expression. Algorithms for use in the finite element method are simplified since the adjoint system gives exactly the same solutions as that of the original system. The second necessary condition for an extremum is satisfied by showing that the second variation is positive semi-definite.

We give a new procedure for generalized factorization and construction of the complete solution of strictly hyperbolic linear partial differential equations or strictly hyperbolic systems of such equations in the plane. This procedure generalizes the classical theory of Laplace transformations of second-order equations in the plane.

We propose a finite difference scheme to simulate solutions to a certain type of hyperbolic stochastic partial differential equation (HSPDE). These solutions can in turn estimate so called volatility modulated Volterra (VMV) processes and L\'{e}vy semistationary (LSS) processes, which is a class of processes that have been employed to model turbulence, tumor growth and electricity forward and spot prices. We will see that our finite difference scheme converges to the solution of the HSPDE as we take finer and finer partitions for our finite difference scheme in both time and space. Finally, we demonstrate our method with an example from the energy finance literature.

In many time dependent simulations, the solution on most of the domain will be fairly smooth, with discontinuities or highly oscillatory phenomena occurring over only a small fraction of the domain. In problems such as these, a mesh refinement approach can be the most efficient, and often the only practical, solution method. Refined grids with smaller and smaller mesh spacing are placed only where they are needed. Since we are solving a time dependent problem, the regions needing refinement will change, and therefore our grids must adapt with time as well. This thesis presents a method based on the idea of multiple, component girds for the solution of hyperbolic partial differential equations(pde) using explicit finite difference techniques. Based upon Richardson-type estimates of the local truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. In addition, this approach is recursive in that fine grids can themselves contain even finer subgrids. Those grids with finer mesh width is space will also have a smaller mesh width in time, making this is a mesh refinement algorithm in time and space.

The hyperbolic partial differential equation (PDE) has important practical uses in science and engineering. This article provides an estimate for solving the Goursat problem in hyperbolic linear PDEs with variable coefficients. The Goursat PDE is transformed into a second kind of linear Volterra in-tegral equation. A convergent algorithm that employs Taylor polynomials is created to generate a collocation solution, and the error using the maxi-mum norm is estimated. The paper includes numerical examples to prove the method’s effectiveness and precision.

Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of non-homogeneous transmission lines. The main result of this paper is a simple test for $C_0$-semigroup generation in terms of the boundary conditions. The result is illustrated with several examples.

"On a System of Linear Partial Differential Equations of the Hyperbolic Type" is an article from The Annals of Mathematics, Volume 20 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967120

A theorem that constructs a path integral solution for general second order partial differential equations is specialized to obtain path integrals that are solutions of elliptic, parabolic, and hyperbolic linear second order partial differential equations with Dirichlet/Neumann boundary conditions. The construction is checked by evaluating several known kernels for regions with planar and spherical boundaries. Some new calculational techniques are introduced.

A theorem that constructs a path integral solution for general second order partial differential equations is specialized to obtain path integrals that are solutions of elliptic, parabolic, and hyperbolic linear second order partial differential equations with Dirichlet/Neumann boundary conditions. The construction is checked by evaluating several known kernels for regions with planar and spherical boundaries. Some new calculational techniques are introduced.

Techniques useful for efficiently time-stepping Galerkin methods for various types of time-dependent partial differential equations are presented and analyzed. Second-order quasilinear hyperbolic problems with smooth solutions are studied as a simple model problem for illustrating the widely applicable techniques. The procedure involves the use of a preconditioned iterative method for approximately solving the different linear systems of equations arising at each time-step in a discrete-time Galerkin method. Optimal order L2 spatial errors and almost optimal order work estimates are obtained for the second-order hyperbolic equation. (Author)

The hyperbolic scheme is used to efficiently generate smoothly varying grids with good step size control near the body. Although only two dimensional applications are presented, the basic concepts are shown to extend to three dimensions.

The objective of this project is to develop and analyze stable time discretizations suitable for the simulation of hyperbolic time-dependent partial differential equations. Implicit and explicit multi-step multi-stage time discretizations with optimal time-step restrictions have been developed, as well as implicit Runge--Kutta methods with downwinding and unconditional stability. A testing suite has been written to test some of these methods with a variety of spatial discretizations. New directions have been explored for implicit-explicit methods, which are useful for problems with convection and diffusion. Finally, novel provably stable multi-step time discretizations for use with Fourier pseudo-spectral spatial approximations of the three dimensional viscous Burgers equations and Navier-Stokes equations have been developed.

We identify the result of the continuous wavelet transform with the difference of solutions of two hyperbolic partial differential equations, for which wavelet's shift and scale are considered as independent variables on 2D plane. The characteristic property, which follows from the introduced representation is is the fact that the transform's values inside the triangle defined by two characteristics (a=const, b=const) and crossecting them slopped line on a scale-shift plane (a,b) are completely and uniquely defined by the value of transform and its derivatives along the last mentioned line.

These notes provide an introduction to and a survey on recent results about the long-time behaviour of solutions to hyperbolic partial differential equations with time-dependent coefficients. Particular emphasis is given also to questions about the sharpness of estimates. The selection of materials is based on the mini-courses taught by the first author at the CRM, Barcelona, and by the second author at Aalto University, Helsinki, both in 2011.

This paper deals with a construction of new algorithm: the modified trigonometric cubic B-Spline differential quadrature (MTB-DQM) for space discretization together with a time integration algorithm" for numerical computation of the hyperbolic equations. Specially, MTB-DQM has been implemented for the initial value system of the telegraph equations together with both Dirichlet and Neumann type boundary conditions. The MTB-DQM is a DQM based on modified trigonometric cubic B-splines as new base functions. The problem has been reduced into an amenable system of ordinary differential equations adopting MTB-DQM. The resulting system of ordinary differential equations is solved using time integration algorithms. Further, the stability of MTB-DQM is studied by computing the eigenvalues of the coefficients matrices for various grid points, which confirmed the stability of MTB-DQM for the telegraphic equations. The accuracy of the method has been illustrated in terms of the various discrete error norms for six test problems of the telegraph equation. A comparison of computed numerical solutions with that obtained by the other methods has been carried out for various time levels considering various space sizes

In this article we show the existence and uniqueness of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise is necessary. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.

An efficient numerical mesh generation scheme capable of creating orthogonal or nearly orthogonal grids about moderately complex three dimensional configurations is described. The mesh is obtained by marching outward from a user specified grid on the body surface. Using spherical grid topology, grids have been generated about full span rectangular wings and a simplified space shuttle orbiter.

The purpose of this research has been to further develop a simple efficient grid generation procedure for external aerodynamics applications. The grid generation scheme is based on solving hyperbolic partial differential equation constraints of grid angularity and mesh incremental volumes. The grid generation scheme has been previously used in two dimensional applications to generate grids about smooth body shapes. The main thrust of this AFOSR supported research has been to extend the hyperbolic partial differential equation procedure to three dimensional applications and to study ways of applying the procedure to body shapes that have discontinuous derivatives. The main part of this report, Part I, is devoted to describing the three dimensional hyperbolic grid generator. This Section first reviews the hyperbolic grid generation procedure in two dimensions and then describes the extension to three dimensions. Part II of this report is both brief and sketchy in its presentation. In this section we describe some of our success in treating bodies with sharp edges and bodies that are exceptionally concave. The last part of this report describes a flow field algorithm development. During the course of this research we had some considerable interaction with AFWAL, and at one point became 'side-tracked' into a successful approach of improving the efficiency of our general implicit Euler and navier-Stokes code.

The authors present an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques. Based upon Richardson-type estimates of the truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. Their approach is recursive in that fine grids can themselves contain even finer grids. The grids with finer mesh width in space also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space. This document includes algorithm, data structures and grid generation procedure, and concludes with numerical examples in one and two space dimensions.

The authors present an adaptive method based on the idea of multiple, component grids for the solution of hyperbolic partial differential equations using finite difference techniques. Based upon Richardson-type estimates of the truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. Their approach is recursive in that fine grids can themselves contain even finer grids. The grids with finer mesh width in space also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space. This document includes algorithm, data structures and grid generation procedure, and concludes with numerical examples in one and two space dimensions.

We present a scalable and efficient iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of hyperbolic partial differential equations. It is an interplay between domain decomposition methods and HDG discretizations. In particular, the method is a fixed-point approach that requires only independent element-by-element local solves in each iteration. As such, it is well-suited for current and future computing systems with massive concurrencies. We rigorously show that the proposed method is exponentially convergent in the number of iterations for transport and linearized shallow water equations. Furthermore, the convergence is independent of the solution order. Various 2D and 3D numerical results for steady and time-dependent problems are presented to verify our theoretical findings.

We present a scalable and efficient iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of hyperbolic partial differential equations. It is an interplay between domain decomposition methods and HDG discretizations. In particular, the method is a fixed-point approach that requires only independent element-by-element local solves in each iteration. As such, it is well-suited for current and future computing systems with massive concurrencies. We rigorously show that the proposed method is exponentially convergent in the number of iterations for transport and linearized shallow water equations. Furthermore, the convergence is independent of the solution order. Various 2D and 3D numerical results for steady and time-dependent problems are presented to verify our theoretical findings.

We derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds. These techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude. Our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales, some of which are sometimes treated exactly. The efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator. This is analyzed and a technique is discussed for reducing this error through a simple change of variables. A procedure for generating the appropriate boundary data for the intermediate solutions is also presented. (Author)

The primary objective was to understand fundamental properties of stochastic partial differential equations. The main results obtained concern properties of level sets of the solution of the one-dimensional wave equation, and regularity properties of the two-dimensional wave equation driven by non-white Gaussian noise. Additional results were obtained in the areas of stochastic optimization and stability of random matrix models. (AN)

The primary objective was to understand fundamental properties of stochastic partial differential equations. The main results obtained concern properties of level sets of the solution of the one-dimensional wave equation, and regularity properties of the two-dimensional wave equation driven by non-white Gaussian noise. Additional results were obtained in the areas of stochastic optimization and stability of random matrix models. (AN)

Thesis (M.S. in A.E.)--University of Michigan, 1950

Thesis (M.S. in A.E.)--University of Michigan, 1950

This work is about the inverse dynamics of underactuated flexible mechanical systems governed by quasi-linear hyperbolic partial differential equations subjected to time-varying Dirichlet boundary conditions that are enforced by unknown, spatially disjunct, hence non-collocated Neumann boundary conditions.