On a linear hyperbolic equation with smooth coefficients without solutions

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in Yuan et al. (SIAM J Sci Comput 38:A2987–A3019, 2016). The DG methods in Yuan et al. (2016) can maintain positivity and high order accuracy, but they rely both on the scaling limiter in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax–Wendroff type theorem is proved in Yuan et al. (2016), guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $$P^k$$ of piecewise polynomials of degree at most k. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu (2010) can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $$P^k$$ or $$Q^k$$ spaces (piecewise kth degree polynomials or piecewise tensor-product kth degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu (2010). Computational results are provided to demonstrate the good performance of our DG schemes.

In this paper we shall derive energy integral inequalities for the linear hyperbolic partial differential equation of any order in several variables with multiple characteristics and constant coefficients. In [1] Garding uses an idea by Leray [6] to develop energy integrals for totally hyperbolic equations (including variable coefficients). Totally hyperbolic equations are subject to the severe restriction that all characteristics remain distinct(2). In his method G'arding multiplies the given operator of order n by another operator of order n 1 whose characteristics separate those of the given operator. Using the Fourier transform and Parseval's formula he arrives at the energy inequality, which is an a priori estimate of the square integral of the solution of the given equation and all the derivatives up to order n -1. In our case of an equation with multiple characteristics we use a variant of G'arding's method. Our energy inequality is of a different form in as much as it provides an a priori estimate of the integral of the square of the solution and certain non-negative integral forms of its derivatives up to order n -1. However, we also show that for totally hyperbolic equations our energy inequalities reduce to the standard form. In hyperbolic equations with multiple characteristics the influence of the lower order terms must be taken into account and hence the form of these lower order terms is suitably restricted. In totally hyperbolic equations no restrictions are imposed on the lower order terms. Our definition of hyperbolicity is therefore such that in the case of distinct characteristics no restrictions are imposed on the lower order terms. In [2] Gairding has given a criterion for hyperbolicity of equations in several variables with multiple characteristics and constant coefficients. A. Lax [4] extended this criterion, in the case of two independent variables, to include variable coefficients. We shall not be concerned here with

from a unified point of view, namely, the extensive use of Green's function. Our main interest is concerned with the first mixed boundary problem (for definition, see ?4) for f linear or nonlinear in u, Harnack theorems, etc. Most of our methods are known. Green's function for the heat equation in rectangular domains was constructed already by E. E. Levi [21] (in particular ?7) in 1907 and was used by him to derive some existence theorems. An extensive use of Green's function for more general parabolic equations was made by Gevrey in his fundamental paper [16] (in particular ??4, 4*, 24, 28, 39, 40, 41). The analogue of the Harnack convergence theorem was first proved for the heat equation in Levi's paper [21, pp. 386-387]. Nonlinear existence problems were considered by Gevrey [16, ?28] by reducing them with the aid of Green's function to integral equations and then applying successive approximations. A more detailed survey on the older literature concerned with Green's function is to be found in the book of Ascoli-BurgattiGiraud [I]. In this paper we use all the above mentioned methods and a few new ones to treat more general problems than those considered in earlier papers. Essential use of Dressel's fundamental solutions for general linear second order parabolic equations [10; 11] enables us to perform this extension. We give below a brief description of our results and their connection to previous papers. In ?1 we construct Green's function for linear second order parabolic equations with smooth coefficients in an (n+1)-dimensional rectangle. In this construction we employ the fundamental solutions constructed by Dressel [11]. Green's function for the heat equation in one dimension was constructed by Levi [21] by reflecting the fundamental solution t-12 exp(_-.2/4t) with respect to the x-variable; our method is an extension of Levi's method.

This article presents a new reliable solver based on polynomial approximation, using the Euler polynomials to construct the approximate solutions of the second-order linear hyperbolic partial differential equations with two variables and constant coefficients. Also, a formula expressing explicitly the Euler expansion coefficients of a function with one or two variables is proved. Another explicit formula, which expresses the two dimensional Euler operational matrix of differentiation is also given. Application of these formulae for reducing the problem to a system of linear algebraic equations with the unknown Euler coefficients, is explained. Hence, the result system can be solved and the unknown Euler coefficients can be found approximately. Illustrative examples with comparisons are given to confirm the reliability of the proposed method. The results show the efficiency and accuracy of the present work.

STUDY OF HYPERBOLIC SYSTEM OF EQUATIONS TO STUDY TRAVEL TIME FOR TRAFFIC MODELLING by Anuj Nayyar 〈Dr. Hongtao Yang〉, Examination Committee Chair Associate Professor of Mathematical Sciences University of Nevada, Las Vegas In this thesis we study the travel time problem based on the known traffic density model. Using the conservation law, we model the travel time function by a boundary value problem of a non homogeneous linear hyperbolic equation. The equation is transformed into an initial value hyperbolic equation, and the well-posedness of the problem is discussed. The mathematical analysis for both density and travel problems are given. We also derive the analytic solutions for several special cases of traffic density. Numerical schemes are proposed for solving for travel time problem. Several numerical examples are presented and error analysis on the solutions obtained is performed to illustrate the rates of convergence of the numerical schemes.

In the case of oscillatory potential, we present some new Lyapunov-type inequalities for linear hyperbolic and elliptic equations on a rectangular domain in ${\mathbb R}^2$. No sign restriction is imposed on the potential function. As applications of the Lyapunov-type inequalities obtained, we give some estimations for disconjugacy of hyperbolic and elliptic Dirichlet boundary value problems.

In this paper, the mathematical formulation for a quadratic optimal control problem governed by a linear hyperbolic integro-differential equation is established. We first show the existence and regularity for the solution of the optimal control problem. The finite element approximation is based on the optimality conditions, which are also derived. Then the a priori error estimates for its finite element approximation are obtained with the optimal convergence order. Furthermore some numerical tests are presented to verify the theoretical results.

In this paper, we consider the controllability and the observability of a networked system of linear hyperbolic partial differential equations with coupled boundary conditions. Using the method of characteristics, we characterize them by the controllability/observability of a discrete-time system which is defined on the boundaries of the PDE system, has low-dimension, and hence the analysis is easy for. We also show the effectiveness of our approach using an simple example.

Fast solution of implicit methods for linear hyperbolic equations

In a previous paper the inverse problem associated with a hyperbolic dispersive partial differential equation with smooth coefficients was considered. The inverse problem (the determination of the coefficients) was formulated in terms of a dual set of integral equations involving measurable quantities, the kernels of the transmission, and reflection operators. These equations contained an unknown parameter which occurs in a linear manner. A better approach to determine this parameter is presented here. It involves an auxiliary equation, which is used to eliminate the unknown parameter from the integral equations. It is shown that the resulting system has a unique solution for a certain class of scattering problems. These uniqueness results are then strengthened when an additional equation is employed to reduce the dual set of integral equations to a single integral equation.

In Huang and Cushman (1981) a high order accurate nonstandard finite element technique was presented for deriving finite difference schemes of ktk-order accuracy for the linear in- viscid Burger’s equation and a second order linear hyperbolic equation. The schemes however were not presented in closed form and as we will see for linear equations the schemes are similar to characteristic schemes. The technique when used on nonlinear equations (e.g. the shallow water equations) requires the inversion of a matrix at each node (the dimension of the matrix increases with increasing order of interpolation). This inversion operation can be a very time-consuming process. We thus propose to present a closed form equation to eliminate the necessity of inverting a matrix at each node.

A sequence of optimal control problems for systems governed by linear hyperbolic equations with the nonhomogeneous Neumann boundary conditions is considered. The integral cost functionals and the differential operators in the equations depend on the param

We consider boundary value problems for the fourth-order linear equation \[ {A^{ijrs}}{u_{ijrs}} + {A^{ijr}}{u_{ijr}} + {A^{ij}}{u_{ij}} - \gamma {({a^{ij}}{u_i})_j} + {A^i}{u_i} + Fu = f\quad {\text {in}}\overline \Omega \] with smooth coefficients. The fourth-order part may degenerate on arbitrary subsets of $\overline \Omega$ i.e., ${A^{ijrs}}(x){m_{ij}}{m_{rs}} \geq 0$ for all $n \times n$ matrices $M$, with no restriction on where equality occurs. We assume the ${a^{ij}}$ part of the operator is uniformly elliptic (of second order) on $\Omega$ while $\gamma$ is a parameter allowing us to increase modulus of ellipticity as much as needed. As in Fichera’s second-order elliptic-parabolic equations [see, for example,

In this paper, we study a new numerical method of order 4 in space and time based on half-step discretization for the solution of three-space dimensional quasi-linear hyperbolic equation of the form [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] subject to prescribed appropriate initial and Dirichlet boundary conditions. The scheme is compact and requires nine evaluations of the function [Formula: see text] For the derivation of the method, we use two inner half-step points each in [Formula: see text]-, [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to three-dimensional (3D) hyperbolic equations with singular coefficients, which is main attraction of our work. We do not require extra grid points for computation. The proposed method when applied to 3D damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve 3D linear hyperbolic equations. Few benchmark problems are solved and numerical results are provided to support the theory which is discussed in this paper.

Radial basis function (RBF) methods have been actively developed in the last decades. RBF methods are global methods which do not require the use of specialized points and that yield high order accuracy if the function is smooth enough. Like other global approximations, the accuracy of RBF approximations of discontinuous problems deteriorates due to the Gibbs phenomenon, even as more points are added. In this paper we show that it is possible to remove the Gibbs phenomenon from RBF approximations of discontinuous functions as well as from RBF solutions of some hyperbolic partial differential equations. Although the theory for the resolution of the Gibbs phenomenon by reprojection in Gegenbauer polynomials relies on the orthogonality of the basis functions, and the RBF basis is not orthogonal, we observe that the Gegenbauer polynomials recover high order convergence from the RBF approximations of discontinuous problems in a variety of numerical examples including the linear and nonlinear hyperbolic partial differential equations. Our numerical examples using multi-quadric RBFs suggest that the Gegenbauer polynomials are Gibbs complementary to the RBF multi-quadric basis.