Fast solution of implicit methods for linear hyperbolic equations

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.

A uniform grid of step size h is superimposed on the space variable x in the first order hyperbolic partial differential equation . The space derivative is approximated by its central difference replacement and the resulting linear system of first order ordinary differential equations is solved employing Pade approximants to the exponential function. A number of stable, two-level difference schemes for solving the hyperbolic equation are thus developed and each is extrapolated to give higher order accuracy. Some of these schemes are unconventional in that they use more than three mesh points at one or both time levels. The resulting increase in computer time is, however, negligible and the improvement in accuracy justifies their use.

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

In this paper, we study the superconvergence properties of the energy-conserving discontinuous Galerkin (DG) method in [18] for one-dimensional linear hyperbolic equations. We prove the approximate solution superconverges to a particular projection of the exact solution. The order of this superconvergence is proved to be $$k+2$$ when piecewise $$\mathbb {P}^k$$ polynomials with $$k \ge 1$$ are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise $$\mathbb {P}^k$$ polynomials with arbitrary $$k \ge 1$$ . Furthermore, we find that the derivative and function value approximations of the DG solution are superconvergent at a class of special points, with an order of $$k+1$$ and $$k+2$$ , respectively. We also prove, under suitable choice of initial discretization, a ( $$2k+1$$ )-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages. Numerical experiments are given to demonstrate these theoretical results.

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.

Bound-preserving neural network method for linear hyperbolic and parabolic equations

This article introduces the hybridizable discontinuous Galerkin (HDG) approach to numerically approximate the solution of a linear hyperbolic integro-differential equation. A priori error estimates for semi-discrete and fully discrete schemes are developed. It is shown that the optimal order of convergence is achieved for both scalar and flux variables. To achieve that, an intermediate projection is introduced for the semi-discrete error analysis, and it also shows that this projection achieves convergence of order $h^{k+3/2}$ for $k ≥ 1.$ Next, superconvergence is achieved for the scalar variable using element-by-element post-processing. For the fully discrete error analysis, the central difference scheme and the mid-point rule approximate the derivative and the integral term, respectively. Hence, the second order of convergence is achieved in the temporal direction. Finally, numerical experiments have been performed to validate the theory developed in this article.

In this work we study exact boundary controllability for a class of hyperbolic linear partial differential equation with constant coefficient which includes the linear Klein-Gordon equation. We consider piecewise smooth domains on the plane, initial state with finite energy and control of Robin type, acting on the whole boundary or only on a part of it.

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.

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.

Solving the tridiagonal linear system and Poisson's equation over a LAN-based computing platform using distributed shared memory

Для дифференциального уравнения гиперболического типа порядка $n$ с некратными характеристиками рассмотрена задача Коши. Приводятся полученные авторами ранее решения задачи Коши для гиперболических уравнений третьего и четвертого порядков с некратными характеристиками в явном виде, аналогичном формуле Даламбера. Получено решение задачи Коши для уравнения гиперболического типа порядка $n$ общего вида. Найденное решение также является аналогом формулы Даламбера. Сформулирована теорема о существовании и единственности регулярного решения задачи Коши для гиперболического уравнения порядка $n$ общего вида с некратными характеристиками.

We consider a non-local boundary value problem for the linear third order equation with hyperbolic operator in the main part. Sufficient conditions were stated to coefficients of the equation and to given functions in order that this non-local boundary value problem has a unique solution. For the proof, we use the Riemann's method.