Linear Hyperbolic Equations Research Articles

The Gradient Discretisation Method for Linear Advection Problems

Abstract We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped ℙ 1 {\mathbb{P}_{1}} conforming and non-conforming finite element and on the hybrid finite volume method.

Adjoint computational methods for 2D inverse design of linear transport equations on unstructured grids

We address the problem of inverse design of linear hyperbolic transport equations in 2D heterogeneous media. We develop numerical algorithms based on gradient-adjoint methodologies on unstructured grids. While the flow equation is compulsorily solved by means of a second order upwind scheme so to guarantee sufficient accuracy, the necessity of using the same order of approximation when solving the sensitivity or adjoint equation is examined. Two test cases, including Doswell frontogenesis, are analysed. We show the convenience of using a low order method for the adjoint resolution, both in terms of accuracy and efficiency. An analytical explanation for this fact is also provided in the sense that, when employing higher order schemes for the adjoint problem, spurious high frequency numerical components slow down the convergence process.

Wave models and dynamical analysis of evolutionary algorithms

By drawing an analogy between the population of an evolutionary algorithm and a gas system (which we call a particle system), we first build wave models of evolutionary algorithms based on aerodynamics theory. Then, we solve the models linear and quasi-linear hyperbolic equations analytically, yielding wave solutions. These describe the propagation of the particle density wave, which is composed of leftward and rightward waves. We demonstrate the convergence of evolutionary algorithms by analyzing the mechanism underlying the leftward wave, and investigate population diversity by analyzing the rightward wave. To confirm these theoretical results, we conduct experiments that apply three typical evolutionary algorithms to common benchmark problems, showing that the experimental and theoretical results agree. These theoretical and experimental analyses also provide several new clues and ideas that may assist in the design and improvement of evolutionary algorithms.

Stability Analysis for a Class of Linear $2\times 2$ Hyperbolic PDEs Using a Backstepping Transform

In this paper, we develop a sufficient stability condition for a class of coupled first-order linear hyperbolic partial differential equations (PDEs) with constant coefficients that appear when considering target systems for backstepping boundary control.Using a backstepping transform, the problem is reformulated as a stability problem for a difference equation with distributed delay. Finding the explicit solution to the backstepping kernels, we derive an explicit sufficient condition depending on the plant coefficients. This stability condition is compared to an existing stability result based on a Lyapunov analysis. Both the proposed and existing sufficient conditions are then contrasted in some examples to a (computationally expensive) numerical approximation of a necessary and sufficient condition for exponential stability to illustrate their conservatism.

Numerical Solution for Semi Linear Hyperbolic Differential Equations

Numerical Solution for Semi Linear Hyperbolic Differential Equations

Fault-tolerant control for a class of switched linear hyperbolic systems

ABSTRACT This paper considers the fault-tolerant control (FTC) problem for switched systems described by linear hyperbolic partial differential equations with the process and boundary faults. The effect of faults is compensated by designing the desirable boundary control gain. Both the accumulative and dissipative characteristics of faults are considered, respectively. Several Lyapunov-based FTC schemes that depend on the system parameters and boundary conditions are proposed which guarantee the exponential stability of the non-switched hyperbolic systems. The new result is further extended to the switched linear hyperbolic systems where the boundary control gain of each mode and the switching law are designed comprehensively. It shows that the FTC goal can be achieved even if only some faulty modes are stabilizable. An air traffic flow example is taken to illustrate the theoretical results.

Fault-tolerant control for a class of linear interconnected hyperbolic systems by boundary feedback

This paper considers a fault-tolerant control problem for a class of interconnected linear hyperbolic partial differential equation systems. Both subsystem faults and coupling faults are considered. Firstly, the well-posedness of the faulty system is analyzed by using semigroup theory. Secondly, for the fault-free case, a stabilizing boundary feedback control based on small-gain theorem is proposed. Consequently, in the presence of faults, fault recoverability conditions are established that maintain the stability of the faulty systems. The fault-tolerant control strategies are also provided. A heat exchanger example is taken to illustrate the effectiveness and practicality of the proposed methods.

A truly forward semi-Lagrangian WENO scheme for the Vlasov-Poisson system

A class of modular high order forward semi-Lagrangian (FSL) schemes with a number of advantages (outlined below) for linear advection equations is extended to have high resolution, in the sense that sharp gradients can be captured without loss of accuracy, by means of weighted essentially non-oscillatory derivative (DWENO) calculations based on finite differences. Principally, our approach (the convected scheme, CS) differs from conventional high order FSL schemes in the means by which it reaches higher order accuracy. The high order CS progressed in this work reserves use of a low order histogram representation of the solution and a mass-conservative volume-weighted projection operator which ordinarily produces a second order accurate solution. A higher order version is obtained by compensating for the amount by which the CS solution and the exact solution disagree at every order in a Taylor series sense. Instead of including these error terms algebraically, which changes the projection operator itself, the novel idea of the CS is to incorporate this information geometrically as a commensurate adjustment to the volume-weighted proportion assigned to each cell. We leverage DWENO approximations to compute these Taylor series error terms more sensitively, and show the corresponding WENO convected schemes (WENO-CS) achieve the desired dual high order and high resolution behavior. Fifth, seventh, and ninth order WENO-CS base solvers are presented for the 1D constant speed advection equation prototype, and are employed in a Strang splitting approach for the numerical solution of linear and nonlinear two-dimensional hyperbolic equations. Results are showcased for 2D linear transport and rigid body rotation, as well as for classic benchmark problems for the (1+1)-dimensional Vlasov–Poisson system including Landau damping, and the two-stream instability. The schemes enjoy the usual benefits of semi-Lagrangian schemes (e.g. no CFL time step restriction) while further claiming distinct advantages afforded to forward schemes including automatic mass conservation, compact support, and therefore straightforward parallelization.

Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations

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.

Multi-dimensional periodic problems for higher-order linear hyperbolic equations

Abstract For higher-order linear hyperbolic equations the problem on periodic solutions is investigated. The concepts of associated problems, and α-well-posedness are introduced. Necessary and sufficient conditions of well-posedness in the two-dimensional case, as well as unimprovable sufficient conditions of well-posedness and α-well-posedness in the multi-dimensional case are established.

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Discontinuous Galerkin Solutions over Nonuniform Meshes: Superconvergence and Optimal Accuracy

Smoothness-increasing accuracy-conserving (SIAC) filtering is an area of increasing interest because it can extract the “hidden accuracy” in discontinuous Galerkin (DG) solutions. It has been shown that by applying a SIAC filter to a DG solution, the accuracy order of the DG solution improves from order k+1 to order 2k+1 for linear hyperbolic equations over uniform meshes. However, applying a SIAC filter over nonuniform meshes is difficult, and the quality of filtered solutions is usually unsatisfactory applied to approximations defined on nonuniform meshes. The applicability to such approximations over nonuniform meshes is the biggest obstacle to the development of a SIAC filter. The purpose of this paper is twofold: to study the connection between the error of the filtered solution and the nonuniform mesh and to develop a filter scaling that approximates the optimal error reduction. First, through analyzing the error estimates for SIAC filtering, we computationally establish for the first time a relation between the filtered solutions and the unstructuredness of nonuniform meshes. Further, we demonstrate that there exists an optimal accuracy of the filtered solution for a given nonuniform mesh and that it is possible to obtain this optimal accuracy by the method we propose, an optimal filter scaling. By applying the newly designed filter scaling over nonuniform meshes, the filtered solution has demonstrated improvement in accuracy order as well as reducing the error compared to the original DG solution. Finally, we apply the proposed methods over a large number of nonuniform meshes and compare the performance with existing methods to demonstrate the superiority of our method.

Error estimates and post-processing of local discontinuous Galerkin method for Schrödinger equations

In this paper, we present L 2 and negative-order norm estimates for the local discontinuous Galerkin (LDG) method solving variable coefficient Schrödinger equations. For these special solutions the LDG method exhibits “hidden accuracy”, and we are able to extract it through the use of a convolution kernel that is composed of a linear combination of B-splines. This technical was initially introduced by Cockburn, Luskin, Shu, and Süli for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving (SIAC) filter. We demonstrate that it is possible to extend the SIAC filter on Schrödinger equations. When polynomials of degree k are used, we can prove theoretically the LDG method solutions are of order k + 1 , whereas the post-processed solutions that convolution with the SIAC filter are of order at least 2 k . Additionally, we present numerical results to confirm that the accuracy of LDG solutions can be improved from O ( h k + 1 ) to at least O ( h 2 k + 1 ) for Schrödinger equations by using alternating numerical fluxes.

Superconvergence of Arbitrary Lagrangian--Eulerian Discontinuous Galerkin Methods for Linear Hyperbolic Equations

In this paper, we study the superconvergence properties of an arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) method with approximations to one-dimensional linear hyperbolic equations. The ALE-DG method is a mesh moving method; we need to deal with the new difficulties brought by the time-dependent test function space and grid velocity field. Since the time derivative cannot commute with the space projections for the ALE-DG method, we will introduce the material derivative in our analysis. With the help of the scaling argument and material derivative, we build a special interpolation function by constructing the correction functions and prove that the numerical solution is superclose to the interpolation function in the $L^2$-norm. The order of the superconvergence is 2k+1 when piecewise polynomials of degree at most $k$ are used. We also rigorously prove a (2k+1)th-order superconvergence rate for the domain and cell average and at the downwind points in the maximal and average norm. Furthermore, we prove that the function value approximation is superconvergent with a rate $k+2$ at the right Radau points and a superconvergence rate $k+1$ for the derivative approximation at all interior left Radau points. All theoretical findings are confirmed by numerical experiments.

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In the paper we consider the following Cauchy problem for the systems of the ordinary dierential equations { ψ′′(t) + K(t,ψ(t),ψ′(t)) = S(t), t ∈ (0,T), ψ(0) = ψ0, ψ′(0) = ψ1, (∗) where K and S are some vector-valued functions, ψ0,ψ1 ∈ Rd are some xed vectors. Such systems appear in various classical mechanics problems because they follow from Newton's second law of the dynamics. If we use the Faedo-Galerkin method for the seeking of the weak solution to the initial boundary value problem for the linear or nonlinear hyperbolic equations, then we also use the problems of type (*) for the construction of the approximation functions. In practice, the Cauchy problems of type (*) often are reduced to the following Cauchy problem of the higher dimension: ψ′(t) = θ(t), t ∈ (0,T), θ′(t) = S(t)−K(t,ψ(t),θ(t)), t ∈ (0,T), ψ(0) = ψ0, θ(0) = ψ1. (∗∗) Then the standard Peano/Caratheodory types existence theorems are used. But these are local theorems and they should be complemented by some extension theorems. On the other hand we can seek the global solutions of problems (*) or (**) directly. In the present paper we and the conditions of the global solvability of the Cauchy problem (*). Corresponding results for problem (**) we got in our previous paper. These conditions contain the Caratheodory existence conditions and the Lasalle extension conditions for (*). Finally, we use these facts to proving the existence theorem for the initial boundary value problem for some nonlinear hyperbolic equations with the variable exponent of the nonlinearity.

Water Hammer Stability in a Hydroelectric Plant with Surge Tank and Throttling

There is considered the dynamics of a hydroelectric power plant with two conduits - the tunnel and the hydraulic turbine penstock - and a surge tank with throttling. The water hammer phenomenon is considered under the assumption that the hydraulic turbine is shut down. Under the standard assumptions of water hammer computations, both the dynamic head and the Darcy-Weisbach losses are neglected. The consequence of all aforementioned assumptions is a model described by linear hyperbolic partial differential equations with linear non-standard boundary conditions. Further, considering the Riemann invariants of the problem along the characteristics, a system of functional differential equations with deviated argument of neutral type is associated to the basic model. This system displays two rationally independent delays. Its stability is studied using the Lyapunov functional approach and taking into account the dissipativeness of the boundary conditions.

Feedback control of bilinear distributed parameter system by input-output linearisation

In this paper, a control law that enforces the tracking of a boundary controlled output for a bilinear distributed parameter system is developed in the framework of geometric control. The dynamic behaviour of the system is described by two weakly coupled linear hyperbolic partial differential equations. The stability of the resulting closed-loop system is investigated based on eigenvalues of the spatial operator of a weakly coupled system of balance equations. It is shown that, under some reasonable assumptions, the stability condition is related to the choice of the tuning parameter of the control law. The performance of the developed control law is demonstrated, through numerical simulation, in the case of a co-current heat exchanger. The control objective is to control the outlet cold fluid temperature by manipulating its velocity. Both tracking and disturbance rejection problems are considered.

On the Turnpike Phenomenon for Optimal Boundary Control Problems with Hyperbolic Systems

We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over the finite time interval $(0,\, T)$ that depends on the boundary traces of the solution. If the time horizon $T$ is sufficiently large, the solution of the dynamic optimal boundary control problem can be approximated by the solution of a steady state optimization problem. We show that for $T\rightarrow\infty$ the approximation error converges to zero in the sense of the norm in $L^2(0,\,1)$ with the rate $1/T$, if the time interval $(0,\, T)$ is transformed to the fixed interval $(0,\, 1)$. Moreover, we show that also for optimal boundary control problems with integer constraints for the controls the turnpike phenomenon occurs. In this case the steady state optimization problem also has the integer constraints. If $T$ is sufficiently large, the integer part of each solution of the dynamic optimal boundary control problem with integer constraints is equal to the integer part of a solution of the static problem. A numerical verification is given for a control problem in gas pipeline operations.

Estimation of Equation Coefficients of Free and Forced String Vibrations in Continuous Medium with Consideration of Dynamic Noise and Measurement Noise

Abstract Linear hyperbolic differential equation of partial derivatives, that sets string vibration, as well as other oscillatory processes in continuous medium and electrodynamics, is also used in many areas of theoretical physics. This article considers estimation problems of equation coefficients of free and forced string vibrations with fixed ends with consideration of additive dynamic and measurement noises. Use of a finite-difference method allowed solving partial differential equations with consideration of additive dynamic and measurement noise by solving systems of linear stationary stochastic, finite-difference and algebraic equations, expressed as state-space models. State-space model representation of string vibration equation makes it possible to use Kalman filter algorithm to make accurate estimates of string vibrations’ coordinates.

The L$^2$-norm Stability Analysis of Runge--Kutta Discontinuous Galerkin Methods for Linear Hyperbolic Equations

In this paper we propose a simple and unified framework to investigate the L$^2$-norm stability of the explicit Runge--Kutta discontinuous Galerkin (RKDG) methods when solving the linear constant-c...

On the Riemann Function

Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in two variables. The first review of Riemann’s method was published by E.T. Copson in 1958. This study extends that work. Firstly, three solution methods were overlooked in Copson’s original paper. Secondly, several new approaches for finding Riemann functions have been developed since 1958. Those techniques are included here and placed in the context of Copson’s original study. There are also numerous equivalences between Riemann functions that have not previously been identified in the literature. Those links are clarified here by showing that many known Riemann functions are often equivalent due to the governing equation admitting a symmetry algebra isomorphic to S L ( 2 , R ) . Alternatively, the equation admits a Lie-Bäcklund symmetry algebra. Combining the results from several methods, a new class of Riemann functions is then derived which admits no symmetries whatsoever.