Linear Hyperbolic PDEs Research Articles

Many-Stage Optimal Stabilized Runge–Kutta Methods for Hyperbolic Partial Differential Equations

A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge–Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach allows the optimization of stability polynomials with more than hundred stages. A potential application of these high degree stability polynomials are problems with locally varying characteristic speeds as found for non-uniformly refined meshes and spatially varying wave speeds. To demonstrate the applicability of the stability polynomials we construct 2N-storage many-stage Runge–Kutta methods that match their designed second order of accuracy when applied to a range of linear and nonlinear hyperbolic PDEs with smooth solutions. These methods are constructed to reduce the amplification of round off errors which becomes a significant concern for these many-stage methods.

Composite Learning Based Adaptive Control of Linear 2×2 Hyperbolic PDE Systems.

This article considers the adaptive stability control of a class of 2×2 linear hyperbolic PDE systems. The PDE model is subject to constant but in-domain and boundary unknown parameters. A novel adaptive controller is developed by leveraging the swapping design technique and composite parameter learning law. With swapping design, several linear and static combinations, including carefully designed filters, unknown parameters, and error terms, are constructed to express the system states. From the static combinations, a composite learning based forgetting-factor least squares law is introduced to guarantee exponential parameter convergence without the persistent excitation (PE). Although inaccurate parameter estimation in the adaptive backstepping control results in asymptotic stability of the system, accurate parameter estimation ensures the exponential convergence of closed-loop system and concomitantly improves the transient performance. Finally, a comparative numerical simulation is performed to validate the effectiveness and advantage of the developed adaptive control scheme.

Tent-pitcher spacetime discontinuous Galerkin method for one-dimensional linear hyperbolic and parabolic PDEs

Tent-pitcher spacetime discontinuous Galerkin method for one-dimensional linear hyperbolic and parabolic PDEs

Regularity of time-periodic solutions to autonomous semilinear hyperbolic PDEs

This paper concerns autonomous boundary value problems for 1D semilinear hyperbolic PDEs. For time-periodic classical solutions, which satisfy a certain non-resonance condition, we show the following: If the PDEs are continuous with respect to the space variable x and C∞-smooth with respect to the unknown function u, then the solution is C∞-smooth with respect to the time variable t, and if the PDEs are C∞-smooth with respect to x and u, then the solution is C∞-smooth with respect to t and x. The same is true for appropriate weak solutions. Moreover, we show examples of time-periodic functions, which do not satisfy the non-resonance condition, such that they are weak, but not classical solutions, and such that they are classical solutions, but not C∞-smooth, neither with respect to t nor with respect to x, even if the PDEs are C∞-smooth with respect to x and u. For the proofs we use Fredholm solvability properties of linear time-periodic hyperbolic PDEs and a result of E. N. Dancer about regularity of solutions to abstract equivariant equations.

DoD Stabilization of linear hyperbolic PDEs on general cut‐cell meshes

AbstractStandard numerical methods for hyperbolic PDEs require for stability a CFL‐condition which implies that the time step size depends on the size of the elements of the mesh. On cut‐cell meshes, elements can become arbitrarily small and thus the time step size cannot take the size of small cut‐cells into account but has to be chosen based on the background mesh elements.A remedy for this is the so called DoD (domain of dependence) stabilization for which several favorable theoretical and numerical properties have been shown in one and two space dimensions [4, 9]. Up to now the method is restricted to stabilization of cut‐cells with exactly one inflow and one outflow face, i.e. triangular cut‐cells with a no‐flow face (see [4]).We extend the DoD stabilization to cut‐cells with multiple in‐ and outflow faces by properly considering the flow distribution inside the cut‐cell. We further prove L2‐stability for the semi‐discrete formulation in space and present numerical results to validate the proposed extension.

A spatio-temporal fully meshless method for hyperbolic PDEs

We introduce a meshless method derived by considering the time variable as a spatial variable without the need to extend further conditions to the solution of linear and non-linear hyperbolic PDEs. The method is based on the moving least squares method, more precisely in the Generalized Finite Difference Method which allows us to select well-conditioned stars. Several 2D and 3D examples including the time variable are shown for both regular and irregular node distributions. The results are compared with explicit GFDM both in terms of errors and execution time.

Chaos of multi-dimensional linear hyperbolic PDEs

This paper deals with the dynamics of a system governed by a multi-dimensional linear hyperbolic PDE. The dynamical behaviors of linear PDEs extremely depend on the selection of space, and a conventional way is to define a infinite-dimensional space with a tuning parameter. Thereby, the linear PDEs can exhibit chaos or stability in the different range of tuning parameter. In this work, the chaos of the C 0 C_0 -semigroup corresponding to the system is established on the Banach space of multivariate analytic functions when the tuning parameter exceeds some given positive number. Based on this, both Devaney and distributional chaos of the system are further obtained. Meanwhile, the C 0 C_0 -semigroup is proved to be uniformly exponentially stable when the tuning parameter is less than a certain positive number, which contributes to showing the global stability of the system. Finally, two examples are given to illustrate effectiveness of our results.

Output regulation for general heterodirectional linear hyperbolic PDEs coupled with nonlinear ODEs

This paper considers general heterodirectional linear hyperbolic PDEs with boundary actuation and collocated measurement, that are bidirectionally coupled with nonlinear ODEs at the unactuated boundary. An output feedback regulator is designed to have the control output track a reference in the presence of disturbances, with a nonlinear signal model generating the reference and the disturbances. This leads to new challenges for both the observer and controller design. The regulator design makes use of results from output regulation theory for nonlinear lumped-parameter and distributed-parameter systems. The derivation of the state feedback regulator requires solving a new type of regulator equations, which consist of a Cauchy problem for linear and semilinear hyperbolic PDEs. A key component of the novel observer design is a nonlinear retarded observer for a finite-dimensional subsystem. Both the controller and the observer design are systematic in nature and derived in several successive steps, that include backstepping transformations and, most notably, predictions of PDE and ODE states. The latter are shown to be the unique solution of general nonlinear Volterra integro-differential equations. Combining the observer with the state feedback regulator is proven to achieve output regulation for the closed-loop system. A numerical example illustrates and confirms the theoretical results.

Output Feedback Synchronization for Networks of Hyperbolic PDE–ODE Systems

This paper considers the output feedback design for the leader-follower synchronization of a network of linear hyperbolic PDE–ODE systems and in particular ODE agents subject to distinct input and output delays. For a communication restricted by a directed graph, a cooperative state feedback is designed using a local backstepping stabilization of the PDE subsystem and a cooperative decoupling transformation. With this, the synchronization problem is traced back to a simultaneous stabilization of the boundary ODEs without introducing additional dynamics in the networked controller. For general PDE–ODE agents an observer using absolute measurements is proposed, while for ODE agents with delays the observer only requires relative measurements. The results of this paper are demonstrated for the output feedback synchronization of ODE agents with distinct input and output delays.

A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs

A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations and switched differential-algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modelled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions.

Hybrid boundary stabilization of linear first-order hyperbolic PDEs despite almost quantized measurements and control input

Abstract We develop a hybrid boundary feedback law for a class of scalar, linear, first-order hyperbolic PDEs, for which the state measurements or the control input are subject to quantization. The quantizers considered are Lipschitz functions, which can approximate arbitrarily closely typical piecewise constant, taking finitely many values, quantizers. The control design procedure relies on the combination of two ingredients—A nominal backstepping controller, for stabilization of the PDE system in the absence of quantization, and a switching strategy, which updates the parameters of the quantizer, for compensation of the quantization effect. Global asymptotic stability of the closed-loop system is established through utilization of Lyapunov-like arguments and derivation of solutions’ estimates, providing explicit estimates for the supremum norm of the PDE state, capitalizing on the relation of the resulting, nonlinear PDE system (in closed loop) to a certain, integral delay equation. A numerical example is also provided to illustrate, in simulation, the effectiveness of the developed design.

Modelling the ecology of phototrophic-heterotrophic biofilms

A mathematical model describing the microbial interactions in phototrophic-heterotrophic biofilms is presented. The main phenomena and factors involved in the model include: biomass growth and decay, substrates production, diffusion and consumption, biological invasion of planktonic species and detachment. In particular, non linear hyperbolic PDEs describe the growth of the microbial species while quasi-linear parabolic PDEs govern the dynamics of substrates and invading species. The whole system of PDEs is considered in a free boundary domain. The following syntrophic interactions are also modelled: the exchange of dissolved oxygen, organic carbon and inorganic carbon produced and released by phototrophs and heterotrophs, respectively. The positive effect of heterotrophic pioneers on the phototrophic growth is modelled by introducing a phototrophic colonization rate depending on the EPS fraction in the biofilm. Numerical simulations are performed to test model accuracy. Simulation results reproduce the main symbiotic mechanisms between phototrophs and heterotrophs reported in literature, such as the positive effect of heterotrophic pioneers and their EPS production on phototrophic growth and the effects of phototrophic organic carbon release on the invasion and growth of heterotrophic bacteria. Furthermore, model results highlight the role played by heterotrophic species under photoinhibition conditions, which provide a positive shading contribution to phototrophic growth. Light is confirmed as the most significant factor in the ecology of phototrophic-heterotrophic biofilms. Such results confirm the accuracy of the model that correctly predicts the evolution of a phototrophic-heterotrophic biofilm and the main phenomena involved, and can be seen as an auxiliary tool in different industrial applications, such as wastewater treatment and bioenergy production.

Analysis and Performance Evaluation of Adjoint-guided Adaptive Mesh Refinement for Linear Hyperbolic PDEs Using Clawpack

Adaptive mesh refinement (AMR) is often used when solving time-dependent partial differential equations using numerical methods. It enables time-varying regions of much higher resolution, which can selectively refine areas to track discontinuities in the solution. The open source Clawpack software implements block-structured AMR to refine around propagating waves in the AMRClaw package. For problems where the solution must be computed over a large domain but is only of interest in a small area, this approach often refines waves that will not impact the target area. We seek a method that enables the identification and refinement of only the waves that will influence the target area. Here we show that solving the time-dependent adjoint equation and using a suitable inner product allows for a more precise refinement of the relevant waves. We present the adjoint methodology in general and give details on the implementation of this method in AMRClaw. Examples and a computational performance analysis for linear acoustics equations are presented. The adjoint method is compared to AMR methods already available in AMRClaw, and the advantages and disadvantages are discussed. The approach presented here is implemented in Clawpack, in Version 5.6.1, and code for all examples presented is archived on Github.

Robust Control Design of Underactuated 2 × 2 PDE-ODE-PDE Systems

In this letter, we design a robust stabilizing controller for a system composed of two sets of linear heterodirectional hyperbolic PDEs, with actuation at one boundary of one of the PDEs, and couplings at the middle boundary with ODEs in a PDE-ODE-PDE configuration. The system is underactuated since only one of the PDE systems is actuated. The design approach employs a backstepping transformation to move the undesired system couplings to the proximal boundary (where the actuation is located). We can then express this target system as a time-delay neutral system for which we can design an appropriate control law to obtain an exponentially stable target system.

Stability analysis of hierarchical tensor methods for time-dependent PDEs

We address the question of whether it is possible to stably integrate time-dependent high-dimensional linear PDEs with hierarchical tensor methods and explicit time stepping schemes. To this end, we develop sufficient conditions for stability and convergence of tensor solutions evolving on tensor manifolds with constant rank. We also argue that the applicability of PDE solvers with explicit time-stepping may be limited by time-step restriction dependent on the dimension of the problem. Numerical applications are presented and discussed for variable coefficients linear hyperbolic and parabolic PDEs.

A space-time discontinuous Galerkin method for linear hyperbolic PDE's with high frequencies

The main purpose of this paper is to describe a space-time discontinuous Galerin (DG) method based on an extended space-time approximation space for the linear first order hyperbolic equation that contains a high frequency component. We extend the space-time DG spaces of tensor-product of polynomials by adding trigonometric functions in space and time that capture the oscillatory behavior of the solution. We construct the method by combining the basic framework of the space-time DG method with the extended finite element method. The basic principle of the method is integrating the features of the partial differential equation with the standard space-time spaces in the approximation. We present error analysis of the space-time DG method for the linear first order hyperbolic problems. We show that the new space-time DG approximation has an improvement in the convergence compared to the space-time DG schemes with tensor-product polynomials. Numerical examples verify the theoretical findings and demonstrate the effects of the proposed method.

Chaotic dynamics of linear hyperbolic PDEs with nonlinear boundary conditions

Abstract The study of chaotic PDEs with variable coefficients involves higher level of complexity than for the ones with constant coefficients. In this paper, we investigate the chaotic oscillations of a one-dimensional second order linear hyperbolic PDE with variable coefficients that is factorizable as a product of two noncommutative first order operators and the boundary conditions at both ends of the PDE are general nonlinear. Numerical simulations are provided to illustrate the effectiveness of our theoretical results.

Two-Sided Boundary Stabilization of Heterodirectional Linear Coupled Hyperbolic PDEs

We solve the problem of stabilizing a general class of linear first-order hyperbolic systems using actuation at both boundaries of the spatial domain. We design a novel Fredholm transformation similarly to backstepping approaches to derive a boundary controller and a boundary observer enabling stabilization by output feedback. This yields an explicit full-state feedback law that achieves the theoretical lower bound for convergence to zero.

Weak vs. 𝒟-solutions to linear hyperbolic first-order systems with constant coefficients

We establish a consistency result by comparing two independent notions of generalized solutions to a large class of linear hyperbolic first-order PDE systems with constant coefficients, showing that they eventually coincide. The first is the usual notion of weak solutions defined via duality. The second is the new notion of [Formula: see text]-solutions which we recently introduced and arose in connection to the vectorial calculus of variations in [Formula: see text] and fully nonlinear elliptic systems. This new approach is a duality-free alternative to distributions and is based on the probabilistic representation of limits of difference quotients.

Control of a 2 × 2 coupled linear hyperbolic system sandwiched between 2 ODEs

SummaryMotivated by an engineering application in cable mining elevators, we address a new problem on stabilization of 2×2 coupled linear first‐order hyperbolic PDEs sandwiched between 2 ODEs. A novel methology combining PDE backstepping and ODE backstepping is proposed to derive a state‐feedback controller without high differential terms. The well‐posedness and invertibility properties of the PDE backstepping transformation are proved. All states, including coupled linear hyperbolic PDEs and 2 ODEs, are included in the closed‐loop exponential stability analysis. Moreover, boundedness and exponential convergence of the designed controller are proved. The performance is investigated via numerical simulation.