scholarly journals Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jacek Banasiak ◽  
Adam Błoch
Keyword(s):  
Boundary Conditions ◽  
Adjacency Matrix ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
General Linear ◽  
First Order ◽  
Metric Graph ◽  
Potential Map ◽  
The Matrix ◽  
Network Language

<p style='text-indent:20px;'>Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port-Hamiltonian with general linear Kirchhoff's boundary conditions can be written as a system of <inline-formula><tex-math id="M1">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> hyperbolic equations on a metric graph <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of <inline-formula><tex-math id="M3">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of <inline-formula><tex-math id="M4">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Recent Advances on Boundary Conditions for Equations in Nonequilibrium Thermodynamics

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1710
Author(s):  
Wen-An Yong ◽  
Yizhou Zhou
Keyword(s):  
Boundary Conditions ◽  
Structural Stability ◽  
Stability Condition ◽  
Hyperbolic Systems ◽  
Moment Closure ◽  
First Order ◽  
Nonequilibrium Phenomena ◽  
Classical Models ◽  
Spatial Domains ◽  
Kreiss Condition

This paper is concerned with modeling nonequilibrium phenomena in spatial domains with boundaries. The resultant models consist of hyperbolic systems of first-order partial differential equations with boundary conditions (BCs). Taking a linearized moment closure system as an example, we show that the structural stability condition and the uniform Kreiss condition do not automatically guarantee the compatibility of the models with the corresponding classical models. This motivated the generalized Kreiss condition (GKC)—a strengthened version of the uniform Kreiss condition. Under the GKC and the structural stability condition, we show how to derive the reduced BCs for the equilibrium systems as the classical models. For linearized problems, the validity of the reduced BCs can be rigorously verified. Furthermore, we use a simple example to show how thus far developed theory can be used to construct proper BCs for equations modeling nonequilibrium phenomena in spatial domains with boundaries.

scholarly journals Linear Hyperbolic Systems on Networks: well-posedness and qualitative properties

2020 ◽  
Author(s):  
Marjeta Kramar ◽  
Delio Mugnolo ◽  
Serge Nicaise
Keyword(s):  
Boundary Conditions ◽  
Evolution Equations ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Order Reduction ◽  
Well Posedness ◽  
Qualitative Properties ◽  
Local Boundary ◽  
First Order ◽  
Necessary And Sufficient

We study hyperbolic systems of one - dimensional partial differential equations under general , possibly non-local boundary conditions. A large class of evolution equations, either on individual 1- dimensional intervals or on general networks , can be reformulated in our rather flexible formalism , which generalizes the classical technique of first - order reduction . We study forward and backward well - posedness ; furthermore , we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first - order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied . p, li { white-space: pre-wrap; }

ON BOUNDARY CONDITIONS FOR FIRST-ORDER SYMMETRIC HYPERBOLIC SYSTEMS WITH CONSTRAINTS

2013 ◽  
Vol 10 (04) ◽  
pp. 725-734 ◽  
Author(s):  
NICOLAE TARFULEA
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Hyperbolic Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
First Order ◽  
The Cauchy Problem ◽  
Initial Boundary Value ◽  
Well Posed

The Cauchy problem for many first-order symmetric hyperbolic (FOSH) systems is constraint preserving, i.e. the solution satisfies certain spatial differential constraints whenever the initial data does. Frequently, artificial space cut-offs are performed for such evolution systems, usually out of the necessity for finite computational domains. However, it may easily happen that boundary conditions at the artificial boundary for such a system lead to an initial boundary value problem which, while well-posed, does not preserve the constraints. Here we consider the problem of finding constraint-preserving boundary conditions for constrained FOSH systems in the well-posed class of maximal non-negative boundary conditions. Based on a characterization of maximal non-negative boundary conditions, we discuss a systematic technique for finding such boundary conditions that preserve the constraints, pending that the constraints satisfy a FOSH system themselves. We exemplify this technique by analyzing a system of wave equations in a first-order formulation subject to divergence constraints.

A numerical method of solving second-order linear differential equations with two-point boundary conditions

1957 ◽  
Vol 53 (2) ◽  
pp. 442-447 ◽  
Author(s):  
E. Cicely Ridley
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Riccati Equation ◽  
Direct Method ◽  
Linear Equations ◽  
Factorization Method ◽  
Order Linear ◽  
First Order ◽  
The Matrix ◽  
Linear Differential

ABSTRACTA direct method of integrating the equation y″ + g(x) y = h(x), with the two-point linear boundary conditions y′(a) + αy(a) = A, y′(b) + βy(b) = B, is based on the factorization of the equation into two first-order linear equations v′ − sv = h and y′ + sy = v, where s is a solution of the Riccati equation s′ − s2 = g. The first-order equations for v and y are integrated in succession, one in the direction of x increasing, and one in the direction of x decreasing, one boundary condition being used in each of these integrations. The appropriate solution of the Riccati equation is determined by the boundary condition at the end of the range from which the integration of the equation for v is started. The process is compared with the matrix factorization method of Thomas and Fox, and its stability discussed.

A Note on the One-Side Exact Boundary Observability for Quasilinear Hyperbolic Systems

2008 ◽  
Vol 15 (3) ◽  
pp. 571-580
Author(s):  
Tatsien Li ◽  
Bopeng Rao ◽  
Zhiqiang Wang
Keyword(s):  
Boundary Conditions ◽  
Hyperbolic Systems ◽  
Exact Boundary Controllability ◽  
Boundary Controllability ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Exact Boundary ◽  
The One ◽  
Boundary Observability ◽  
Exact Boundary Observability

Abstract The known theory on the one-side exact boundary observability for first order quasilinear hyperbolic systems requires that the unknown variables are suitably coupled or satisfy the Group Property in boundary conditions on the non-observation side (see [Tatsien, C. R. Math. Acad. Sci. 342: 937–942, 2006]–[Tatsien, ESAIM Control Optim. Calc. Var.], [Russell, SIAM Rev. 20: 639–739, 1978]). In this paper we illustrate, with an inspiring example, that the one-side exact boundary observability can be realized by means of a suitable coupling of the unknown variables in quasilinear hyperbolic system itself instead of in boundary conditions. Moreover, an implicit duality between the one-side exact boundary controllability and one-side exact boundary observability is also revealed in this situation.

A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws: Application to Shallow Water Equations

2012 ◽  
Vol 12 (4) ◽  
pp. 1183-1214 ◽  
Author(s):  
Guglielmo Stecca ◽  
Annunziato Siviglia ◽  
Eleuterio F. Toro
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Shallow Water Equations ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Unstructured Meshes ◽  
Numerical Comparison ◽  
First Order ◽  
Systems Of Conservation Laws ◽  
First Order Method

AbstractWe construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of the UFORCE method developed by Stecca, Siviglia and Toro, in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan. The proposed first-order method is shown to be identical to the Godunov upwind method in applications to a 2 x 2 linear hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations. Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes. Finally, numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.

Chaotic Oscillations of Second Order Linear Hyperbolic Equations with Nonlinear Boundary Conditions: A Factorizable but Noncommutative Case

2015 ◽  
Vol 25 (11) ◽  
pp. 1530032 ◽  
Author(s):  
Liangliang Li ◽  
Yu Huang ◽  
Goong Chen ◽  
Tingwen Huang
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Hyperbolic Equations ◽  
Nonlinear Boundary Conditions ◽  
Second Order ◽  
Chaotic Oscillations ◽  
Van Der Pol ◽  
Order Linear ◽  
First Order

If a second order linear hyperbolic partial differential equation in one-space dimension can be factorized as a product of two first order operators and if the two first order operators commute, with one boundary condition being the van der Pol type and the other being linear, one can establish the occurrence of chaos when the parameters enter a certain regime [Chen et al., 2014]. However, if the commutativity of the two first order operators fails to hold, then the treatment in [Chen et al., 2014] no longer works and significant new challenges arise in determining nonlinear boundary conditions that engenders chaos. In this paper, we show that by incorporating a linear memory effect, a nonlinear van der Pol boundary condition can cause chaotic oscillations when the parameter enters a certain regime. Numerical simulations illustrating chaotic oscillations are also presented.

Chaotic Oscillations of Solutions of First Order Hyperbolic Systems in 1D with Nonlinear Boundary Conditions

2014 ◽  
Vol 24 (05) ◽  
pp. 1450072 ◽  
Author(s):  
Xiongping Dai ◽  
Tingwen Huang ◽  
Yu Huang ◽  
Goong Chen
Keyword(s):  
Boundary Conditions ◽  
Chaos Theory ◽  
Nonlinear Boundary ◽  
Dimensional Space ◽  
Hyperbolic Systems ◽  
Nonlinear Boundary Conditions ◽  
Chaotic Oscillations ◽  
One Dimensional ◽  
First Order ◽  
Delay Effects

We study chaotic oscillations of solutions of a first order hyperbolic system in one-dimensional space, where the governing equation is linear but the boundary condition contains nonlinearity with nonlocal and possibly time-delay effects. The main thrust of the paper is the advancement of existing chaos theory to multicomponent hyperbolic PDEs that allows a unified treatment of a general class of nonlinear, nonlocal and time-delayed boundary conditions where components of waves travel with several different speeds.

scholarly journals Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen
Keyword(s):  
Large Time ◽  
Hyperbolic Equations ◽  
Strong Stability ◽  
Matrix Exponential ◽  
Linear Component ◽  
Nonlinear Hyperbolic Equations ◽  
Time Step ◽  
Integrating Factor ◽  
The Matrix ◽  
Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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