The basic tool is the auxiliary linear problem for the wave function. Let utbe the exact solution to the semi discrete equation. The sc hr o ding er w av e equati on macquarie university. This approach is commonly termed the beam propagation method bpm. The particledistributions can be derived from numerical simulations, e.
Elastic wave equations the equation system can be written as. An integrable semidiscrete analogue of the onedimensional coupled yajima oikawa system, which is comprised of multicomponent short waves and one component long wave, is proposed by using a bilinear technique. We then discretize in time using any standard numerical method for systems of ordinary differential equations. Numerical approximation of a general class of nonlinear unidirectional wave equations with a convolutiontype nonlocality in space is considered. Later, in section 4, following 1, we study the decay properties of the energy for semi discrete approximation schemes of 1d damped wave equations. Comparison of the continuous, semidiscrete and fully. Since the equation is linear, any linear combinations of these solutions will also be solutions. Let utbe the exact solution to the semidiscrete equation.
Energy conservation issues in the numerical solution of the. Pdf boundary controllability of a linear semidiscrete 1. In this article one discusses the controllability of a semi discrete system obtained by discretizing in space the linear 1d wave equation with a boundary control at one extremity. Pde wave equation on semiinfinite string mathematics. It works for light, sound, waves on the surface of water and a great deal more. We prove that, if the high modes of the discrete initial data have been filtered out, there exists a sequence of uniformly bounded controls and any. It arises in fields like acoustics, electromagnetics, and fluid dynamics historically, the problem of a vibrating string such as that of a musical. Moreover, the algebraic form in which we cast the semi. It has been applied to solve a time relay 2d wave equation. In section 7 we present some numerical examples, comparing our method. This is the motivation for the application of the semigroup theory to cauchys problem. Pdf elliptic solutions of the semidiscrete bkp equation. The article gives a semi discrete method for solving highdimension wave equationby the method, highdimension wave equation is converted by, means of diseretizationinto id wave equation system which is wellposed. Notes on the algebraic structure of wave equations steven g.
Notes on the algebraic structure of wave equations. The goal of this article is to analyze the observability properties for a space semidiscrete approximation scheme derived from a mixed. The wave equation governs a wide range of phenomena, including gravitational waves, light waves, sound waves, and even the oscillations of strings in string theory. The basic idea of our approach is that we first discretize the spatial dimensions in a highly accurate way, so that the resulting semidiscrete problem is a hamiltonian system of odes to which galerkin spectral variational integrators can be applied directly. Boundary stabilization for 1 d semidiscrete wave equation. As for a single wave equation, as well as for the direct complete observability of the coupled wave equations, we prove the lack of the numerical. The wave equation is the universal equation of physics.
Lectures on semigroup theory and its application to cauchys. We present a new method for solving the wave equation implic. Observability properties of a semidiscrete 1d wave. The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e. The 1d wave equation hyperbolic prototype the 1dimensional wave equation is given by. For the simulation of the propagation of optical waves in open wave guiding structures of integrated optics the parabolic approximation of the scalar wave equation is commonly used. Controllability of partial differential equations and its. There are many ways to discretize the wave equation.
Depending on the medium and type of wave, the velocity v v v can mean many different things, e. A numerical scheme for the controlled semi discrete 1d wave equation is considered. In these notes we analyze some problems related to the controllability and observability of partial differential equations and its space semidiscretizations. Numerical integration of partial differential equations pdes. Some scientists 3 use amplitude or peak amplitude to mean semiamplitude.
Bancroft abstract a new method of migration using the finite element method fem and the finite difference method fdm is jointly used in the spatial domain. The article gives a semidiscrete method for solving highdimension wave equationby the method, highdimension wave equation is converted by, means of diseretizationinto id wave equation system which is wellposed. An implicit solution to the wave equation matthew causley andrew christlieb benjamin ong lee van groningen november 6, 2012. The goal of this article is to analyze the observability properties for a space semidiscrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform. Semidiscretization of the hamiltonian wave equation. A numerical scheme for the controlled semidiscrete 1d wave equation is considered. Plane wave semicontinuous galerkin method for the helmholtz. Uniform boundary controllability of a semidiscrete 1d wave. Pdf boundary controllability of a linear semidiscrete 1d. The controllability for the semidiscrete wave equation with. For this purpose, let us introduce the space nitedi erence scheme of equation 1. A semi discrete numerical method based on both a uniform space discretization and the discrete convolution operator is introduced to solve the cauchy problem.
Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. Nicolson method of so lving the parabolic wave equation. An integrable semidiscretization of the coupled yajimaoikawa. This leads to a system of ordinary differential equations in time, called the semidiscrete equations. Uniform boundary controllability of a semidiscrete 1d. We assume that only one of the two components of the unknown is observed.
Our method will give an explanation why in the case of. Pdf comparison of the continuous, semidiscrete and. Pdf comparison of the continuous, semidiscrete and fully. Lectures on semigroup theory and its application to.
Moments of electron and iondistribution in spaceplasma. Observability properties of a semidiscrete 1d wave equation. Numerical approximation schemes for multidimensional wave. The convergence of the semidijcrete method is given. Boundary stabilization for 1 d semidiscrete wave equation by. It is known that the semidiscrete models obtained with finite. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Boundary stabilization for 1d semidiscrete wave equation by filtering technique.
This kind of function is known as a plane wave along k. Pdf uniform boundary controllability of a semidiscrete 1d. First, we prove that the exponential decay of the semidiscrete energy is. Transparent boundary conditions, beam propagation method, parabolic wave equation. The method is proved to be uniformly convergent as the mesh size goes to zero. We analyze the convergence of the boundary controls of the semi discrete equations to a control of the continuous wave equation when the mesh size tends to zero. The basic idea of our approach is that we first discretize the spatial dimensions in a highly accurate way, so that the resulting semi discrete problem is a hamiltonian system of odes to which galerkin spectral variational integrators can be applied directly. Hence, a discontinuous galerkin scheme is used to discretize the problem in space and appropriate time stepping schemes are then used to solve the resulting system of ordinary. We perform a gaussian beam construction at the semi discrete level showing the existence of exponentially concentrated waves we perform the fourier analysis of the discontinuous galerkin methods for the wave equation.
Three different formulations continuous, semi discrete and fully discrete of the nonlocal transparent boundary conditions are described and compared here. Then, consider perturbation etto the exact solution such that the perturbed solution, vt, is. Semiamplitude means half of the peaktopeak amplitude. Three different formulations continuous, semidiscrete and fullydiscrete of the nonlocal transparent boundary conditions are described and compared here. Energy conservation issues in the numerical solution of. A semidiscrete numerical method for convolutiontype. Suppose we have the wave equation in the semi plane. In this paper, we study the controllability problem of the semidiscrete internally controlled onedimensional wave equation with the finite element method.
C is the key parameter in the discrete wave equation. We analyze the convergence of the boundary controls of the semidiscrete equations to a control of the continuous wave equation when the mesh size tends to zero. We prove that, if the high modes of the discrete initial data have been filtered out, there exists a sequence of uniformly bounded controls and. This technique is used in 9 on the context of boundary observability for 1d wave. We also know that since i12 1 then u e kx must also be a solution. The wave equation is an important hyperbolic partial di. We derive the observability inequality and prove the exact controllability for the semi discrete internally controlled wave equation, with the controls taken from a finite dimensional space. Analysis and discretization of semilinear stochastic wave.
We analyze the convergence of the boundary controls of the semi discrete equations to a control of the. In this paper, we study the controllability problem of the semi discrete internally controlled onedimensional wave equation with the finite element method. Introduction for the computer modelling of the propagation. We shall see that by modifying the semidiscrete equation with some terms. In this article one discusses the controllability of a semidiscrete system obtained by discretizing in space the linear 1d wave equation with a boundary control at one extremity. We derive the observability inequality and prove the exact controllability for the semidiscrete internally controlled wave equation, with the controls taken from a finite dimensional space. Later, in section 4, following 1, we study the decay properties of the energy for semidiscrete approximation schemes of 1d damped wave equations. A semidiscrete numerical method based on both a uniform space discretization and the discrete convolution operator is introduced to solve the cauchy problem. Finite difference discretization of hyperbolic equations. This approach of reducing a pde to a system of odes, to which we then apply an ode solver, is often called the method of lines. The presented analysis is based on the representation of its solution in form of fourierseries expansions along the eigenfunctions of laplace operator with. When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k.
We shall discuss the basic properties of solutions to the wave equation 1. Controlling discrete equations resulting from numerical approximations of the contin. It is the most widely used measure of orbital wobble in astronomy and the measurement of small radial velocity semiamplitudes of nearby stars is important in the search for exoplanets see doppler spectroscopy. It is the relativistic schrodinger equation that describes the quantum mechanical evolution of the wave function of a single particle with zero rest mass 7. Onedimensional wave equations with cubic power law perturbed by qregular additive spacetime random noise are considered. Semi discretization of the hamiltonian wave equation. It is known that the semi discrete models obtained with finite. We show that the same negative results have to be expected. Wave equation dg modal basis semi discrete scheme ferienakademie 2014 4 cauchy kowalewski taylor series ader fully discrete. But there are nontrivial examples as, for instance, the. The goal of this article is to analyze the observability properties for a space semi discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform.
Plane wave semicontinuous galerkin method for the helmholtz equation anders matheson. We analyze the problem of boundary observability, i. Carlos castro sorin micu july 22, 2005 abstract in this article one discusses the controllability of a semidiscrete system obtained by discretizing in space the linear 1d wave equation with a boundary control at one extremity. Wave equation dg modal basis semidiscrete scheme ferienakademie 2014 4 cauchy kowalewski taylor series ader fullydiscrete update scheme. Finite difference methods for wave motion hans petter. We analyze the convergence of the boundary controls of the semidiscrete equations to a control of the. Wave equations, examples and qualitative properties. This technique is used in 9 on the context of boundary observability for 1d wave equation with dirichlet boundary conditions. These models describe the displacement of nonlinear strings excited by stateindependent random external forces. Pdf uniform boundary controllability of a semidiscrete. Energy conservation issues in the numerical solution of the semilinear wave equation.
Siam journal on numerical analysis siam society for. We consider elliptic solutions of the semidiscrete bkp equation and derive equations of motion for their poles. Boundary stabilization for 1d semidiscrete wave equation by. Matthias ehrhardt for the simulation of the propagation of optical waves in open wave guiding structures of integrated optics the parabolic approximation of the scalar wave equation is. Wave equation electric charges and currents on right side of waveequation can be computed from other sources. In this paper, we propose a semidiscrete numerical approach based on a uniform spatial discretization and truncated discrete convolution sums for the computation of solutions to the cauchy problem associated to the onedimensional nonlocal nonlinear wave equation, which is a regularized conservation law, 1. First we present the problems under consideration in the classical examples of the wave and heat equations and recall some well known results. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. We consider space semidiscretizations of the 1d wave equation in a bounded interval with homogeneous dirichlet boundary conditions. Controllability, wave equation, heat equation, navierstokes equations, semidiscrete approximations. Semidiscrete and fullydiscrete transparent boundary conditions tbc for the parabolic wave equation 1 theory lubomr sumichrast.
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