Lectures on semigroup theory and its application to. In this paper, we study the controllability problem of the semidiscrete internally controlled onedimensional wave equation with the finite element method. Transparent boundary conditions, beam propagation method, parabolic wave equation. The discussions for the multidimensional schr odinger equation on cartesian meshes are given in section 4. 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. Wave equation dg modal basis semi discrete scheme ferienakademie 2014 4 cauchy kowalewski taylor series ader fully discrete.
This leads to a system of ordinary differential equations in time, called the semidiscrete equations. Analysis and discretization of semilinear stochastic wave. An integrable semidiscretization of the coupled yajimaoikawa. 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. 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. Numerical integration of partial differential equations pdes. Wave equation dg modal basis semidiscrete scheme ferienakademie 2014 4 cauchy kowalewski taylor series ader fullydiscrete update scheme. 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.
It works for light, sound, waves on the surface of water and a great deal more. Three different formulations continuous, semidiscrete and fullydiscrete of the nonlocal transparent boundary conditions are described and compared here. Wave equation electric charges and currents on right side of waveequation can be computed from other sources. Pde wave equation on semiinfinite string mathematics. 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. 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. First, we prove that the exponential decay of the semidiscrete energy is. The controllability for the semidiscrete wave equation. 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.
Three different formulations continuous, semi discrete and fully discrete of the nonlocal transparent boundary conditions are described and compared here. We analyze the convergence of the boundary controls of the semidiscrete equations to a control of the. This technique is used in 9 on the context of boundary observability for 1d wave equation with dirichlet boundary conditions. This approach is commonly termed the beam propagation method bpm. Energy conservation issues in the numerical solution of the semilinear wave equation. 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. Semi discretization of the hamiltonian wave equation. Semiamplitude means half of the peaktopeak amplitude. In this paper, we study the controllability problem of the semi discrete internally controlled onedimensional wave equation with the finite element method. Energy conservation issues in the numerical solution of. Uniform boundary controllability of a semidiscrete 1d. The particledistributions can be derived from numerical simulations, e.
C is the key parameter in the discrete wave equation. Pdf comparison of the continuous, semidiscrete and fully. A semidiscrete numerical method based on both a uniform space discretization and the discrete convolution operator is introduced to solve the cauchy problem. 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. 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. Since the equation is linear, any linear combinations of these solutions will also be solutions. Onedimensional wave equations with cubic power law perturbed by qregular additive spacetime random noise are considered. This technique is used in 9 on the context of boundary observability for 1d wave. 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. Finite difference methods for wave motion hans petter. It is known that the semidiscrete models obtained with finite. 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.
Let utbe the exact solution to the semi discrete equation. Numerical approximation schemes for multidimensional wave. Uniform boundary controllability of a semidiscrete 1d wave. It arises in fields like acoustics, electromagnetics, and fluid dynamics historically, the problem of a vibrating string such as that of a musical. The 1d wave equation hyperbolic prototype the 1dimensional wave equation is given by. The wave equation is the universal equation of physics. 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. 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. Boundary stabilization for 1d semidiscrete wave equation by. We show that the same negative results have to be expected.
The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. An implicit solution to the wave equation matthew causley andrew christlieb benjamin ong lee van groningen november 6, 2012. This is the motivation for the application of the semigroup theory to cauchys problem. 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.
When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k. The controllability for the semidiscrete wave equation with. Boundary stabilization for 1d semidiscrete wave equation by filtering technique. Controllability of partial differential equations and its. A numerical scheme for the controlled semi discrete 1d wave equation is considered. Nicolson method of so lving the parabolic wave equation. 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. Finite difference discretization of hyperbolic equations. Controllability, wave equation, heat equation, navierstokes equations, semidiscrete approximations. 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.
Plane wave semicontinuous galerkin method for the helmholtz. The convergence of the semidijcrete method is given. We analyze the convergence of the boundary controls of the semi discrete equations to a control of the. But there are nontrivial examples as, for instance, the. We present a new method for solving the wave equation implic.
In these notes we analyze some problems related to the controllability and observability of partial differential equations and its space semidiscretizations. 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. The presented analysis is based on the representation of its solution in form of fourierseries expansions along the eigenfunctions of laplace operator with. Let utbe the exact solution to the semidiscrete equation. The goal of this article is to analyze the observability properties for a space semidiscrete approximation scheme derived from a mixed.
Observability properties of a semidiscrete 1d wave. Comparison of the continuous, semidiscrete and fully. For this purpose, let us introduce the space nitedi erence scheme of equation 1. Observability properties of a semidiscrete 1d wave equation. Pdf boundary controllability of a linear semidiscrete 1.
In section 7 we present some numerical examples, comparing our method. We shall see that by modifying the semidiscrete equation with some terms. Some scientists 3 use amplitude or peak amplitude to mean semiamplitude. There are many ways to discretize the wave equation. 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. Pdf comparison of the continuous, semidiscrete and.
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. 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 has been applied to solve a time relay 2d wave equation. Moreover, the algebraic form in which we cast the semi. We shall discuss the basic properties of solutions to the wave equation 1. 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. We assume that only one of the two components of the unknown is observed. Introduction for the computer modelling of the propagation. We consider space semidiscretizations of the 1d wave equation in a bounded interval with homogeneous dirichlet boundary conditions.
We then discretize in time using any standard numerical method for systems of ordinary differential equations. The wave equation is an important hyperbolic partial di. Moments of electron and iondistribution in spaceplasma. Pdf uniform boundary controllability of a semidiscrete 1d. Semidiscrete and fullydiscrete transparent boundary conditions tbc for the parabolic wave equation 1 theory lubomr sumichrast. Pdf elliptic solutions of the semidiscrete bkp equation. A numerical scheme for the controlled semidiscrete 1d wave equation is considered. Pdf boundary controllability of a linear semidiscrete 1d.
Lectures on semigroup theory and its application to cauchys. 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. This kind of function is known as a plane wave along k. These models describe the displacement of nonlinear strings excited by stateindependent random external forces. 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. Then, consider perturbation etto the exact solution such that the perturbed solution, vt, is. Boundary stabilization for 1 d semidiscrete wave equation by. We also know that since i12 1 then u e kx must also be a solution. At times it is useful to consider the discretization process in two stages, first discretizing only in space, leaving the problem continuous in time. Plane wave semicontinuous galerkin method for the helmholtz equation anders matheson. Our method will give an explanation why in the case of. Elastic wave equations the equation system can be written as.
The basic tool is the auxiliary linear problem for the wave function. A semidiscrete numerical method for convolutiontype. Siam journal on numerical analysis siam society for. Pdf uniform boundary controllability of a semidiscrete. The sc hr o ding er w av e equati on macquarie university. 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. The method is proved to be uniformly convergent as the mesh size goes to zero. 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. Thanks for contributing an answer to mathematics stack exchange. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. 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 semi discrete problem is a hamiltonian system of odes to which galerkin spectral variational integrators can be applied directly. Notes on the algebraic structure of wave equations. Depending on the medium and type of wave, the velocity v v v can mean many different things, e.
Suppose we have the wave equation in the semi plane. 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. Boundary stabilization for 1 d semidiscrete wave equation. Controlling discrete equations resulting from numerical approximations of the contin. Energy conservation issues in the numerical solution of the. We analyze the problem of boundary observability, i. 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. Semidiscretization of the hamiltonian wave equation. We perform a gaussian beam construction at the semidiscrete level showing the existence of exponentially concentrated waves we perform the fourier analysis of the discontinuous galerkin methods for the wave equation. It is known that the semi discrete models obtained with finite. Later, in section 4, following 1, we study the decay properties of the energy for semidiscrete approximation schemes of 1d damped wave equations. Wave equations, examples and qualitative properties.