A variational formulation for the navierstokes equation. Variational formulation an overview sciencedirect topics. In this way the equations of fluid and geophysical fluid dynamics are rederived making use of a unifying principle, that is hamiltons principle of least action. For deterministic obstacles, we will focus on the case which is directly related to stochastic equations. Pdf the ultraweak variational formulation for elastic wave. The wave equation is solved by using a fourthorder variant of the staggeredgrid nite di erence scheme virieux, 1986. Variational methods for timedependent wave propagation problems patrick joly inria rocquencourt, bp105 le chesnay, france patrick. Keywords dual variational methodsemilinear wave equation dissipation. I variational formulation of problems and variational methods brigitte lucquindesreux encyclopedia of life support systems eolss force f xxd presses on each surface elementdx x xdd1 2. Finally, numerical results are presented in section 4.
A hilbertspace formulation of and variational principle for spontaneous wiggler radiation. Jan 22, 2005 in this paper we prove a new variational principle for the navierstokes equation which asserts that its solutions are critical points of a stochastic control problem in the group of areapreserving diffeomorphisms. Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. Variational methods for timedependent wave propagation problems 11. Solving timeharmonic scattering problems by the ultra. A fractal derivative model is established, and a variational principle is established for the anisotropic wave traveling. This transformation is a change of gauge, of which there are several possible choices, as discussed in rs99. In nal, we will apply this variational formulation to a system, i. Stochastic variational inequalities for a wave equation. That is, when a family of minimizing curves is constructed, the values of the optical length satisfy the characteristic equation corresponding the wave equation. If a function on the product of a domain and the interval is continuous and has continuous partial derivatives, while the function is continuously differentiable. When g0, it is naturally called a homogeneous neumann boundary condition. The vertical membrane displacement is represented by a real valued function u,which is the.
Clearly, from m one can compute u by using the leray projection on the divergence. A new technique to solve elliptic linear pdes, called ultra weak variational formulation uwvf in this paper, is introduced in b. Solving timeharmonic scattering problems by the ultra weak. We will focus on one approach, which is called the variational approach. This should be done without losing any order of accuracy. Variational iteration method for solving wave equation. Wu 20 obtained a variational formulation for higherorder waterwave equations. Variational principles for nonlinear partial differential equations have come to play an important role in mathematics and physics. Numerical solution of the wave equation with acoustic boundary conditions. The vertical membrane displacement is represented by a real valued function u,which is the solution of laplaces equation. Pdf a variational formulation of a stabilized unsplit. Functions that maximize or minimize functionals may be found. Variational methods for timedependent wave propagation problems. The equation itself is also the eulerlagrange equation of a.
In section 2 we formulate the wave problem, then provide its variational formulation and discretization, assuming that there is no. Variational equations once we have obtained the right hand side of the equations of motion, we can solve also for the variational equation. Variational formulation for the stationary fractional. Chapter 3 the variational formulation of elliptic pdes we now begin the theoretical study of elliptic partial differential equations and boundary value problems. Functionals are often expressed as definite integrals involving functions and their derivatives. We summarize the various model asymptotic equations in table 1. Numerical solution of the wave equation with acoustic. We propose a new variant of the variational theory of complex rays vtcr method introduced in 15, 16. Most of you have seen the derivation of the 1d wave equation from newtons and. John paul roop y abstract in this paper a theoretical framework for the galerkin. Derivation and physical interpretation of general boundary conditions. In this paper, the variational iteration method has been successfully applied for finding the solution of the wave equation. We shall discuss the basic properties of solutions to the wave equation 1. A hilbertspace formulation of and variational principle.
Variational spacetime methods for the elastic wave equation and. However, it is well known that not every nonlinear partial differential equation admits a variational formula. Keywordsmethod of lines, variational formulation, wave propagation in rods. In the present study we will extensively exploit the lagrangian variational formalism.
It is applied to a model wave problem, the twodimensional helmholtz problem. Weak and variational formulations for bies related to the wave equation a. Introduction the method of lines is a simple and reliable method of. Practical use of variational principles for modeling water. The solution obtained by the variational iteration method is an infinite sin or cos series for appropriate initialboundary conditions, which can, in turn, be expressed in a closed form to give the desired exact solution. In a weak formulation, an equation is no longer required to hold absolutely and this is not even well defined and has instead weak solutions only with respect to certain test vectors or test functions. Another approach is to transform the original problem into a system of. Equation 1 is called the variational equation for along the solution in the literature a weaker form of this theorem is more often quoted where instead of frechet differentiability a weaker sense of differentiability is used. Wave equations, examples and qualitative properties. Variational formulation of problems and variational methods. Variational principles for some nonlinear wave equations. All physical laws is believed that they can be derived from action using variational principle. Application of an ultra weak variational formulation of. Variational methods in semilinear wave equation with.
Also, most of the important variational statements for problems in engineering and physics have been known for over 200 years. In 21 wazwaz concluded that the km,n equations could not be derived from a. In the next section, we will give this variational formulation to 1. Due to the di erences, our variational formulation relates the wave velocity to the in mum, which enables us to obtain some new results. The ultraweak variational formulation has been used effectively to solve timeharmonic acoustic and electromagnetic wave propagation in inhomogeneous media.
We discuss evolution variational inequalities for a wave equation with unilateral constraint on the velocity. Variational principle approach to general relativity. On advanced variational formulation of the method of lines. Mathematically, a partial differential equation is an equation which contains partial derivatives, such as the wave equation.
Chapter 3 the variational formulation of elliptic pdes. An anisotropic wave in a porous medium is a hot topic in the coastal protection. Hamiltonian formulation for water wave equation author. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u tt. A weighted variational formulation based on planewave. This paper is devoted to an evaluation of the potentialities of this technique. Variational formulations for scattering in a three. The shortest path problem is formulated in terms of the following cost function, r, which. Variational formulation for the stationary fractional advection dispersion equation vincent j. It may also provide an example of a nonlocal surface wave equation that is easier to analyze than the hiz equation. The equations are analyzed within the framework of lagrangian and hamiltonian mechanics for continuous systems. Poisson equation in one dimension c 20182019 masayuki yano.
Together with parabolic differential equations in general, heatconduction equations occur with such regularity in important applications that variational principles leading to these equations have been an important topic for. Afterwards, we present the derivation of the linear elastic wave equation. A variational formulation of kinematic waves 3 r x x 0 k q qmax. Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem. The mathematics of pdes and the wave equation mathtube. In this paper we consider a dirichlet or neumann problem of onedimensional wave. There are one way wave equations, and the general solution to the two way equation could be done by forming linear combinations of such solutions. Unesco eolss sample chapters computational methods and algorithms vol. The quadrature formula must be sufficiently precise in order to preserve the accuracy of the finite element method appropriate criteria can be found in. We develop the ultraweak variational formulation for elastic wave propagation in two space dimensions. A weighted variational formulation based on planewave basis.
This nonlinear formulation will be successfully applied in chapter five to many useful physical problems. This principle is a natural extension of the results by arnold, ebin, and marsden concerning the euler equation. In section 2 we formulate the wave problem, then provide its variational formulation and discretization, assuming that there is no obstacle. Variational formulations for direct timeharmonic scattering problems in a.
Asymptotic variational wave equations springerlink. Prepared for aer1418 variational methods for pdes taught at the university of toronto. Regarding the wave equation, there are two basic solution concepts. In this paper we prove a new variational principle for the navierstokes equation which asserts that its solutions are critical points of a stochastic control problem in the group of areapreserving diffeomorphisms. The ultraweak variational formulation for elastic wave problems article pdf available in siam journal on scientific computing 255 january 2004 with 155 reads how we measure reads. The variational principle reveals an energy conservation law during the traveling process. A variational formulation for the navierstokes equation 3 the scalar function kx,t is arbitrary at t 0 and its evolution is chosen conveniently. A variational formulation of a stabilized unsplit convolutional perfectly matched layer for the isotropic or anisotropic seismic wave equation. The nonlinear variational wave equation helge holden norwegian university of science and technology trondheim, norway 1 joint work with xavier raynaud bilbao, january 20 bcam, january 20 archive for rational mechanics and analysis 201 2011 871964 doi. One approach is to deal directly with the second order problem in the sense of a single. Generalized variational formulation water wave problem possesses several variational structures 36, 57, 60. Generalized variational principle for long waterwave. Variational principle is a method which is powerful in physics. Both deterministic and stochastic obstacle problems will be considered.
Weak and variational formulations for bies related to the. One method of solution is so simple that it is often overlooked. Variational methods in semilinear wave equation with nonlinear boundary conditions and stability questions a. Special attention is devoted to a formulation based on a natural energy identity that leads to a spacetime weak formulation of the corresponding bie with robust theoretical properties. This system of equation has been used by stoker 17, debnath 18 for the investigation of the linearized initial value problem for the generation and propagation of water waves. Variational formulation of fluid and geophysical fluid. The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. Partial differential equation transform variational.
We consider an inviscid irrotational flow of constant density. Another important feature of variational methods is that often dual principles exist that allow one to. General relativity theory is a theory for gravity which galilean relativity fails to explain. An alternative approach is to apply the socalled mass lumping procedure. Variational formulation of fluid and geophysical fluid dynamics. To simplify the discussion, we assume the plasma to be in mechanical and electrostatic equilibrium in the absence of the wave. Practical use of variational principles for modeling water waves. Variational principles in classical mechanics download book. The variational formulation of elliptic pdes beforewestartdelvingintheabstract,letusgiveacouplemoremodelproblems of a different kind. The solutions of the one wave equations will be discussed in the next section, using characteristic lines ct. A weighted variational formulation based on plane wave basis for discretization of helmholtz equations qiya hu and long yuany april 25, 20 abstract in this paper we are concerned with numerical methods for solving helmholtz equations. This section discusses the rationale and variational formulation of pde transforms.
The ultraweak variational formulation for elastic wave. This consists of replacing the exact mass matrix by an approximation, the lumped mass matrix, which is diagonal. This is equivalent to formulating the problem to require a solution in the sense of a distribution. Variational methods for timedependent wave propagation. A spacetime discontinuous galerkin method for the solution.
987 1519 971 535 558 324 584 841 1247 846 716 784 709 745 1432 40 107 1271 887 1314 1068 386 1254 850 1488 927 1292 682 1007 870 897 213 746 116 1198 484 1096 885