a direct approach to finding exact invariants for one dimensional time dependent classical hamiltonians for a classical hamiltonian h 1 2 p2 v q t with an arbitrary time dependent potential v q t exact invariants that can be expressed as series in positive powers of p i q p t n 0pnfn q t are examined the method is based on direct use of the equation di dt i t i h 0 a recursion relation for the coefficients fn q t is obtained all potentials that admit an invariant quadratic in p are found and for those potentials all invariants quadratic in p are determined the feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed the systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation

the frequency decomposition multi grid method a new variant of the multi grid algorithms is presented it uses multiple coarse grid corrections with particularly associated prolongations and restrictions in this paper the robustness with respect to anisotropic problems is considered

schrodinger equation with time dependent boundary conditions a schrodinger equation for a well potential with varying width is studied generalized canonical transformations are shown to transform the problem into a time dependent harmonic oscillator problem submitted to fixed boundary conditions this transformed problem is solved by a perturbation technique and gives the evolution of the average energy of the system according to the motion of the well motions corresponding to a renormalization or compaction group are shown to be solvable by separation of variables

stable three point wavelet bases on general meshes recently we introduced a wavelet basis on general possibly locally refined linear finite element spaces each wavelet is a linear combination of three nodal basis functions independently of the number of space dimensions in the present paper we show h s stability of this basis for a range of s that in any case includes s 1 which means that the corresponding additive schwarz preconditioner is optimal for second order problems furthermore we generalize the construction of the wavelet basis to manifolds we show that the wavelets have at least one and in areas where the manifold is smooth and the mesh is uniform even two vanishing moments because of these vanishing moments apart from preconditioning the basis can be used for compression purposes for a class of integral equation problems the stiffness matrix with respect to the wavelet basis will be close to a sparse one in the sense that a priori it can be compressed to a sparse matrix without the order of convergence being reduced

finite difference schemes for the parabolic equation in a variable depth environment with a rigid bottom boundary condition we consider a linear schrodinger type partial differential equation the parabolic equation of underwater acoustics in a layer of water bounded below by a rigid bottom of variable topography using a change of depth variable technique we transform the problem into one with horizontal bottom for which we establish an a priori h1 estimate and prove an optimal order error bound in the maximum norm for a crank nicolson type finite difference approximation of its solution we also consider the same problem with an alternative rigid bottom boundary condition due to abrahamsson and kreiss and prove again a priori h1 estimates and optimal order error bounds for a crank nicolson scheme

wavelet multigrid analysis of squeeze film characteristics of poroelastic bearings the combined study of effects of poroelasticity and couple stresses on the performance of lubrication aspects of porelastic bearings in general and that of synovial joints in particular are analyzed the modified form of reynolds equation which incorporates the elastic nature of cartilage and stokes couple stress fluid as lubricant is derived and solved using a recently developed wavelet multigrid method this method has greatest advantage of minimizing the errors using wavelet transforms in obtaining accurate solution as grid size tends to zero it is found that 6 7 cycles are required to obtain a reasonably accurate solution in the multigrid scheme whereas only one cycle is required to obtain the solution in the wavelet multigrid method also matrix of discrete wavelet transform acts as a natural preconditioner producing rapid convergence it is observed that the poroelastic bearings with couple stress fluid as lubricant provide enhancement in pressure and ensure the increased load carrying capacity compared with viscous fluids this may be one of the reasons in the efficient lubrication and proper functioning of synovial joints

a new multigrid approach to convection problems abstract a new multigrid method for convection problems is presented it is designed to overcome the problem of alignment the flow being aligned with the grid the technique employs semi coarsening in several directions simultaneously and gives rise to multiple coarser grids with the same total number of points per grid level but with different sizes in each co ordinate direction the amount of work per multigrid cycle is still o n as an example the method is applied to the nonlinear upwind differenced euler equations of gas dynamics in two dimensions convergence rates are estimated by two level fourier analysis for the linearised equations numerical experiments on the nonlinear equations confirm these estimates

a fast wavelet multigrid method to solve elliptic partial differential equations abstract in this paper we present a wavelet based multigrid approach to solve elliptic boundary value problems encountered in mathematical physics the system of equations arising from finite difference discretization is represented in wavelet basis these equations are solved using multiresolution properties of wavelets characterized by sparse matrices having condition number o 1 together with a multigrid strategy for accelerating convergence the filter coefficients of d 2 k k 2 3 4 from daubechies family of wavelets are used to demonstrate the effectiveness and efficiency of the method the distinguishing feature of the method is it works as both solver and preconditioner as a consequence it avoids instability minimizes error and speeds up convergence compared to the classical multigrid method this approach requires substantially shorter computation time at the same time meeting accuracy requirements it is found that just one cycle is enough for the convergence of wavelet multigrid scheme whereas normally 7 8 cycles are required in classical multigrid schemes to meet the same accuracy numerical examples show that the scheme offers a fast and robust technique for elliptic pde s

the frequency decomposition multilevel method a robust additive hierarchical basis preconditioner hackbusch s frequency decomposition multi level method is characterized by the application of three additional coarse grid corrections in parallel to the standard one each coarse grid correction was designed to damp errors from a different part of the frequency spectrum in this paper we introduce a cheap variant of this method partly based on semi coarsening which demands less recursive calls than the original version using the theory of the additive schwarz methods we will prove robustness of our method as a preconditioner applied to anisotropic equations

fast and robust sixth order multigrid computation for the three dimensional convection diffusion equation we present a sixth order explicit compact finite difference scheme to solve the three dimensional 3d convection diffusion equation we first use a multiscale multigrid method to solve the linear systems arising from a 19 point fourth order discretization scheme to compute the fourth order solutions on both a coarse grid and a fine grid then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equations with a high reynolds number we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact soc scheme compared with the previously published fourth order compact foc scheme

a gmres based plane smoother in multigrid to solve 3d anisotropic fluid flow problems for a discretization of the 3d steady incompressible navier stokes equations a solution method is presented for solving flow problems on stretched grids the discretization is a vertex centered finite volume discretization with a flux splitting approach for the convective terms second order accuracy is obtained with the well known defect correction technique b koren j comput phys 87 25 1990 the solution method used is multigrid for which a plane smoother is presented for obtaining good convergence in flow domains with severely stretched grids a matrix is set up in a plane which is solved iteratively with a preconditioned gmres method here a stop criterion for gmres is tested which reduces the number of inner iterations compared to an exact plane solver without affecting the multigrid convergence rates the performance of the solution method is shown for a poisson model problem and for 3d incompressible channel flow examples

direct multi grid methods for linear systems with harmonic aliasing patterns multi level numerical methods that obtain the exact solution of a linear system are presented the methods are devised by combining ideas from the full multi grid algorithm and perfect reconstruction filters the problem is stated as whether a direct solver is possible in a full multi grid scheme by avoiding smoothing iterations and using different coarse grids at each step the coarse grids must form a partition of the fine grid and thus establishes a strong connection with domain decomposition methods an important analogy is established between the conditions for direct solution in multi grid solvers and perfect reconstruction in filter banks furthermore simple solutions of these conditions for direct multi grid solvers are found by using mirror filters as a result different configurations of direct multi grid solvers are obtained and studied

a new multigrid formulation for high order finite difference methods on summation by parts form multigrid schemes for high order finite difference methods on summation by parts form are studied by comparing the effect of different interpolation operators by using the standard linear prolonga

flexible multiple semicoarsening for three dimensional singularly perturbed problems we present robust parallel multigrid based solvers for 3d scalar partial differential equations the robustness is obtained by combining multiple semicoarsening strategies matrix dependent transfer operators and a krylov subspace acceleration the basis for the 3d preconditioner is a 2d method with multiple semicoarsened grids based on the mg s method from c w oosterlee appl numer math 19 1995 pp 115 128 and t washio and c w oosterlee gmd arbeitspapier 949 gmd st augustin germany 1995 the 2d method is generalized to three dimensions with a line smoother in the third dimension the method based on semicoarsening has been parallelized with the grid partitioning technique j linden b steckel and k stuben parallel comput 7 1988 pp 461 475 o a mcbryan et al impact comput sci engrg 3 1991 pp 1 75 and is evaluated as a solver and as a preconditioner on a mimd machine the robustness of the 3d method is shown for finite volume and finite difference discretizations of 3d anisotropic diffusion equations and convection dominated convection diffusion problems

the improved robustness of multigrid elliptic solvers based on multiple semicoarsened grids multigrid convergence rates degenerate on problems with stretched grids or anisotropic operators unless one uses line or plane relaxation for three dimensional problems only plane relaxation suffices in general while line and plane relaxation algorithms are efficient on sequential machines they are quite awkward and inefficient on parallel machines this paper presents a new multigrid algorithm based on the use of multiple coarse grids that eliminates the need for line or plane relaxation in anisotropic problems this algorithm is developed and the standard multigrid theory is extended to establish rapid convergence for this class of algorithms the new algorithm uses only point relaxation allowing easy and efficient parallel implementation yet achieves robustness and convergence rates comparable to line and plane relaxation multigrid algorithms the algorithm described here is a variant of mulder s multigrid algorithm w mulder j compact phys 83 1989 pp 303 323 for hyperbolic problem

an energy minimizing interpolation for robust multigrid methods we propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation it can handle pde coefficients of various types on structured or unstructured grids under one framework the formulation is general it can be applied to any dimension we demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem an oscillatory coefficient problem and an anisotropic problem empirically the convergence rate is independent of the coefficients of the underlying pde in addition to being independent of the mesh size the proposed method is primarily designed for second order elliptic pdes with possible extensions to other classes of problems such as integral equations

on additive schwarz preconditioners for sparse grid discretizations based on the framework of subspace splitting and the additive schwarz scheme we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems for a bxp like preconditioner we derive an estimate of the optimal ordero 1 and for a hb like variant we obtain an estimate of the ordero k 2 2 k 2 wherek denotes the number of levels employed furthermore we confirm these results by numerically computed condition numbers

an exact invariant for the time dependent double well anharmonic oscillators lie theory and quasi invariance groups the author derives an exact energy type invariant for the double well anharmonic oscillators this invariant holds for certain time dependent potential parameters a nonlinear superposition principle is used to solve the nonlinear constraint equations which appear in the resolution

invariants for dissipative nonlinear systems by using rescaling a rescaling transformation of space and time is introduced in the study of nonlinear dissipative systems that are described by a second order differential equation with a friction term proportional to the velocity t v the transformation is of the form x t where x c t t d dt a2 t this rescaling is used to find each potential for which there exists an exact invariant quadratic in the velocity and to find the invariant the invariants are found explicitly for a power law potential t xm 1 m 1 and an arbitrary coefficient of friction t we show in an example how the rescaling transformation can be chosen to give an asymptotic solution of the equation in cases where the exact invariant does not exist for certain parameters the asymptotic solution is a self similar solution that is an attractor for all initial conditions the technique of applying a rescaling transformation has been useful in other problems and may have additional practical applications

robust multigrid methods for nonsmooth coefficient elliptic linear systems we survey the literature on robust multigrid methods which have been developed in recent years for solving second order elliptic pdes with nonsmooth coefficients we highlight the key ideas of designing robust multigrid methods which are able to recover the usual multigrid efficiency for nonsmooth coefficient pdes on structured or unstructured grids in particular we shall describe various approaches for constructing the interpolation and the smoothing operators and the coarse grid points selections

the frequency decomposition multi grid method ii convergence analysis based on the additive schwarz method the convergence analysis of a special variant of the additive schwarz iteration is presented it can be applied to the frequency decomposition multigrid method and yields robust convergence

the gerber shiu discounted penalty function in the delayed renewal risk process with random income abstract this paper considers the delayed renewal risk model with random premium income the gerber shiu discounted penalty function in the delayed renewal model is expressed in terms of the corresponding gerber shiu function in the ordinary renewal model the obtained results can be viewed as the discrete analogy of the classical sparre anderson risk model

a robust hierarchical basis preconditioner on general meshes in this paper we introduce a multi level direct sum space decomposition of general possibly locally refined linear or multi linear finite element spaces the resulting additive schwarz preconditioner is optimal for symmetric second order elliptic problems moreover it turns out to be robust with respect to coefficient jumps over edges in the coarsest mesh perturbations with positive zeroth order terms and after a further decomposition of the spaces also with respect to anisotropy along the grid lines important for an efficient implementation is that stable bases of the subspaces defining our decomposition consisting of functions having small supports can be easily constructed

a boundary condition capturing multigrid approach to irregular boundary problems we propose a geometric multigrid method for solving linear systems arising from irregular boundary problems involving multiple interfaces in two and three dimensions in this method we adopt a matrix free approach i e we do not form the fine grid matrix explicitly and we never form nor store the coarse grid matrices as many other robust multigrid methods do the main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function numerical results are given to compare our multigrid method with black box and algebraic multigrid methods