the existence and uniqueness of solution to wavelet collocation the study of the numerical solutions of pdes with wavelet collocation has yielded a number of substantial results however the existence and uniqueness of solution has not been discussed yet in this paper the existence and uniqueness of solution to wavelet collocation for elliptic equations is established and discussed moreover wavelet collocation is applied to numerical example to examine its appropriateness according to numerical example and analysis it is seen that the existence and uniqueness of solution of this paper is feasible and the new theory is meaningful for developing wavelet collocation

a fast adaptive diffusion wavelet method for burger s equation abstract a fast adaptive diffusion wavelet method is developed for solving the burger s equation the diffusion wavelet is developed in 2006 coifman and maggioni 2006 and its most important feature is that it can be constructed on any kind of manifold classes of operators which can be used for construction of the diffusion wavelet include second order finite difference differentiation matrices the efficiency of the method is that the same operator is used for the construction of the diffusion wavelet as well as for the discretization of the differential operator involved in the burger s equation the diffusion wavelet is used for the construction of an adaptive grid as well as for the fast computation of the dyadic powers of the finite difference matrices involved in the numerical solution of burger s equation in this paper we have considered one dimensional and two dimensional burger s equation with dirichlet and periodic boundary conditions for each test problem the cpu time taken by fast adaptive diffusion wavelet method is compared with the cpu time taken by finite difference method and observed that the proposed method takes lesser cpu time we have also verified the convergence of the given method

the differentiation matrix for daubechies based wavelets on an interval the differentiation matrix for a daubechies based wavelet basis defined on an interval is constructed it is shown that the differentiation matrix based on the currently available boundary constructions does not maintain the superconvergence encountered under periodic boundary conditions

a one time truncate and encode multiresolution stochastic framework in this work a novel adaptive strategy for stochastic problems inspired from the classical email protected s framework is presented the proposed algorithm allows building in a very general manner stochastic numerical schemes starting from a whatever type of deterministic schemes and handling a large class of problems from unsteady to discontinuous solutions its formulations permits to recover the same results concerning the interpolation theory of the classical multiresolution approach but with an extension to uncertainty quantification problems the present strategy permits to build numerical scheme with a higher accuracy with respect to other classical uncertainty quantification techniques but with a strong reduction of the numerical cost and memory requirements moreover the flexibility of the proposed approach allows to employ any kind of probability density function even discontinuous and time varying without introducing further complications in the algorithm the advantages of the present strategy are demonstrated by performing several numerical problems where different forms of uncertainty distributions are taken into account such as discontinuous and unsteady custom defined probability density functions in addition to algebraic and ordinary differential equations numerical results for the challenging 1d kraichnan orszag are reported in terms of accuracy and convergence finally a two degree of freedom aeroelastic model for a subsonic case is presented though quite simple the model allows recovering some physical key aspect on the fluid structure interaction thanks to the quasi steady aerodynamic approximation employed the injection of an uncertainty is chosen in order to obtain a complete parameterization of the mass matrix all the numerical results are compared with respect to classical monte carlo solution and with a non intrusive polynomial chaos method

simultaneous space time adaptive wavelet solution of nonlinear parabolic differential equations dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time these methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time the same time step is used for all spatial locations and all scales this approach clearly does not fully exploit space time intermittency we propose an adaptive wavelet collocation method for solving highly intermittent problems e g turbulence on a simultaneous space time computational domain which naturally adapts both the space and time resolution to match the solution besides generating a near optimal grid for the full space time solution this approach also allows the global time integration error to be controlled the efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent 1d t dimensional and 2d t dimensional test problems in particular we found that the space time method uses roughly 18 times fewer space time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method while achieving similar global accuracy

multiresolution based on weighted averages of the hat function i linear reconstruction techniques in this paper we analyze a particular example of the general framework developed in a harten it siam j numer anal 33 1996 pp 1205 1256 the case in which the discretization operator is obtained by taking local averages with respect to the hat function we consider a class of reconstruction procedures which are appropriate for this multiresolution setting and describe the associated prediction operators that allow us to climb up the ladder from coarse to finer levels of resolution in part i we use data independent linear reconstruction techniques as our approximation tool we show how to obtain multiresolution transforms in bounded domains and analyze their stability with respect to perturbations

parallel adaptive wavelet collocation method for pdes a parallel adaptive wavelet collocation method for solving a large class of partial differential equations is presented the parallelization is achieved by developing an asynchronous parallel wavelet transform which allows one to perform parallel wavelet transform and derivative calculations with only one data synchronization at the highest level of resolution the data are stored using tree like structure with tree roots starting at a priori defined level of resolution both static and dynamic domain partitioning approaches are developed for the dynamic domain partitioning trees are considered to be the minimum quanta of data to be migrated between the processes this allows fully automated and efficient handling of non simply connected partitioning of a computational domain dynamic load balancing is achieved via domain repartitioning during the grid adaptation step and reassigning trees to the appropriate processes to ensure approximately the same number of grid points on each process the parallel efficiency of the approach is discussed based on parallel adaptive wavelet based coherent vortex simulations of homogeneous turbulence with linear forcing at effective non adaptive resolutions up to 20483 using as many as 2048 cpu cores

numerical solution of evolution equations by the haar wavelet method an efficient numerical method for solution of nonlinear evolution equations based on the haar wavelets approach is proposed the method is tested in the case of burgers and sine gordon equations numerical results obtained by computer simulation are compared with other available solutions these calculations demonstrate that the accuracy of the haar wavelet solutions is quite high even in the case of a small number of grid points

adaptive multiresolution weno schemes for multi species kinematic flow models multi species kinematic flow models lead to strongly coupled nonlinear systems of first order spatially one dimensional conservation laws the number of unknowns the concentrations of the species may be arbitrarily high models of this class include a multi species generalization of the lighthill whitham richards traffic model and a model for the sedimentation of polydisperse suspensions their solutions typically involve kinematic shocks separating areas of constancy and should be approximated by high resolution schemes a fifth order weighted essentially non oscillatory weno scheme is combined with a multiresolution technique that adaptively generates a sparse point representation spr of the evolving numerical solution thus computational effort is concentrated on zones of strong variation near shocks numerical examples from the traffic and sedimentation models demonstrate the effectiveness of the resulting weno multiresolution weno mrs scheme

a dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain a dynamically adaptive multilevel wavelet collocation method is developed for the solution of partial differential equations the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where sharp transitions occur the scheme handles general boundary conditions the method is applied to the solution of the one dimensional burgers equation with small viscosity a moving shock problem and a nonlinear thermoacoustic wave problem the results indicate that the method is very accurate and efficient

wavelet optimized finite difference mesh for mhd flow in a circular duct abstract there is a large class of important problems in scientific and engineering applications where the solution shows very little variation in the core region but very rapid changes close to the boundary of the domain of interest magnetohydrodynamic flow problems belong to this category treatment of such problems numerically therefore requires a non uniform mesh which is very fine near the boundary the determination of the size of the grid in different parts of the domain is therefore a challenging problem some recent studies suggest that wavelet methods can be effectively applied to create an optimized mesh the method has mostly been used for one dimensional problems the present paper investigates the problem for fully developed mhd duct flow problems which are basically two dimensional the boundary layer character is more pronounced in one direction as compared to the other the governing equations comprise a coupled system of two partial differential equations with dirichlet boundary conditions wavelet based adaptive mesh is first created and then the system is discretized over this mesh computations have been carried out at different precision levels for various hartmann numbers for a pipe of circular section contours are also drawn to depict the core and the boundary layer region

wavelet optimized finite difference method with non static re gridding the main aim of wavelet based numerical methods for solving partial differential equations is to develop adaptive schemes in order to achieve accuracy and computational efficiency the wavelet optimized finite difference method wofd uses wavelets to generate appropriate grids to apply finite difference method its standard implementation carries out static re griddings after a fixed number of time steps we present an effective implementation of wofd method that reduces the number of static re griddings hence leading to reduction of flops without significant loss of accuracy numerical experiments are performed on different cases of heat and schrodinger equations

generalized multiresolution schemes abstract the concept of a multiresolution scheme as introduced by a harten see for instance cam report 93 03 1993 cam report 94 10 1994 icase report no 94 59 1994 j comput phys 115 1994 319 338 is generalized this generalization is based on a categorical setting which provides a much clearer and a more comprehensive understanding of the subject examples show the broad spectrum of applicability of generalized multiresolution schemes

multiresolution schemes for the reactive euler equations we present multiresolution mr schemes for the efficient numerical solution of the one dimensional system of the reactive euler equations which has possibly stiff source terms the original version of the method was developed by a harten 1995 comm pure appl math 48 12 1305 for homogeneous hyperbolic conservation laws by computing the cell average mr representation of the solution we obtain much information about the solution s regularity this description of smoothness is then used to reduce the number of direct flux computations as well as the expensive high order eno essentially nonoscillatory reconstruction both of which are now performed only near discontinuities thereby the numerical solution procedure becomes considerably more efficient in the present case of the reactive euler equations the average efficiency factor measured by counting the number of actual flux computations ranges from about 5 to 12 this is on the same order of and in some cases comes reasonably close to actual speed up factors obtained by code timings which were between 3 to 5 the mr overhead rate was about 10 for the eno and 36 for tvd schemes respectively the quality of the solution is shown to be the same as that of the finest grid detailed numerical and performance results are shown for up to fourth order accuracy for source terms ranging from moderate to extremely stiff

fully adaptive multiresolution finite volume schemes for conservation laws the use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by a harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation in this approach the solution is still represented at each time step on the finest grid resulting in an inherent limitation of the potential gain in memory space and computational time the present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes in which the solution is represented and computed in a dynamically evolved adaptive grid a crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages several solutions to this problem are proposed analyzed and compared in terms of accuracy and complexity

domain decomposition spectral approximations for an eigenvalue problem with a piecewise constant coefficient consider a model eigenvalue problem with a piecewise constant coefficient we split the domain at the discontinuity of the coefficient function and define the multidomain variational formulation for the eigenproblem the discrete multidomain variational formulations are defined for legendre galerkin and legendre collocation methods the spectral rate of convergence of the approximate eigensolutions is proven for the legendre galerkin method the minmax principle is used for the convergence analysis the legendre collocation chebyshev collocation legendre collocation penalty and chebyshev collocation penalty methods are also defined by using the multidomain approach and their numerical results applied to the eigenproblem are demonstrated the spectral convergence for the eigenvalues and eigenfunctions is confirmed for all the multidomain spectral techniques presented here

multiresolution representation in unstructured meshes in this paper we describe techniques to represent data which originate from discretization of functions in unstructured meshes in terms of their local scale components to do so we consider a nested sequence of discretization which corresponds to increasing levels of resolution and we define the scales as the difference in information between any two successive levels we obtain data compression by eliminating scale coefficients which are sufficiently small this capability for data compression can be used to reduce the cost of numerical schemes by solving for the more compact representation of the numerical solution in terms of its significant scale coefficients

application of generalized wavelets an adaptive multiresolution scheme abstract in this paper we add a new member to the family of adaptive multiresolution schemes these schemes employ adaptive data dependent reconstruction techniques which use the expensive eno essentially nonoscillatory interpolation only near discontinuities and a simple central stencil everywhere else in addition by recursive diadic coarsening of the original fine grid in smooth regions the fluxes can be interpolated from the coarser mesh instead of being computed directly we first present semi discrete versions of harten s multiresolution schemes harten 1994 to appear compare their performance in the nonlinear cases to their fully discrete counterparts and then test them in the linear case as well for the latter we find an unexpected oscillatory behavior which after a closer examination is shown to be related to the linear smearing of discontinuities we shall therefore discuss numerical issues related to the original adaptive multiresolution schemes in doing so we propose a new switch between eno and central interpolation which provides a truly tolerably oscillatory interpolation the resulting algorithm is termed the modified adaptive multiresolution scheme the new second order central eno switch is also proved to ensure that wherever the central stencil is used it will still result is an essentially nonoscillatory interpolation

multiresolution representation of data a general framework in this paper we present a general framework for a multiresolution representation of data which is obtained by the discretization of mappings this framework which can be viewed as a generalization of the theory of wavelets includes discretization corresponding to unstructured grids in several space dimensions and thus is general enough to enable us to embed most numerical problems in a multiresolution setting furthermore this framework allows for nonlinear data dependent multiresolution representation schemes and thus enables us to design adaptive data compression algorithms in this paper we also study the stability of linear schemes for a multiresolution representation and derive sufficient conditions for existence of a multiresolution basis

on the use of biorthogonal interpolating wavelets for large eddy simulation of turbulence recent work in the field of turbulence modelling has demonstrated the benefits of the wavelet based multiresolution analysis technique as a tool for the formulation of the large eddy simulation les equations in this formalism the les equations are obtained by projecting the navier stokes equations onto a hierarchy of wavelet spaces this paper investigates the use of biorthogonal interpolating wavelets as a basis for this projection placing special emphasis on the wavelet based differential operators that define this mapping a detailed analysis of their convergence properties is presented and compared to those of their orthogonal counterpart the daubechies wavelets based on this study we highlight the weaknesses of the unlifted interpolating wavelet representation for les sub grid modelling finally we establish a link between the unlifted framework and the sampling based les approach recently proposed in the literature

using rationalized haar wavelet for solving linear integral equations the main purpose of this paper is to demonstrate that using rationalized haar wavelet for solving linear fredholm integral equation of the second kind we convert the integral equation to a system of linear equations and by using numerical examples we show our estimation have a good degree of accuracy

fitted mesh b spline collocation method for solving self adjoint singularly perturbed boundary value problems in this paper we develop b spline collocation method for solving a class of self adjoint singularly perturbed equations given by 0 1 email protected a x y b x y x g x a x a 0 a x 0 b x b 0 y 0 email protected y 1 email protected use the fitted mesh technique to generate piecewise uniform mesh and use b spline method which leads to a tridiagonal linear system the convergence analysis is given and the method is shown to have uniform convergence of second order numerical illustrations are given in the end to demonstrate the efficiency of our method

stochastic discrete equation method sdem for two phase flows a new scheme for the numerical approximation of a five equation model taking into account uncertainty quantification uq is presented in particular the discrete equation method dem for the discretization of the five equation model is modified for including a formulation based on the adaptive semi intrusive asi scheme thus yielding a new intrusive scheme sdem for simulating stochastic two phase flows some reference test cases are performed in order to demonstrate the convergence properties and the efficiency of the overall scheme the propagation of initial conditions uncertainties is evaluated in terms of mean and variance of several thermodynamic properties of the two phases

multiresolution schemes for conservation laws with viscosity this paper presents multiresolution schemes for the efficient numerical solution of one dimensional conservation laws with viscosity the method originally developed by a harten commun pure appl math to appear for hyperbolic conservation laws computes the cell average multiresolution representation of the solution which provides much information about the solution s regularity as a consequence the possibly expensive eno essentially nonoscillatory reconstruction as well as numerous flux computations are performed only near discontinuities and thereby the numerical solution procedure becomes considerably more efficient the multiresolution scheme is also expected to follow possibly unsteady irregularities from one time step to the next when viscosity is added predicting the location of the irregularity becomes a problem of estimating the change in shock thickness to this end we derive shock width estimates for our 1d prototype equations which when combined with the stability restriction of the numerical scheme provide a reliable mechanism for enlarging the original multiresolution stencil the numerical experiments for scalar conservation laws indicate the feasibility of multiresolution schemes for the viscous case as well

biorthogonal multiwavelets on the interval for numerical solutions of burgers equation to numerically solve the burgers equation in this paper we propose a general method for constructing wavelet bases on the interval 0 1 derived from symmetric biorthogonal multiwavelets on the real line in particular we obtain wavelet bases with simple structures on the interval 0 1 from the hermite cubic splines in comparison with all other known constructed wavelets on the interval 0 1 our constructed wavelet bases on the interval 0 1 from the hermite cubic splines not only have good approximation and symmetry properties with extremely short supports but also employ a minimum number of boundary wavelets with a very simple structure these desirable properties make them to be of particular interest in numerical algorithms our constructed wavelet bases on the interval 0 1 are then used to solve the nonlinear burgers equation our method is based on the finite difference formula combined with the collocation method therefore our proposed numerical scheme in this paper is abbreviated as mfdcm mixed finite difference and collocation method some numerical examples are provided to demonstrate the validity and applicability of our proposed method which can be easily implemented to produce a desired accuracy

solving second kind integral equations by galerkin methods with continuous orthogonal wavelets abstract in this paper we use the continuous wavelets on the interval constructed by cohen et al appl comput harm anal 1 1993 54 81 to solve the second kind integral equations to this end we give the decomposition and reconstruction algorithm for these wavelets and construct the quadrature formulae for the calculation of inner products of any functions and the scaling functions which are required in the wavelet galerkin methods for integral equations in this method the integral kernels are represented in these wavelet bases as sparse matrices to high precision thus we present an efficient algorithm for numerical solution of second kind integral equations

second generation wavelet collocation method for the solution of partial differential equations an adaptive numerical method for solving partial differential equations is developed the method is based on the whole new class of second generation wavelets wavelet decomposition is used for grid adaptation and interpolation while a new o n hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the treatment of nonlinear terms and general boundary conditions is a straightforward task due to the collocation nature of the algorithm in this paper we demonstrate the algorithm for one particular choice of second generation wavelets namely lifted interpolating wavelets on an interval with uniform regular sampling the main advantage of using second generation wavelets is that wavelets can be custom designed for complex domains and irregular sampling thus the strength of the new method is that it can be easily extended to the whole class of second generation wavelets leaving the freedom and flexibility to choose the wavelet basis depending on the application

convergence analysis of planewave expansion methods for 2d schrodinger operators with discontinuous periodic potentials in this paper we consider the problem of computing the spectrum of a two dimensional schrodinger operator with discontinuous periodic potential in two dimensions using fourier or planewave expansion methods problems of this kind are currently of great interest in the design of new optical devices to determine band gaps and to compute localized modes in photonic crystal materials although fourier methods may not be every applied mathematician s first choice for this problem because of the discontinuities in the potential we will show here that even though as expected the convergence is not exponential the method has several desirable features that make it competitive with other discretization techniques such as finite element methods both with respect to implementation and convergence properties in particular we will prove that simple preconditioners for the system matrix are optimal leading to a computational complexity of mathcal o n log n in the number of planewaves n using the fast fourier transform moreover we derive sharp error estimates that show that the method is essentially third order in the eigenvalues and of order frac 3 2 in the eigenfunctions in the h 1 norm and frac 5 2 in the l 2 norm to improve the planewave expansion method in the case of discontinuous potentials it has been proposed in the physics literature to replace the discontinuous potential with an effective potential that is smooth despite the additional error this incurs we will here answer the question whether this smoothing is worth it in fact our convergence analysis of the modified method provides an optimal choice for the smoothing parameter but it also shows that the overall rate of convergence is no faster than before and so smoothing does not seem to be worth it all the theoretical results are confirmed in our numerical experiments

explicit a posteriori error estimates for eigenvalue analysis of heterogeneous elastic structures an a posteriori error estimator is developed for the eigenvalue analysis of heterogeneous elastic structures it constitutes an extension of a well known explicit estimator to heterogeneous structures we prove that our estimates are independent of the variations in material properties and independent of the polynomial degree of finite elements finally we study numerically the effectivity of this estimator on several model problems

uniformly convergent non standard finite difference methods for self adjoint singular perturbation problems we design non standard finite difference schemes for self adjoint singularly perturbed two point boundary value problems essential physical properties e g dissipativity of the solutions of such problems are captured in the schemes by an appropriate renormalization of the denominator of the discrete derivative the schemes are analyzed for e uniform convergence several numerical examples are given to support the predicted theory

filter bank methods for hyperbolic pdes we use biorthogonal filter banks to solve hyperbolic pdes adaptively with a sparse multilevel representation of the solution the methods described are of finite difference type and the filter banks are used to give a sparse representation of signals and to transform between grids on different scales we derive bounds for the error and number of coefficients in the sparse representation these bounds also apply for filter banks that are not associated with any wavelets we develop algorithms for fast differentiation and multiplication in detail the strength of the method is shown in various test problems

wavelet moment method for solving fredholm integral equations of the first kind in this paper we suggest an efficient method for solving fredholm integral equations of the first kind using wavelets as basis functions in the moment method and reducing the order of the linear equation rather than making the matrix sparse in the end we give some numerical examples

on rubin s variant of the p adic birch and swinnerton dyer conjecture ii let e q be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field k in 1991 by studying a certain special value of the katz two variable p adic l function lying outside the range of p adic interpolation k rubin formulated a p adic variant of the birch and swinnerton dyer conjecture when e k is infinite and he proved that his conjecture is true for e k of rank one when e k is finite however the statement of rubin s original conjecture no longer applies and the relevant special value of the appropriate p adic l function is equal to zero in this paper we extend our earlier work and give an unconditional proof of an analogue of rubin s conjecture when e k is finite

numerical solution of fredholm integral equations of the first kind by using linear legendre multi wavelets in this paper we suggest an efficient method for solving fredholm integral equations of the first kind the continuous legendre multi wavelets constructed on 0 1 are utilized as a basis in galerkin method to reduce the solution of linear integral equations to a system of algebraic equations illustrative examples are included to demonstrate the efficiency and the application of the technique

wavelet optimized finite difference method using interpolating wavelets for self adjoint singularly perturbed problems we design a wavelet optimized finite difference wofd scheme for solving self adjoint singularly perturbed boundary value problems the method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids small dissipation of the solution is captured significantly using an adaptive grid the adaptive feature is performed automatically by thresholding the wavelet coefficients numerical examples have been solved and compared with non standard finite difference schemes in j m s lubuma k c patidar uniformly convergent non standard finite difference methods for self adjoint singular perturbation problems j comput appl math 191 2006 228 238 the proposed method outperforms the non standard finite difference for studying singular perturbation problems for small dissipations very small e and effective grid generation therefore the proposed method is better for studying the more challenging cases of singularly perturbed problems

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

an adaptive multilevel wavelet collocation method for elliptic problems an adaptive multilevel wavelet collocation method for solving multi dimensional elliptic problems with localized structures is described the method is based on multi dimensional second generation wavelets and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems int j comp fluid dyn 17 2003 151 wavelet decomposition is used for grid adaptation and interpolation while a hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid in order to accelerate the convergence of the solver an iterative procedure analogous to the multigrid algorithm is developed the overall computational complexity of the solver is o n where n is the number of adapted grid points the accuracy and computational efficiency of the method are demonstrated for the solution of two and three dimensional elliptic test problems

a wavelet optimized very high order adaptive grid and order numerical method wavelets detect information at different scales and at different locations throughout a computational domain furthermore wavelets can detect the local polynomial content of computational data numerical methods are most efficient when the basis functions of the method are similar to the data present by designing a numerical scheme in a completely adaptive manner around the data present in a computational domain one can obtain optimal computational efficiency this paper extends the numerical wavelet optimized finite difference wofd method to arbitrarily high order so that one obtains in effect an adaptive grid and adaptive order numerical method which can achieve errors equivalent to errors obtained with a spectrally accurate numerical method

a wavelet optimized adaptive multi domain method the formulation and implementation of wavelet based methods for the solution of multi dimensional partial differential equations in complex geometries is discussed utilizing the close connection between daubechies wavelets and finite difference methods on arbitrary grids we formulate a wavelet based collocation method well suited for dealing with general boundary conditions and nonlinearities to circumvent problems associated with completely arbitrary grids and complex geometries we propose to use a multi domain formulation in which to solve the partial differential equation with the ability to adapt the grid as well as the order of the scheme within each subdomain besides supplying the required geometric flexibility the multi domain formulation also provides a very natural load balanced data decomposition suitable for parallel environments the performance of the overall scheme is illustrated by solving two dimensional hyperbolic problems

using high order methods on adaptively refined block structured meshes derivatives interpolations and filters block structured adaptive mesh refinement amr is used for efficient resolution of partial differential equations pdes solved on large computational domains by clustering mesh points only where required by large gradients previous work has indicated that fourth order convergence can be achieved on such meshes by using a suitable combination of high order derivative interpolation and filter stencils and can deliver significant computational savings over conventional second order methods at engineering error tolerances in this paper we explore the interactions between the errors introduced by the discretized operators for derivatives interpolations and filters we develop general expressions for high order derivative interpolation and filter stencils applicable in multiple dimensions using a fourier approach to facilitate high order block structured amr implementations these stencils are derived under the assumption that the fields that they are applied to are smooth for a given derivative stencil and thus a given order of accuracy the necessary interpolation order is found to be dependent on the highest spatial derivative in the pde being solved this is demonstrated empirically using one and two dimensional model equations by observing the increase in delivered accuracy as the order of accuracy of the interpolation stencil is increased we also examine the empirically observed order of convergence as the effective resolution of the mesh is increased by successively adding levels of refinement with different orders of derivative interpolation and filter stencils the procedure devised here is modular and its various pieces i e the derivative interpolation and filter stencils can be extended independently to higher orders and dimensions in the case of interpolation stencils the application of these methods often requires nontrivial logic especially near domain boundaries this logic along with the stencils used in this study are available as a freely downloadable software library

numerical expansion methods for solving integral equations by interpolation and gauss quadrature rules in this paper we introduce a numerical method for solving linear integral equations the main idea based on interpolation for unknown function where is interpolated in the zeros of the chebyshev s polynomials next we use gauss quadrature rules as gauss chebyshev or clenshaw curtis the technique is very effective and simple specially for integral equations of first kind as fredholm s and volterra s types in the end for showing efficiency of this method we use numerical examples

finite element analysis of magnetohydrodynamic pipe flow abstract this paper deals with magnetohydrodynamic flows in pipes with arbitrary cross section and arbitrary wall conductivities under the influence of a transverse magnetic field the equations studied are of considerable practical interest since they model induction flow meters electromagnetic pumps magnetic propulsion devices and generators and have application in medicine in power generation and in nuclear reactor technology we are primarily interested in the questions of existence uniqueness and finite element approximation of solutions to the equations of steady state magnetohydrodynamics with arbitrary boundary conditions which describe this phenomenon we derive error estimates for the approximate solution and show that one obtains optimal accuracy when using the finite element method to construct approximate solutions we illustrate these results with some numerical experiments

interpolating wavelet collocation method of time dependent maxwell s equations characterization of electrically large optical waveguide discontinuities biorthogonal interpolating wavelets have been applied to electromagnetic field modeling through the wavelet collocation method in time domain yielding a versatile first principle algorithm for the solution of time dependent maxwell s equations with inhomogeneous media the resulting scheme maintains high accuracy while by virtue of its sub gridding capability significant reduction of the computational expenditure has been obtained the proposed method has been applied to the analysis of two dimensional dielectric waveguide discontinuities particularly for the modeling of electrically large optical waveguides where the dimension of the analyzed structure is much larger than the wavelength of the highest frequency content of the transmitted signal the proposed method has been proven to be highly efficient compared to the standard finite difference method

fourier spectral simulations and gegenbauer reconstructions for electromagnetic waves in the presence of a metal nanoparticle we describe fourier pseudospectral time domain simulations carried out in order to study light interacting with a metallic nanoscale object the difficulty of using fourier methods to accurately predict the electromagnetic scattering in such problems arises from the discontinuity in the dielectric function along the surface of the metallic object standard fourier methods lead to oscillatory behavior in approximating solutions that are nonsmooth or that have steep gradients by applying the gegenbauer reconstruction technique as a postprocessing method to the fourier pseudospectral solution we successfully reduce the oscillations after postprocessing our computational results including comparison with finite difference time domain simulations demonstrate the efficiency and accuracy of the method

a fast bie iteration method for an arbitrary body in a flow of incompressible inviscid fluid the paper is concerned with the new iteration algorithm to solve boundary integral equations arising in boundary value problems of mathematical physics the stability of the algorithm is demonstrated on the problem of a flow around bodies placed in the incompressible inviscid fluid with a discrete numerical treatment we approximate the exact matrix by a certain toeplitz one and then apply a fast algorithm for this matrix on each iteration step we illustrate the convergence of this iteration scheme by a number of numerical examples both for hard and soft boundary conditions it appears that the method is highly efficient for hard boundaries being much less efficient for soft boundaries

very high order compact finite difference schemes on non uniform grids for incompressible navier stokes equations this article presents a family of very high order non uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible navier stokes equations the high order compact schemes on non uniform grids developed in shukla and zhong r k shukla x zhong derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation j comput phys 204 2005 404 for linear model equations are extended to the full navier stokes equations in the vorticity and streamfunction formulation two methods for the solution of helmholtz and poisson equations using high order compact schemes on non uniform grids are developed the schemes are constructed so that they maintain a high order of accuracy not only in the interior but also at the boundary second order semi implicit temporal discretization is achieved through an implicit backward differentiation scheme for the linear viscous terms and an explicit adam bashforth scheme for the non linear convective terms the boundary values of vorticity are determined using an influence matrix technique the resulting discretized system with boundary closures of the same high order as the interior is shown to be stable when applied to the two dimensional incompressible navier stokes equations provided enough grid points are clustered at the boundary the resolution characteristics of the high order compact finite difference schemes are illustrated through their application to the one dimensional linear wave equation and the two dimensional driven cavity flow comparisons with the benchmark solutions for the two dimensional driven cavity flow thermal convection in a square box and flow past an impulsively started cylinder show that the high order compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary

on the superconvergence of galerkin methods for hyperbolic ibvp finite element galerkin methods using b splines of order r for periodic first order hyperbolic equations exhibit superconvergence on uniform grids mesh size h at the nodes i e there is an error estimate cal o h 2r instead of the expected convergence rate cal o h r in this paper it will be shown that no matter how the approximating subspace s h is modified in a boundary layer 0 s 1 h s arbitrary but fixed the superconvergence property is lost for the hyperbolic model problem u t u x 0 leq x infty t geq 0 we shall also discuss the implications of this result when constructing compact implicit difference schemes

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

convergence of an adaptive semi lagrangian scheme for the vlasov poisson system an adaptive semi lagrangian scheme for solving the cauchy problem associated to the periodic 1 1 dimensional vlasov poisson system in the two dimensional phase space is proposed and analyzed a key feature of our method is the accurate evolution of the adaptive mesh from one time step to the next one based on a rigorous analysis of the local regularity and how it gets transported by the numerical flow the accuracy of the scheme is monitored by a prescribed tolerance parameter e which represents the local interpolation error at each time step in the l metric the numerical solutions are proved to converge in l towards the exact ones as e and t tend to zero provided the initial data is lipschitz and has a finite total curvature or in other words that it belongs to w 1 infty cap w 2 1 the rate of convergence is mathcal o delta t 2 varepsilon delta t which should be compared to the results of besse who recently established in siam j numer anal 42 1 350 382 2004 similar rates for a uniform semi lagrangian scheme but requiring that the initial data are in mathcal c 2 several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive discretizations

solving hyperbolic pdes using interpolating wavelets a method is presented for adaptively solving hyperbolic pdes the method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids the adaptability is performed automatically by thresholding the wavelet coefficients operations such as differentiation and multiplication are fast and simple due to the one to one correspondence between point values and wavelet coefficients in the interpolating basis treatment of boundary conditions is simplified in this sparse point representation spr numerical examples are presented for one and two dimensional problems it is found that the proposed method outperforms a finite difference method on a uniform grid for certain problems in terms of flops

multiresolution schemes for the numerical solution of 2 d conservation laws i a generalization of harten s multiresolution algorithms to two dimensional 2 d hyperbolic conservation laws is presented given a cartesian grid and a discretized function on it the method computes the local scale components of the function by recursive diadic coarsening of the grid since the function s regularity can be described in terms of its scale or multiresolution analysis the numerical solution of conservation laws becomes more efficient by eliminating flux computations wherever the solution is smooth instead in those locations the divergence of the solution is interpolated from the next coarser grid level first the basic 2 d essentially nonoscillatory eno scheme is presented then the 2 d multiresolution analysis is developed and finally the subsequent scheme is tested numerically the computational results confirm that the efficiency of the numerical scheme can be considerably improved in two dimensions as well

on the convergence of the fourier approximation for eigenvalues and eigenfunctions of discontinuous problems in this paper we consider a model eigenvalue problem with discontinuous coefficients in order to study the convergence of the fourier methods applied to this problem we prove that the rate of convergence of the fourier galerkin method is third order for the eigenvalues and order 2 5 for the eigenfunctions for the fourier collocation method we obtained only second order accuracy we also show that the fourier collocation method can be improved by a preprocessing of the coefficients the theory is confirmed by numerical results

an adaptive meshfree diffusion wavelet method for partial differential equations on the sphere abstract an adaptive meshfree diffusion wavelet method for solving partial differential equations pdes on the sphere is developed approximation formulae for laplacian beltrami 2 and gradient operators are derived using radial basis functions rbfs and the convergence of these approximations to 2 and is verified for two test functions the matrix approximating the laplace beltrami operator is used for the construction of the diffusion wavelet the diffusion wavelet is used for the adaptation of node arrangement as well as for the fast computation of dyadic powers of the matrices involved in the numerical solution of the pde the efficiency of the method is that the same operator is used for the construction of the diffusion wavelet and for the approximation of the differential operators as a part of the wavelet method the behaviour of the compression error with respect to different parameters involved in the construction of the diffusion wavelet is tested the cpu time taken by the proposed method is compared with the cpu time taken by the rbf based collocation method and it is observed that the proposed method performs better the method is tested on three test problems namely spherical diffusion equation linear problem of computing a moving steep front nonlinear and problem of turing patterns system of nonlinear reaction diffusion equations

time splitting finite difference method with the wavelet adaptive grids for semiclassical gross pitaevskii equation in supercritical case abstract the gross pitaevskii equation is the model equation of the single particle wave function in a bose einstein condensation a computation difficulty of the gross pitaevskii equation comes from the semiclassical problem in supercritical case in this paper we apply a diffeomorphism to transform the original one dimensional gross pitaevskii equation into a modified equation the adaptive grids are constructed through the interpolating wavelet method then we use the time splitting finite difference method with the wavelet adaptive grids to solve the modified gross pitaevskii equation where the approximation to the second order derivative is given by the lagrange interpolation method at last the numerical results are given it is shown that the obtained time splitting finite difference method with the wavelet adaptive grids is very efficient for solving the one dimensional semiclassical gross pitaevskii equation in supercritical case and it is suitable to deal with the local high oscillation of the solution

a bubble stabilized least squares finite element method for steady mhd duct flow problems at high hartmann numbers in this paper we devise a stabilized least squares finite element method using the residual free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow we convert the original system of second order partial differential equations into a first order system formulation by introducing two additional variables then the least squares finite element method using c 0 linear elements enriched with the residual free bubble functions for all unknowns is applied to obtain approximations to the first order system the most advantageous features of this approach are that the resulting linear system is symmetric and positive definite and it is capable of resolving high gradients near the layer regions without refining the mesh thus this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the hartmann number before incoorperating the bubble functions into the global problem we apply the galerkin least squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements therefore we indeed introduce a two level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual free bubble functions numerical results confirming theoretical findings are presented for several examples including the shercliff problem

a wavelet method for solving singular integral equation of mhd this paper presents haar wavelet approximation to solve a singular integral equation which has singularities on a diagonal of the domain r 1 1 x 1 1 the singularities arise basically due to modified bessel function k 0 which appears as a part of the kernel thus the integral equation is weakly logarithmically singular only the problem is solved considering all the singularities of the kernel and results are examined for approximations of different orders our interest to solve the problem using haar wavelet basis is due to its simplicity and efficiency in numerical approximation the results show convergence trend as mesh is refined comparison is made with some available results obtained earlier by partial consideration of the singularities

finite element method for solving mhd flow in a rectangular duct a finite element method is given to obtain the solution in terms of velocity and induced magnetic field for the steady mhd magnetohydrodynamic flow through a rectangular pipe having arbitrarily conducting walls linear and then quadratic approximations have been taken for both velocity and magnetic field for comparison and it is found that with the quadratic approximation it is possible to increase the conductivity and hartmann number m m 100 a special solution procedure has been used for the resulting block tridiagonal system of equations computations have been carried out for several values of hartmann number 5 m 100 and wall conductivity it is also found that if the wall conductivity increases the flux decreases the same is the effect of increasing the hartmann number selected graphs are given showing the behaviour of the velocity field and induced magnetic field

fundamental solution for coupled magnetohydrodynamic flow equations in this paper a fundamental solution for the coupled convection diffusion type equations is derived the boundary element method bem application then is established with this fundamental solution for solving the coupled equations of steady magnetohydrodynamic mhd duct flow in the presence of an external oblique magnetic field thus it is possible to solve mhd duct flow problems with the most general form of wall conductivities and for large values of hartmann number the results for velocity and induced magnetic field is visualized in terms of graphics for values of hartmann number m 300

a high accuracy spectral method based on min max principle for biharmonic eigenvalue problems on a spherical domain abstract in this study we develop a high accuracy spectral method based on the min max principle for biharmonic eigenvalue problems in the spherical domain by analyzing the orthogonal spherical harmonic and approximation and using the min max principle we first deduce the error estimates of approximate eigenvalues then we construct an appropriate set of orthogonal spherical functions contained in h 0 2 and establish the matrix formulations for the discrete variational form whose mass matrix and stiff matrix are all sparse so that we can solve the numerical solutions efficiently finally we provide some numerical experiments to validate that the theoretical results are correct

a multiresolution finite volume scheme for two dimensional hyperbolic conservation laws in this paper high resolution finite volume schemes are combined with an adaptive mesh technique inspired by multiresolution analysis to improve the computational efficiency for two dimensional hyperbolic conservation laws the method is conservative moreover it is stable which is proven numerically in this paper the computational grid is dynamically adapted so that higher spatial resolution is automatically allocated to regions where strong gradients are observed using this proposed scheme we compute several two dimensional model problems and a compressive rate ranging from about 5 10 is observed in all simulations

multiresolution schemes on triangles for scalar conservation laws this paper proposes a multiresolution procedure adapted to triangular cell averages to improve the performance of finite volume schemes by reducing flux evaluation cost using the approach introduced by a harten a specific coarse to fine prediction scheme is proposed that ensures the stability of the computations even when a large number of scales are involved numerical tests are presented that illustrate the computational gain as well as the order of accuracy of the scheme

numerical simulation of shock cylinder interactions i resolution abstract we apply two different high order shock capturing schemes to the study of a two dimensional unsteady inviscid flow in particular we study the interaction of a planar shock with o cylindrical volume of a light gas helium or hydrogen contained in air the two schemes used are the chebyshev collocation method and the eno finite difference scheme of osher and shu and they are applied to a physical model consisting of the euler equations with a real gas equation of state and multiple chemical species the parallel implementation and low level coding of the eno scheme on the thinking machines cm 5 results in much higher performance than is possible on a standard serial or vector machine the eno code is compared with an existing experimental result and agrees well with it the results of spectral and eno calculations are then compared with each other at different resolutions for a mach 2 interaction the spectral scheme though highly oscillatory in nature for discontinuous problems gibbs accurately predicts both large and fine scale structures of the interaction between the shock and the light gas cylinder good results can be recovered from the spectral results by post processing the raw numerical data to remove the gibbs phenomena these results are compared with the eno schemes the comparison is progressively better as the grid refinement and numerical order of the eno scheme is increased this demonstrates definitively the applicability and value of high order schemes to flows with shocks and complicated non linear physics

adaptive multiresolution schemes for shock computations in this paper we present adaptive multiresolution schemes for the computation of discontinuous solutions of hyperbolic conservation laws starting with the given grid we consider the grid averages of the numerical solution for a hierarchy of nested grids which is obtained by diadic coarsening and we compute its equivalent multiresolution representation this representation of the numerical solution consists of the grid averages of the coarsest grid and the set of errors in predicting the grid averages of each level of resolution from the next coarser one these errors depend on the size of the grid and the local regularity of the solution at a jump discontinuity they remain the size of the jump independent of the level of resolution this observation enables us to identify the location of discontinuities in the numerical solution in a region of smoothness once the numerical solution is resolved to our satisfaction at a certain locality of some grid then the prediction errors there for this grid and all finer ones are small this enables us to obtain data compression by setting to zero the terms of the multiresolution representation that fall below a specified tolerance the numerical flux of the adaptive scheme is taken to be that of a standard centered scheme unless it corresponds to an identified discontinuity in which case it is taken to be the flux of an eno scheme we use the data compression of the numerical solution in order to reduce the number of numerical flux evaluations as follows we start with the computation of the exact numerical fluxes at the few grid points of the coarsest grid and then we proceed through diadic refinement to the given grid at each step of refinement we add values for the numerical flux at the new grid points which are the centers of the coarser cells wherever the solution is locally well resolved i e the corresponding prediction error is below the specified tolerance the costly exact value of the numerical flux function is replaced by an accurate enough approximate value which is obtained by an inexpensive interpolation from the values of the coarser grid

approximation of the laplace and stokes operators with dirichlet boundary conditions through volume penalization a spectral viewpoint we report the results of a study on the spectral properties of laplace and stokes operators modified with a volume penalization term designed to approximate dirichlet conditions in the limit when a penalization parameter eta tends to zero the eigenvalues and eigenfunctions are determined either analytically or numerically as functions of eta both in the continuous case and after applying fourier or finite difference discretization schemes for fixed eta we find that only the part of the spectrum corresponding to eigenvalues lambda lesssim eta 1 1 approaches dirichlet boundary conditions while the remainder of the spectrum is made of uncontrolled spurious wall modes the penalization error for the controlled eigenfunctions is estimated as a function of eta and lambda surprisingly in the stokes case we show that the eigenfunctions approximately satisfy with a precision o eta o navier slip boundary conditions with slip length equal to sqrt eta moreover for a given discretization we show that there exists a value of eta corresponding to a balance between penalization and discretization errors below which no further gain in precision is achieved these results shed light on the behavior of volume penalization schemes when solving the navier stokes equations outline the limitations of the method and give indications on how to choose the penalization parameter in practical cases

a numerical strategy to discretize and solve the poisson equation on dynamically adapted multiresolution grids for time dependent streamer discharge simulations we develop a numerical strategy to solve multi dimensional poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts the method is an extension of the multiresolution finite volume scheme used to solve hyperbolic and parabolic time dependent pdes such an approach guarantees a numerical solution of the poisson equation within a user defined accuracy tolerance most adaptive meshing approaches in the literature solve elliptic pdes level wise and hence at uniform resolution throughout the set of adapted grids here we introduce a numerical procedure to represent the elliptic operators on the adapted grid strongly coupling inter grid relations that guarantee the conservation and accuracy properties of multiresolution finite volume schemes the discrete poisson equation is solved at once over the entire computational domain as a completely separate process the accuracy and numerical performance of the method are assessed in the context of streamer discharge simulations