energy based comparison between the fourier galerkin method and the finite element method abstract the fourier galerkin method in short ffth has gained popularity in numerical homogenisation because it can treat problems with a huge number of degrees of freedom because the method incorporates the fast fourier transform fft in the linear solver it is believed to provide an improvement in computational and memory requirements compared to the conventional finite element method fem here we systematically compare these two methods using the energetic norm of local fields which has the clear physical interpretation as being the error in the homogenised properties this enables the comparison of memory and computational requirements at the same level of approximation accuracy we show that the methods effectiveness relies on the smoothness regularity of the solution and thus on the material coefficients thanks to its approximation properties fem outperforms ffth for problems with jumps in material coefficients while ambivalent results are observed for the case that the material coefficients vary continuously in space ffth profits from a good conditioning of the linear system independent of the number of degrees of freedom but generally needs more degrees of freedom to reach the same approximation accuracy more studies are needed for other fft based schemes non linear problems and dual problems which require special treatment in fem but not in ffth

a reconstruction based chebyshev collocation method for the poisson equation an accurate treatment of the gibbs wilbraham phenomenon on irregular interfaces abstract an accurate numerical solution of the laplace equation and poisson equation is desirable in physical problems related to fluid flows and spectral methods are well suited towards this goal however in the presence of an interface the solution is discontinuous owing to the interface jump conditions and thus the numerical approximation is susceptible to the gibbs wilbraham phenomenon leading to a reduced solution accuracy in order to resolve the discontinuity at the interface we propose a reconstruction technique wherein the approximate solution is expressed as the sum of an infinitely differentiable smooth function and a modified heaviside function the smooth function is composed of chebyshev polynomials while the modified heaviside function is expressed in a weak form using the jump conditions across the interface the reconstruction framework allows us to impose the interfacial conditions exactly to motivate further application of the technique we first present the underlying rationale by revisiting the work of wilbraham 1848 13 followed by the numerical implementation we present numerical solution of four equations to illustrate the application of the proposed technique i an ordinary differential equation ode ii the euler bernoulli equation iii the laplace equation and iv the poisson equation the laplace and poisson equation are solved on a square domain embedded with complex interfacial geometries in two dimensions and on a cube with a spherical interface in three dimensions the method achieves error in the maximum norm o 10 14 using approximately 30 modes in each direction and requires a computational time of less than a second for all the problems we also conduct a sensitivity analysis to demonstrate the robustness of the algorithm under sub grid scale perturbations to interface location an alternate derivation for the modified heaviside function is proposed that is distinct from the taylor series expansion used to treat interface conditions in difference methods the pointwise error plots are subsequently presented to demonstrate that the gibbs wilbraham phenomenon is accurately resolved finally we present concluding thoughts on the potential applications of the reconstruction technique

mappings and accuracy for chebyshev pseudo spectral approximations abstract the effect of mappings on the approximation by chebyshev collocation of functions which exhibit localized regions of rapid variation is studied a general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials a particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in applications it is shown that the mapped function can be approximated much more accurately by chebyshev polynomial approximations than the original function the effect on the approximation of introducing subdomains is also studied the accuracy of the pseudo spectral approximation is very sensitive to the location of the interface although this sensitivity is reduced when mappings are employed within the subdomains

a strategy for choosing gegenbauer reconstruction parameters for numerical stability the gegenbauer reconstruction method was first proposed in 1992 but in early studies no attempts were made to optimize the relevant parameters of this method these parameters were allowed to grow proportionally with the number of nodes which in many cases resulted in exponential convergence for a selected range of the proportionality constants early studies also made clear that very large error bounds could be expected if these key parameters were not chosen carefully subsequent studies then pointed out that although unrelated to the method s analytically predictable domains of poor accuracy round off errors could also sabotage the method s accuracy the challenge of successfully implementing a gegenbauer reconstruction then rests on understanding the performance trade offs we can expect when choosing the key parameters in accordance with different objectives in this study we propose a new strategy for choosing optimal parameters in the chebyshev gegenbauer reconstruction method specifically to achieve numerical stability this strategy is based on asymptotic analysis as well as minimization problems in one and two dimensions the effectiveness of our approach which could also be applied to a wider selection of polynomials is then illustrated with results from numerical experiments

flux corrected pseudospectral method for scalar hyperbolic conservation laws abstract the pseudospectral method has under used advantages in problems involving shocks and discontinuities these emerge from superior accuracy in phase and group velocities as compared to finite difference schemes of all orders dispersion curves for finite difference schemes suggest that group velocity error typically outranks gibbs error as a cause of numerical oscillation a flux conservative form of the pseudospectral method is derived for compatibility with flux limiters used to preserve monotonicity the resulting scheme gives high quality results in linear advection and shock formation propagation examples

cell averaging chebyshev methods for hyperbolic problems this paper describes a cell averaging method for the chebyshev approximations of first order hyperbolic equations in conservation form we present formulas for transforming between pointwise data at the collocation points and cell averaged quantities and vice versa this step trivial for the finite difference and fourier methods is nontrivial for the global polynomials used in spectral methods we then prove that the cell averaging methods presented are stable for linear scalar hyperbolic equations and present numerical simulations of shock density wave interaction using the new cell averaging chebyshev methods

essentially nonoscillatory spectral fourier methods for shock wave calculations in this paper we present an essentially nonoscillatory spectral fourier method for the solution of hyperbolic partial differential equations the method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the fourier method the high accuracy away from the shock is enhanced by using filters numerical results confirm that essentially no oscillations develop in the solution also the accuracy of the spectral solution of the inviscid burgers equation is shown to be higher than a fixed order

asymptotic analysis for singularly perturbed convection diffusion equations with a turning point turning points occur in many circumstances in fluid mechanics when the viscosity is small very complex phenomena can occur near turning points which are not yet well understood a model problem corresponding to a linear convection diffusion equation e g suitable linearization of the navier stokes or benard convection equations is considered our analysis shows the diversity and complexity of behaviors and boundary or interior layers which already appear for our equations simpler than the navier stokes or benard convection equations of course the diversity and complexity of these structures will have to be taken into consideration for the study of the nonlinear problems in our case at this stage the full theoretical asymptotic analysis is provided this study is totally new to the best of our knowledge numerical treatment and more complex problems will be considered elsewhere

random convex programs random convex programs rcps are convex optimization problems subject to a finite number n of random constraints the optimal objective value j of an rcp is thus a random variable we study the probability with which j is no longer optimal if a further random constraint is added to the problem violation probability v it turns out that this probability rapidly concentrates near zero as n increases we first develop a theory for rcps leading to explicit bounds on the upper tail probability of v then we extend the setup to the case of rcps with r a posteriori violated constraints rcpvs a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control explicit and nonasymptotic bounds are derived also in this case the upper tail probability of v is upper bounded by a multiple of a beta distribution irrespective of the distribution on the random constraints all results are derived under no feasibility assumptions on the problem further the relation between rcpvs and chance constrained problems ccp is explored showing that the optimal objective j of an rcpv with the generic constraint removal rule provides with arbitrarily high probability an upper bound on the optimal objective of a corresponding ccp moreover whenever an optimal constraint removal rule is used in the rcpvs then appropriate choices of n and r exist such that j approximates arbitrarily well the objective of the ccp

approximation of the discontinuities of a function by its classical orthogonal polynomial fourier coefficients in the present paper we generalize the method suggested in an earlier paper by the author and overcome its main deficiency first we modify the well known prony method which subsequently will be utilized for recovering exactly the locations of jump discontinuities and the associated jumps of a piecewise constant function by means of its fourier coefficients with respect to any system of the classical orthogonal polynomials next we will show that the method is applicable to a wider class of functions namely to the class of piecewise smooth functions for functions which piecewise belong to c 2 1 1 the locations of discontinuities are approximated to within o 1 n by means of their fourier jacobi coefficients unlike the previous one the generalized method is robust since its success is independent of whether or not a location of the discontinuity coincides with a root of a classical orthogonal polynomial in addition the error estimate is uniform for any c d c 1 1 to the end we discuss the accuracy stability and complexity of the method and present numerical examples

classroom note converting matrix riccati equations to second order linear ode the matrix riccati equation may be transformed into the system of linear second order ordinary differential equations odes known in differential geometry as the jacobi equation generalizing a familiar result in the scalar case

gegenbauer approximation in certain hilbert spaces and its applications to singular differential equations gegenbauer approximation in certain hilbert spaces is investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using gegenbauer polynomials this method keeps the spectral accuracy the main idea and techniques used in this paper are also applicable to some singular problems in multiple dimensional spaces

detecting the singularities of a function of vp class by its integrated fourier series in the present paper we pursue the general idea suggested in our previous work namely we utilize the truncated fourier series as a tool for the approximation of the points of discontinuities and the magnitudes of jumps of a 2 periodic bounded function earlier we used the derivative of the partial sums while in this work we use integrals first we obtain new identities which determine the jumps of a 2 periodic function of vp 1 p 2 class with a finite number of discontinuities by means of the tails of its integrated fourier series next based on these identities we establish asymptotic expansions for the approximations of the location of the discontinuity and the magnitude of the jump of a 2 periodic piecewise smooth function with one singularity by an appropriate linear combination obtained via integrals of different order we significantly improve the accuracy of the initial approximations then we apply richardson s extrapolation method to enhance the approximation results for a function with multiple discontinuities we use simple formulae which eliminate all discontinuities of the function but one then we treat the function as if it had one singularity finally we give the description of a programmable algorithm for the approximation of the discontinuities investigate the stability of the method study its complexity and present some numerical results

approximating the jump discontinuities of a function by its fourier jacobi coefficients in the present paper we generalize eckhoff s method i e the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its fourier chebyshev coefficients a new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function which belongs to a restricted class of the piecewise smooth functions by means of its fourier jacobi coefficients for arbitrary indices approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations it is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities in addition we study the accuracy of the approximations and present some numerical examples

spectral methods for the euler equations ii chebyshev methods and shock fitting on exprime chaque composante de la vitesse dans les equations d euler par une somme de polynomes de tchebycheff qui constitue une fonction d interpolation on utilise les expressions spectrales ainsi obtenues pour calculer plusieurs ecoulements gazeux bidimensionnels par la methode d ajustement d ondes de choc selon cette methode les conditions initiales correspondent a une onde de choc plane normale a un ecoulement supersonique et on determine la forme veritable de l onde a l aide des formules de rankine hugoniot qui expriment les conditions aux limites

pseudospectral solution of near singular problems using numerical coordinate transformations based on adaptivity the work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions the approach makes use of the adaptive finite difference method presented by huang and sloan siam j sci comput 15 1994 pp 776 797 this method is applied on a coarse grid to obtain a rough approximation of the solution and a suitable adapted mesh the adaptive finite difference solution permits the construction of a smooth coordinate transformation that relates the computational space to the physical space the map between the spaces is based on chebyshev polynomial interpolation finally the standard pseudospectral ps method is applied to the transformed differential problem to obtain highly accurate nonoscillatory numerical solutions numerical results are presented for steady problems in one and two space dimensions

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

on discontinuous solutions of hyperbolic equations abstract a modified fourier method which is applicable to numerical simulations of discontinuous solutions of hyperbolic equations is presented truncated fourier series representations are de truncated at every timestep by utilising step functions in the reconstruction of the discontinuous solutions the de truncated expansions are used to eliminate the numerical dispersion and diffusion which normally ruin the discontinuous solutions in a few timesteps when traditional fourier methods are applied