spectral convergence for orthogonal polynomials on triangles we study the behavior of orthogonal polynomials on triangles and their coefficients in the context of spectral approximations of partial differential equations in these spectral approximations one studies series expansions u sum k 0 infty hat u k phi k where the phi k are orthogonal polynomials our results show that for any function u in c infty the series expansion converges faster than with polynomial order

on the construction of suitable solutions to the navier stokes equations and questions regarding the definition of large eddy simulation abstract this paper discusses the notion of large eddy simulation les as a mathematical concept and proposes to use the concept of suitable weak solutions as a building block for the development of a mathematical theory of les various techniques for constructing suitable weak solutions to the three dimensional navier stokes equations are reviewed and many features of these mathematical constructions are shown to comply with well accepted heuristic characterizations of les

adaptive spectral viscosity for hyperbolic conservation laws spectral approximations to nonlinear hyperbolic conservation laws require dissipative regularization for stability the dissipative mechanism must on the other hand be small enough in order to retain the spectral accuracy in regions where the solution is smooth we introduce a new form of viscous regularization which is activated only in the local neighborhood of shock discontinuities the basic idea is to employ a spectral edge detection algorithm as a dynamical indicator of where in physical space to apply numerical viscosity the resulting spatially local viscosity is successfully combined with spectral viscosity where a much higher than usual cut off frequency can be used numerical results show that the new adaptive spectral viscosity scheme significantly improves the accuracy of the standard spectral viscosity scheme in particular results are improved near the shocks and at low resolutions examples include numerical simulations of burgers equation shallow water with bottom topography and the isothermal euler equations we also test the schemes on a nonconvex scalar problem finding that the new scheme approximates the entropy solution more reliably than the standard spectral viscosity scheme

a spectral vanishing viscosity for the les of turbulent flows within rotating cavities a spectral vanishing viscosity technique svv is presented for the simulation of 3d turbulent incompressible flows within a rotor stator cavity one characteristic of this technique is that the svv is active only for the short length scales a feature which is reminiscent of large eddy simulation models the spectral vanishing viscosity first introduced by e tadmor for the inviscid burgers equation e tadmor convergence of spectral methods for nonlinear conservation laws siam j numer anal 26 1 1989 30 is incorporated into the cylindrical navier stokes equations written in velocity pressure formulation the second order operator involved in the svv method is implemented in a chebyshev collocation fourier galerkin pseudo spectral code the svv is shown to lead to stable discretizations without sacrificing the formal accuracy i e exponential convergence in the proposed discretization les results are presented here for rotational reynolds numbers ranging from re 7 i 10 4 to re 7 i 10 5 turbulent quantities are shown to compare very favorably with results of direct numerical simulation dns and experimental measurements

the legendre burgers equation when artificial dissipation fails abstract artificial viscosity is a common device for stabilizing flows with shocks and fronts the computational diffusion smears the frontal zone over a small distance where is chosen so that the discretization has a couple of grid points in the front and thus is able to resolve the shock spectral element methods use a legendre spectral viscosity whose effect is to damp the coefficient of p n x by some amount that depends only on the degree n of the legendre polynomial legendre viscosity is better than ordinary diffusion because it does not require spurious boundary conditions does not increase the temporal stiffness of the differential equations and can be applied locally on an element by element basis unfortunately legendre diffusion is equivalent to a diffusion with a spatially varying coefficient that goes to zero at the boundaries using the simplest example one in which the second derivative of burgers equation is replaced by the legendre operator to give the legendre burgers equation u t uu x 1 x 2 u x x we show that the width of the computational front can similarly tend to zero at the endpoints causing a numerical catastrophe

chebyshev legendre super spectral viscosity method for nonlinear conservation laws in this paper a super spectral viscosity method using the chebyshev differential operator of high order d s sqrt 1 x 2 opx s is developed for nonlinear conservation laws the boundary conditions are treated by a penalty method compared with the second order spectral viscosity method the super one is much weaker while still guaranteeing the convergence of the bounded solution of the chebyshev galerkin chebyshev collocation or legendre galerkin approximations to nonlinear conservation laws which is proved by compensated compactness arguments

spectral simulation of supersonic reactive flows we present in this paper numerical simulations of reactive flows interacting with shock waves we argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long term integrations of complicated flows even in the presence of shock waves a spectral code is described in detail and the theory that leads to each of its components is explained results of interactions of hydrogen jets with shock waves are presented and analyzed and comparisons with eno finite difference schemes are carried out

the use of a legendre pseudospectral viscosity technique to solve a class of nonlinear dynamic hamilton jacobi equations in this research we study the problem of finding the approximate solution of a class of hamilton jacobi equations namely the eikonal equation we employ the legendre pseudospectral viscosity method to solve this problem this method basically consists of adding a spectral viscosity to the equation this spectral viscosity which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme several test problems are considered and the numerical results are given to show the efficiency of the proposed method

filtering non periodic functions abstract we consider a non periodic function u with legendre expansion k 0 uk lk the best approximation of u in the l2 sense by n degree polynomials is the truncated series nu nk 0 uk lk when the function u is discontinuous the polynomial approximation nu is oscillatory we present a filter that acts on nu to give a good approximation of u

a spectral viscosity method for correcting the long term behavior of pod models low dimensional flow dynamical systems may converge to erroneous states after long time integration even if they are initialized with the correct state in this paper we investigate the accuracy of such two dimensional models constructed from karhunen loeve expansions for flows past a circular cylinder we first demonstrate that although the short term dynamics may be predicted accurately with only a handful of modes retained drifting of the solution may arise after a few hundred vortex shedding cycles we then propose a dissipative model based on a spectral viscosity sv diffusion convolution operator the parameters of the sv model are selected rigorously based on bifurcation analysis our results show that this is an effective way of improving the accuracy of long term predictions of low dimensional galerkin systems

a spectral viscosity method based on hermite functions for nonlinear conservation laws we consider the approximation by a spectral method of the solution of the cauchy problem for a scalar conservation law in one dimension posed in the whole real line we analyze a spectral viscosity method in which the orthogonal basis considered is the one of hermite functions we prove the convergence of the approximate solution to the unique entropy solution of the problem by using compensated compactness arguments

parallel depth first search in general directed graphs a directed cycle separator of an n vertex directed graph is a vertex simple directed cycle such that when the vertices of the cycle are deleted the resulting graph has no strongly connected component with more than n 2 vertices it is shown that the problem of finding a directed cycle separator is in randomized nc it is also proved that computing cycle separators and conducting depth first search in directed graphs are deterministically nc equivalent these two results together yield the first randomized nc algorithm for depth first search in general directed graphs

fractional burgers equation with nonlinear non locality we consider the viscous burgers equation with a fractional nonlinear term as a model involving non local nonlinearities in conservation laws which surprisingly has an analytical solution obtained by a fractional extension of the hopfcole transformation we use this model and its inviscid limit to develop stable spectral and discontinuous galerkin spectral element methods by employing the concept of spectral vanishing viscosity svv for the global spectral method svv is very effective and the computational cost is o n2 which is essentially the same as for the standard burgers equation we also develop a local discontinuous galerkin ldg spectral element method to improve the accuracy around discontinuities and we again stabilize the ldg method with the svv operator finally we solve numerically the inviscid fractional burgers equation both with the spectral and the spectral element ldg methods we study systematically the stability and convergence of both methods and determine the effectiveness of each method for different parameters

a globally convergent newton gmres subspace method for systems of nonlinear equations newton krylov methods are variants of inexact newton methods where the approximate newton direction is taken from a subspace of small dimension here we introduce a new hybrid newton gmres method where a global strategy restricted to a low dimensional subspace generated by gmres is performed the obtained process is consistent with preconditioning and with matrix free implementation computational results indicate that our proposal enhances the classical backtracking inexact method

filtering in legendre spectral methods we discuss the impact of modal filtering in legendre spectral methods both on accuracy and stability for the former we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity computational results confirm that less strict necessary conditions appear to be adequate we proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability the results are illustrated by computational experiments

a constrained optimization approach to solving certain systems of convex equations abstract this research presents a new constrained optimization approach for solving systems of nonlinear equations particular advantages are realized when all of the equations are convex for example a global algorithm for finding the zero of a convex real valued function of one variable is developed if the algorithm terminates finitely then either the algorithm has computed a zero or determined that none exists if an infinite sequence is generated either that sequence converges to a zero or again no zero exists for solving n dimensional convex equations the constrained optimization algorithm has the capability of determining that the system of equations has no solution global convergence of the algorithm is established under weaker conditions than previously known and in this case the algorithm reduces to newton s method together with a constrained line search at each iteration it is also shown how this approach has led to a new algorithm for solving the linear complementarity problem

the efficient implementation of a finite element multi resolution viscosity method for hyperbolic conservation laws it is well known that the classic galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the euler equations for compressible flows adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use an elegant solution developed in the context of spectral methods by eitan tadmor and coworkers is to add diffusion only to the high frequency modes of the solution and can lead to stabilization without sacrificing accuracy we incorporate this idea into the finite element framework by using hierarchical functions as a multi frequency basis the result is a new finite element method for solving hyperbolic conservation laws for this method convergence for a one dimensional scalar conservation law has previously been proved here the method is described in detail several issues connected with its efficient implementation are considered and numerical results for several examples involving one and two dimensional hyperbolic conservation laws are provided several advantageous features of the method are discussed including the ease for which discontinuities can be detected and artificial diffusion can be applied anisotropically and locally in physical as well as frequency space

stabilized spectral element computations of high reynolds number incompressible flows the possibility of using the spectral vanishing viscosity method for the spectral element computation of high reynolds number incompressible flows is investigated an exponentially accurate stabilized formulation is proposed and then applied to the computation of the 2d wake of a cylinder such a formulation call be easily implemented in existing spectral element solvers since only modifying the computation of the viscous term while preserving the symmetry of the corresponding bilinear form

a finite improvement algorithm for the linear complementarity problem abstract this paper is concerned with the development of an improvement algorithm for the linear complementarity problem lcp our approach to find a solution to lcp is to solve the equivalent constrained optimization problem cop of maximizing the sum of the minimum of each complementary pair of variables subject to the constraints that each such minimum is nonpositive the algorithm ascent in nature is similar to the simplex method in the sense that it moves between basic points of an associated system of linear equations these basic points are feasible to our cop whose objective function is piecewise linear and concave classes of matrices are characterized for which our algorithm processes lcp for every right hand side vector and every matrix in the class a computational study shows that our algorithm is clearly superior to a previos improvement algorithm computational comparisons with lemke s well known algorithm are also presented

detection of edges in spectral data ii nonlinear enhancement we discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities where f x f x f x neq 0 our approach is based on two main aspects localization using appropriate concentration kernels and separation of scales by nonlinear enhancement to detect such edges one employs concentration kernels k epsilon cdot depending on the small scale epsilon it is shown that odd kernels properly scaled and admissible in the sense of having small w 1 infty moments of order cal o epsilon satisfy k epsilon f x f x cal o epsilon thus recovering both the location and amplitudes of all edges as an example we consider general concentration kernels of the form k sigma n t sum sigma k n sin kt to detect edges from the first 1 epsilon n spectral modes of piecewise smooth f s here we improve in generality and simplicity over our previous study in a gelb and e tadmor appl comput harmon anal 7 1999 pp 101 135 both periodic and nonperiodic spectral projections are considered we identify in particular a new family of exponential factors sigma exp cdot with superior localization properties the other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges where k epsilon f x sim f x neq 0 and the smooth regions where k epsilon f cal o epsilon sim 0 numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors

adapting the spectral vanishing viscosity method for large eddy simulations in cylindrical configurations during the last decade the spectral vanishing viscosity svv method has been adopted successfully for large eddy type simulations les with high order discretizations in both cartesian and cylindrical coordinate systems for the latter case however previous studies were confined to annular domains in the present work we examine the applicability of svv in cylindrical coordinates to flows in which the axis region is included within the setting of an exponentially convergent spectral element fourier discretization in addition to the standard svv viscosity kernel two modified kernels with enhanced stabilization in the axis region are considered three fluid flow examples are considered including turbulent pipe flow the results on the one hand show a surprisingly small influence of the svv kernel while on the other they reveal the importance of spatial resolution in the axis region

parallel search algorithms for graphs and trees abstract this paper surveys parallel algorithms for tree traversal and graph search techniques the principal model of computation considered here is the parallel random access machine in this model each processor can access any cell of a shared memory in unit time thus the issue of interprocessor communication do not get intermixed with the logical structure of parallel algorithms the paper is organized into four major sections section 2 deals with tree traversals there are three main algorithms that differ completely from one another in approach in absolute terms however all require a specific input form or designed to preprocess the input in desired form section 3 presents parallel algorithms for breadth first search of general graphs there are two major techniques for developing efficient parallel algorithms for breadth first search of a graph one is based on computation of shortest paths in a uniformly positive edge weighted graph the other is based on tree merging technique section 4 is devoted to parallel algorithms for depth first search of graphs considering the importance of depth first search as a general paradigm for algorithmic solution of several graph problem there have been lot of efforts to design efficient parallel algorithms for depth first search but the success so far has been restricted to small classes of graphs like directed acyclic graphs dag or planar graphs for dags there are two different techniques to obtain depth first spanning trees or forests viz lexicographic sorting of paths and tree merging in the case of the planar graph the technique is to recursively divide the input graph into smaller graphs by path separators and then run the algorithm in parallel on smaller graphs produced by the separators section 5 talks about car decomposition search of graphs and efficient parallel algorithms for the problem in fact ear decomposition search is considered as an alternative to depth first search in designing parallel algorithms for complicated graph problems the strategy used for car decomposition of a graph is almost identical to that used for orientation of the edges of a connected bridgeless undirected graph so that the resultant graph is strongly connected

legendre pseudospectral viscosity method for nonlinear conservation laws in this paper the legendre spectral viscosity sv method for the approximate solution of initial boundary value problems associated with nonlinear conservation laws is studied the authors prove that by adding a small amount of sv bounded solutions of the legendre sv method converge to the exact scalar entropy solution the convergence proof is based on compensated compactness arguments and therefore applies to certain 2 times 2 systems finally numerical experiments for scalar as well as the one dimensional system of gas dynamics equations are presented which confirm the convergence of the legendre sv method moreover these numerical experiments indicate that by post processing the sv approximation one can recover the entropy solution within spectral accuracy

stabilisation of spectral hp element methods through spectral vanishing viscosity application to fluid mechanics modelling in this paper we present a formulation of spectral vanishing viscosity svv for the stabilisation of spectral hp element methods applied to the solution of the incompressible navier stokes equations we construct the svv around a filter with respect to an orthogonal expansions and prove that this methodology provides a symmetric semi positive definite svv operator after providing a few simple one and two dimensional examples to demonstrate the utility of the svv we examine how it can be applied to a spectral hp element discretisation of the navier stokes equations using a velocity correction splitting scheme we provide three fluid flow examples to help illustrate the pros and cons of this approach on stability and accuracy 2005 elsevier b v all rights reserved

spatial eigenanalysis of spectral hp continuous galerkin schemes and their stabilisation via dg mimicking spectral vanishing viscosity for high reynolds number flows abstract this study considers the spatial eigensolution analysis of spectral hp continuous galerkin cg schemes complementing a recent work by moura et al 2016 15 which addressed cg s temporal analysis while the latter assumes periodic boundary conditions the spatial approach presumes inflow outflow type conditions and therefore provides insights for a different class of problems the linear advection diffusion problem is considered for a wide range of peclet numbers allowing for viscous effects at different intensities the inviscid linear advection case receives particular attention owing to the manifestation of peculiar characteristics previously observed for discontinuous galerkin dg schemes in the limit of strong over upwinding these effects are discussed in detail due to their potential to negatively affect solution quality and numerical stability of under resolved simulations at high reynolds numbers the spectral vanishing viscosity svv technique is subsequently considered as a natural stabilization strategy in the context of linear advection an optimization procedure is employed to match svv diffusion levels to those of dg at appropriate polynomial orders the resulting cg svv discretisations are tested against under resolved computations of spatially developing vortex dominated flows and display excellent robustness at high reynolds numbers along with superior eddy resolving characteristics at higher polynomial orders this highlights the importance of appropriate stabilization techniques to improve the potential of spectral hp cg methods for high fidelity simulations of transitional and turbulent flows including implicit les under resolved dns approaches

a spectral vanishing viscosity method for large eddy simulations a new simulation approach for high reynolds number turbulent flows is developed combining concepts of monotonicity in nonlinear conservation laws with concepts of large eddy simulation the spectral vanishing viscosity svv first introduced by e tadmor siam j numer anal 26 30 1989 is incorporated into the navier stokes equations for controlling high wavenumber oscillations unlike hyperviscosity kernels the svv approach involves a second order operator which can be readily implemented in standard finite element codes in the work presented here discretization is performed using hierarchical spectral hp methods accommodating effectively an ab initio intrinsic scale separation the key result is that monotonicity is enforced via svv leading to stable discretizations without sacrificing the formal accuracy i e exponential convergence in the proposed discretization several examples are presented to demonstrate the effectiveness of the new approach including a comparison with eddy viscosity spectral les of turbulent channel flow in its current implementation the svv approach for controlling the small scales is decoupled from the large scales but a procedure is proposed that will provide coupling similar to the classical les formulation

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

sparse low energy decomposition for viscous conservation laws for viscous conservation laws solutions contain smooth but high contrast features which require the use of fine grids to properly resolve on coarse grids these high contrast jumps resemble shocks rather than their true viscous profiles which could lead to issues in the numerical approximation of their underlying dynamics in many cases the equations of motion emit traveling wave solutions which can be used to represent the viscous profiles analytically the traveling wave solutions can be thought of as a lower dimensional representation of the motion since they contain information from the evolution equation but are constant along certain time space curves using a parameterized basis involving the traveling waves along with the sparse low energy decompositions found in imaging sciences we propose an approximation to viscous conservation laws which separates the coarse smooth component from the sharp fine one our method provides an appropriate approximation to the solution on a coarse grid thereby accurately under resolving the viscous profile this is similar to the philosophy of shock capturing methods in the sense that we want to capture the viscous front without needing to resolve the profile theoretical results on the consistency of our method are shown in general we provide several computational examples for convex and non convex fluxes

accuracy of high order and spectral methods for hyperbolic conservation laws with discontinuous solutions higher order and spectral methods have been used with success for elliptic and parabolic initial and boundary value problems with smooth solutions on the other hand higher order methods have been applied to hyperbolic problems with less success as higher order approx imations of discontinuous solutions suffer from the gibbs phenomenon we extend past work and show that spectral methods yield spectral convergence of moments even when applied to problems with discontinuous solutions besides spectral fourier methods for periodic domains we also prove high order convergence for adjoint consistent non periodic numerical methods exemplified by the discontinuous galerkin finite element method

a finite element multiresolution viscosity method for hyperbolic conservation laws it is well known that the classic galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the euler equations for compressible flow it is also well known that naively adding artificial diffusion to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use an elegant approach referred to as spectral viscosity methods has been developed for spectral methods in which one adds diffusion only to the high frequency modes of the solution the result being that stabilization is effected without sacrificing accuracy we extend this idea into the finite element framework by using hierarchical finite element functions as a multifrequency basis the result is a new finite element method for solving hyperbolic conservation laws in which artificial diffusion can be applied selectively only to the high frequency modes of the approximation as for spectral viscosity methods this results in stability without compromising accuracy in the context of a one dimensional scalar hyperbolic conservation law we prove the convergence of approximate solutions obtained using the new method to the entropy solution of the conservation law to illustrate the method the results of a computational experiment for a one dimensional hyperbolic conservation law are provided

accurate reconstructions of functions of finite regularity from truncated fourier series expansions kowledge of a truncated fourier series expansion for a 2 periodic function of finite regularity which is assumed to be piecewise smooth in each period is used to accurately reconstruct the corresponding function an algebraic equation of degree m is constructed for the m singularity locations in each period for the function in question the m coefficients in this algebraic equation are obtained by solving an algebraic system of m equations determined by the coefficients in the known truncated expansion if discontinuities in the derivatives of the function are considered in addition to discontinuities in the function itself that algebraic system will be nonlinear with respect to the m unknown coefficients the degree of the algebraic system will depend on the desired order of accuracy for the reconstruction i e a higher degree will normally lead to a more accurate determination of the singularity locations by solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations we are able to reconstruct the 2 periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiab1e up to the specified order

an interior penalty stabilised incompressible discontinuous galerkin fourier solver for implicit large eddy simulations abstract we present an implicit large eddy simulation iles h p high order 2 unstructured discontinuous galerkin fourier solver with sliding meshes the solver extends the laminar version of ferrer and willden 2012 34 to enable the simulation of turbulent flows at moderately high reynolds numbers in the incompressible regime this solver allows accurate flow solutions of the laminar and turbulent 3d incompressible navier stokes equations on moving and static regions coupled through a high order sliding interface the spatial discretisation is provided by the symmetric interior penalty discontinuous galerkin ip dg method in the x y plane coupled with a purely spectral method that uses fourier series and allows efficient computation of spanwise periodic three dimensional flows since high order methods e g discontinuous galerkin and fourier are unable to provide enough numerical dissipation to enable under resolved high reynolds computations i e as necessary in the iles approach we adapt the laminar version of the solver to increase controllably the dissipation and enhance the stability in under resolved simulations the novel stabilisation relies on increasing the penalty parameter included in the dg interior penalty ip formulation the latter penalty term is included when discretising the linear viscous terms in the incompressible navier stokes equations these viscous penalty fluxes substitute the stabilising effect of non linear fluxes which has been the main trend in implicit les discontinuous galerkin approaches the ip dg penalty term provides energy dissipation which is controlled by the numerical jumps at element interfaces e g large in under resolved regions such as to stabilise under resolved high reynolds number flows this dissipative term has minimal impact in well resolved regions and its implicit treatment does not restrict the use of large time steps thus providing an efficient stabilization mechanism for iles the ip dg stabilisation is complemented with a spectral vanishing viscosity svv method in the z direction to enhance stability in the continuous fourier space the coupling between the numerical viscosity in the dg plane and the svv damping provides an efficient approach to stabilise high order methods at moderately high reynolds numbers we validate the formulation for three turbulent flow cases a circular cylinder at re 3900 a static and pitch oscillating naca 0012 airfoil at re 10000 and finally a rotating vertical axis turbine at re 40000 with reynolds based on the circular diameter airfoil chord and turbine diameter respectively all our results compare favourably with published direct numerical simulations large eddy simulations or experimental data we conclude that the dg fourier high order solver with ip svv stabilisation proves to be a valuable tool to predict turbulent flows and associated statistics for both static and rotating machinery

a multidomain spectral method for supersonic reactive flows this paper has a dual purpose it presents a multidomain chebyshev method for the solution of the two dimensional reactive compressible navier stokes equations and it reports the results of the application of this code to the numerical simulations of high mach number reactive flows in recessed cavity the computational method utilizes a newly derived interface boundary conditions as well as an adaptive filtering technique to stabilize the computations the results of the simulations are relevant to recessed cavity flame holders

spectral methods for 2d riemann problems we consider riemann problems for a linear hyperbolic equation at time t 0 the solution is known outside the disk d o t inside this disk we make coordinate transformations and obtain an elliptic problem which is solved by a spectral tau method we compare the result with the spectral viscosity method applied to the original hyperbolic equation

the shallow water equations in lagrangian coordinates recent advances in the collection of lagrangian data from the ocean and results about the well posedness of the primitive equations have led to a renewed interest in solving flow equations in lagrangian coordinates we do not take the view that solving in lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted rather the grid in lagrangian coordinates represents the initial position of particles and it does not change with time we apply numerical methods traditionally used to solve differential equations in eulerian coordinates to solve the shallow water equations in lagrangian coordinates the difficulty with solving in lagrangian coordinates is that the transformation from eulerian coordinates results in solving a highly nonlinear partial differential equation the non linearity is mainly due to the jacobian of the coordinate transformation which is a precise record of how the particles are rotated and stretched the inverse jacobian must be calculated thus lagrangian coordinates cannot be used in instances where the jacobian vanishes for linear spatial flows we give an explicit formula for the jacobian and describe the two situations where the lagrangian shallow water equations cannot be used because either the jacobian vanishes or the shallow water assumption is violated we also prove that linear in space steady state solutions of the lagrangian shallow water equations have jacobian equal to one in the situations where the shallow water equations can be solved in lagrangian coordinates accurate numerical solutions are found with finite differences the chebyshev pseudospectral method and the fourth order runge kutta method the numerical results shown here emphasize the need for high order temporal approximations for long time integrations

a stable penalty method for the compressible navier stokes equations i open boundary conditions the purpose of this paper is to present asymptotically stable open boundary conditions for the numerical approximation of the compressible navier stokes equations in three spatial dimensions the treatment uses the conservation form of the navier stokes equations and utilizes linearization and localization at the boundaries based on these variables the proposed boundary conditions are applied through a penalty procedure thus ensuring correct behavior of the scheme as the reynolds number tends to infinity the versatility of this method is demonstrated for the problem of a compressible flow past a circular cylinder

eigensolution analysis of spectral hp continuous galerkin approximations to advection diffusion problems this study addresses linear dispersion diffusion analysis for the spectral hp continuous galerkin cg formulation in one dimension first numerical dispersion and diffusion curves are obtained for the advection diffusion problem and the role of multiple eigencurves peculiar to spectral hp methods is discussed from the eigencurves behaviour we observe that cg might feature potentially undesirable non smooth dispersion diffusion characteristics for under resolved simulations of problems strongly dominated by either convection or diffusion subsequently the linear advection equation augmented with spectral vanishing viscosity svv is analysed dispersion and diffusion characteristics of cg with svv based stabilization are verified to display similar non smooth features in flow regions where convection is much stronger than dissipation or vice versa owing to a dependency of the standard svv operator on a local peclet number first a modification is proposed to the traditional svv scaling that enforces a globally constant peclet number so as to avoid the previous issues in addition a new svv kernel function is suggested and shown to provide a more regular behaviour for the eigencurves along with a consistent increase in resolution power for higher order discretizations as measured by the extent of the wavenumber range where numerical errors are negligible the dissipation characteristics of cg with the svv modifications suggested are then verified to be broadly equivalent to those obtained through upwinding in the discontinuous galerkin dg scheme nevertheless for the kernel function proposed the full upwind dg scheme is found to have a slightly higher resolution power for the same dissipation levels these results show that improved cg svv characteristics can be pursued via different kernel functions with the aid of optimization algorithms

newton s method for linear complementarity problems this paper presents an iterative newton type method for solving a class of linear complementarity problems this class was discovered by mangasarian who had established that these problems can be solved as linear programs cottle and pang characterized solutions of the problems in terms of least elements of certain polyhedral sets the algorithms developed in this paper are shown to converge to the least element solutions some applications and computational results are also discussed

determination of optimal parameters for the chebyshev gegenbauer reconstruction method we propose a new strategy for choosing the optimal parameters for the gegenbauer reconstruction method based on chebyshev spectral coefficients under different assumptions on the smoothness of the function f these parameters are optimal in the sense that the bounds on the truncation and regularization errors were forced to be equal this strategy is independent on the number of terms n in the chebyshev expansion of the function f and guarantees exponential convergence as n rightarrow infty of the gegenbauer series to f on the intervals of smoothness the effectiveness of this strategy and exponential convergence are confirmed by numerical examples for functions with varying degrees of smoothness

efficient parallel algorithms for a class of graph theoretic problems in this paper we present efficient parallel algorithms for the following graph problems finding the lowest common ancestors for vertex pairs of a directed tree finding all fundamental cycles a directed spanning forest all bridges all bridge connected components all separation vertices all biconnected components and testing the biconnectivity of an undirected graph all these algorithms achieve the o lg 2 n time bound with the first two algorithms using n lceil n lg n rceil processors and the remaining algorithms using n lceil n lg 2 n rceil processors in all cases our algorithms are better than the previously known algorithms and in most cases reduce the number of processors used by a factor of n lg n moreover our algorithms are optimal with respect to the time processor product for dense graphs with the exception of the first two algorithms the machine model we use is the pram which is a simd model allowing simultaneous reads but not simultaneous writes to the same memory

spectral vanishing viscosity method for nonlinear conservation laws we propose a new spectral viscosity sv scheme for the accurate solution of nonlinear conservation laws it is proved that the sv solution converges to the unique entropy solution under appropriate reasonable conditions the proposed sv scheme is implemented directly on high modes of the computed solution this should be compared with the original nonperiodic sv scheme introduced by maday ould kaber and tadmor in siam j numer anal 30 1993 321 342 where sv is activated on the derivative of the sv solution the new proposed sv method could be viewed as a correction of the former and it offers an improvement which is confirmed by our numerical experiments a postprocessing method is implemented to greatly enhance the accuracy of the computed sv solution the numerical results show the efficiency of the new method

a subgrid scale deconvolution approach for shock capturing we develop a method for the modeling of flow discontinuities which can arise as weak solutions of inviscid conservation laws due to its similarity with recently proposed approximate deconvolution models for large eddy simulation the method potentially allows for a unified treatment of flow discontinuities and turbulent subgrid scales a filtering approach is employed since for the filtered evolution equations the solution is smooth and can be solved for by standard central finite difference schemes without special consideration of discontinuities a sufficiently accurate representation of the filtered nonlinear combination of discontinuous solution components which arise from the convection term can be obtained by a regularized deconvolution applied to the filtered solution for stable integration the evolution equations are supplemented by a relaxation regularization based on a secondary filter operation and a relaxation parameter an estimate for the relaxation parameter is provided the method is related to the spectral vanishing viscosity method and the regularized chapman enskog expansion method for conservation laws we detail the approach and demonstrate its efficiency with the inviscid and viscous burgers equations the isothermal shock problem and the one dimensional euler equations

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

numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size