the design of local navier stokes preconditioning for compressible flow a family of navier stokes preconditioners is presented that may reduce the stiffness due to complicated interaction between convection and diffusion in viscous flows navier stokes preconditioning is developed based on a fourier analysis of the discretized equations and a dispersion analysis of the differential equations navier stokes preconditioning can be extended from the euler technique with two methods a by using block jacobi preconditioning for the viscous terms b by introducing analytic dependence on the cell reynolds number in the preconditioner with these techniques it is possible to produce a local navier stokes preconditioner effective for all mach and cell reynolds numbers these techniques and a combined method are analyzed with respect to condition number and linear wave propagation and are illustrated with some numerical results

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

artificial boundary conditions for one dimensional cubic nonlinear schrodinger equations this paper addresses the construction of nonlinear integro differential artificial boundary conditions for one dimensional nonlinear cubic schrodinger equations several ways of designing such conditions are provided and a theoretical classification of their accuracy is given semidiscrete time schemes based on the method developed by duran and sanz serna ima j numer anal 20 2000 pp 235 261 are derived for these unusual boundary conditions stability results are stated and several numerical tests are performed to analyze the capacity of the proposed approach

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

physics based gmres preconditioner for compressible and incompressible navier stokes equations abstract this paper presents the implementation of a local physics preconditioning mass matrix 8 for an unified approach of 3d compressible and incompressible navier stokes equations using an supg finite element formulation and gmres implicit solver during the last years a lot of effort has been dedicated to finding a unified approach for compressible and incompressible flow in order to treat fluid dynamic problems with a very wide range of mach and reynolds numbers 10 26 37 on the other hand supg finite element formulation and gmres implicit solver is one of the most robust combinations to solve state of the art cfd problems 1 6 9 22 29 30 31 the selection of a good preconditioner and its performance on parallel architecture is another open problem in cfd community the local feature of the preconditioner presented here means that no communication among processors is needed when working on parallel architectures due to these facts we consider that this research can make some contributions towards the development of a unified fluid dynamic model with high rates of convergence for any combination of mach and reynolds numbers being very suitable for massively parallel computations finally it is important to remark that while this kind of preconditioning produces stabilized results in nearly incompressible regimes the standard version exhibits some numerical drawbacks that lead to solutions without physical meaning

the importance of eigenvectors for local preconditioners of the euler equations the design of local preconditioners to accelerate the convergence to a steady state for the compressible euler equations has so far been solely based on eigenvalue analysis however numerical evidence exists that the eigenvector structure also has an influence on the performance of preconditioners and should therefore be included in the design process in this paper we present the mathematical framework for the eigenvector analysis of local preconditioners for the multi dimensional euler equations the non normality of the preconditioned system is crucial in determining the potential for transient amplification of perturbations several existing local preconditioners are shown to possess a highly non normal structure for low mach numbers this non normality leads to significant robustness problems at stagnation points a modification to these preconditioners which eliminates the non normality is suggested and numerical results are presented showing the marked improvement in robustness

error propagation when approximating multi solitons the kdv equation as a case study abstract the paper studies the influence of the time discretizations when simulating some phenomena involving more than one solitary wave taking the kdv equation as a case study we obtain some conditions on the numerical method in order to get a more correct simulation of multi soliton solutions they are related to the evolution of the conserved quantities of the problem through the numerical integration it is shown that when approximating n solitons a method that preserves n invariants of the problem shows a better time propagation of the error than that of a general scheme as a consequence of this the simulation of some physical parameters that characterize the waves is more suitable when using conservative integrators we also show how these results can be extended to the approximation to multi solitons of any equation of the kdv hierarchy and more generally other integrable equations

a multi scale formulation for compressible turbulent flows suitable for general variational discretization techniques based on the recently introduced variational multi scale vms approach to large eddy simulation les as introduced in t j r hughes l mazzei k e jansen large eddy simulation and the variational multiscale method comput visual sci 3 2001 47 59 s s collis monitoring unresolved scales in multiscale turbulence modeling phys fluids 13 6 2001 1800 1806 we present a vms formulation which can be used in the simulation of compressible flows special attention is given to obtain a vms formulation which is suitable for complex flow domains and general variational discretization techniques a generalization of the favre averaging procedure is introduced such that the formulation resembles the favre filtered navier stokes equations traditionally used in les of compressible flow and no explicit subgrid terms arise in the continuity equation also we show that with the use of discretization methods other than fourier spectral methods the vms projection no longer commutes with differentiation this results in additional subgrid scale terms which resemble the commutator error as encountered in the traditional filtering approach to les

advantages of using the vertical gradient of gravity for 3 d interpretation gravity gradiometric data or gravity data transformed into vertical gradient can be efficiently processed in three dimensions for delineating density discontinuities model studies performed with the combined use of maxima of analytic signal and of horizontal gradient and the euler deconvolution techniques on the gravity field and its vertical gradient demonstrate the superiority of the latter in locating density contrasts particularly in the case of interfering anomalies where the use of gravity alone fails the gravity gradient is able to provide useful information with satisfactory accuracy

error growth in the numerical integration of periodic orbits with application to hamiltonian and reversible systems we analyze in detail the growth with time of the coefficients of the asymptotic expansion of the error in the numerical integration with one step methods of periodic solutions of systems of ordinary differential equations variable stepsizes are allowed we successively consider general hamiltonian and reversible problems for hamiltonian and reversible systems and under fairly general hypotheses on the orbit being integrated numerical methods with relevant geometric properties symplecticness energy conservation reversibility are proved to have better error growth than general methods

magnetic interpretation in three dimensions using euler deconvolution magnetic survey data in grid form may be interpreted rapidly for source positions and depths by deconvolution using euler s homogeneity relation the method employs gradients either measured or calculated data need not be pole reduced so that remanence is not an interfering factor geologic constraints are imposed by use of a structural index model studies show that the method can locate or outline confined sources vertical pipes dikes and contacts with remarkable accuracy a field example using data from an intensively studied area of onshore britain shows that the method works well on real data from structurally complex areas and provides a series of depth labeled euler trends which mark magnetic edges notably faults with good precision

the application of preconditioning in viscous flows a time derivative preconditioning algorithm that is effective over a wide range of flow conditions from inviscid to very diffusive flows and from low speed to supersonic flows has been developed the algorithm uses a preconditioning matrix that introduces well conditioned eigenvalues while simultaneously avoiding nonphysical time reversals for viscous flows the resulting algorithm also provides a mechanism for controlling the inviscid and viscous time step parameters at very diffusive flows thereby ensuring rapid convergence for very viscous flows as well as for inviscid flows computational capabilities are demonstrated through computation of a wide variety of problems convergence rates are shown to be accelerated by as much as two orders of magnitudes while providing solutions that are indentical to those obtained without preconditioning method

supg finite element computation of compressible flows with the entropy and conservation variables formulations abstract supg stabilized finite element formulations of compressible euler equations based on the conservation and entropy variables are investigated and compared the formulation based on the conservation variables consists of the formulation introduced by tezduyar and hughes plus a shock capturing term the formulation based on the entropy variables is the same as the one by hughes franca and mallet which has a shock capturing term built in these formulations are tested on several subsonic transonic and supersonic compressible flow problems it is shown that the stabilized formulation based on the conservation variables gives solutions which are just as good as those obtained with the entropy variables furthermore the solutions obtained using the two formulations are very close and in some cases almost indistinguishable consequently it can be deduced that the relative merits of these two formulations will continue to remain under debate

simple stabilizing matrices for the computation of compressible flows in primitive variables the key ingredient that balances stability and accuracy in stabilized formulations is the parameter of intrinsic time scales for multi dimensional hyperbolic systems of equations this parameter is a matrix and the available expressions for its computation involve the solution of an eigenvalue problem which can be tedious or cpu time consuming thus for formulations based on primitive variables including pressure a couple of simple stabilizing matrices are presented which are easy to implement and cpu economic numerical evaluations show the performance of the various choices

asymptotic expansions of the global error of fixed stepsize methods in his fundamental paper on general fixed stepsize methods skeel 6 studied convergence properties but left the existence of asymptotic expansions as an open problem in this paper we give a complete answer to this question for the special cases of one step and linear multistep methods our proof is shorter than the published ones asymptotic expansions are the theoretical base for extrapolation methods

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

the fourier pseudospectral method with a restrain operator for the korteweg de vries equation abstract in this paper we develop a fourier pseudospectral method with a restrain operator for the korteweg de vries equation we prove the generalized stability of the schemes and give convergence estimations depending on the smoothness of the solution of the p d e

fourier stability analysis and local courant number of the preconditioned variational multiscale stabilization p vms for euler compressible flow abstract the results of a fourier stability analysis of the preconditioned variational multiscale stabilization p vms method introduced in moragues et al 2015 are presented in this paper p vms combines a variational multiscale stabilized finite elements discretization together with local preconditioning in this work we deal with the p vms method using van leer lee roe s vanleer et al 1991 and choi merkle s choi and merkle 1993 local preconditioners we solve the euler equations of compressible flow for steady problems we concentrate on explicit time integration schemes the stability analysis is performed on a two dimensional simplified problem with a structured mesh and its conclusions are applied to two and three dimensional general problems with unstructured meshes as a result of this analysis a local courant friedrichs lewy number is defined for the computation of the time step the convergence rate is evaluated and compared with the traditional constant courant friedrichs lewy number for various test cases spanning a large range of mach numbers

conserved quantities of some hamiltonian wave equations after full discretization hamiltonian pdes have some invariant quantities which would be good to conserve with the numerical integration in this paper we concentrate on the nonlinear wave and schrodinger equations under hypotheses of regularity and periodicity we study how a symmetric space discretization makes that the space discretized system also has some invariants or nearly invariants which well approximate the continuous ones we conjecture some facts which would explain the good numerical approximation of them after time integration when using symplectic runge kutta methods or symmetric linear multistep methods for second order systems

a variational multiscale stabilized finite element method for the solution of the euler equations of nonhydrostatic stratified flows we present a compressible version of the variational multiscale stabilization vms method applied to the finite element fe solution of the euler equations for nonhydrostatic stratified flows this paper is meant to verify how the algorithm performs when solving problems in the framework of nonhydrostatic atmospheric dynamics this effort is justified by the previously observed good performance of vms and by the advantages that a compact galerkin formulation offers on massively parallel architectures a paradigm for both computational fluid dynamics cfd and numerical weather prediction nwp practitioners we also propose a simple technique to construct a well balanced approximation of the dominant hydrostatics that if not properly discretized may cause unacceptable vertical oscillations this is a relevant problem in nwp especially in the proximity of steep topography to evaluate the performance of the method for stratified environments six standard 2d and two 3d test cases are selected of these two admit a semi analytic solution while the remaining six are non steady and non linear thermal problems with dominant buoyancy effects that challenge the algorithm in terms of stability

the fourier pseudospectral method with a restrain operator for the rlw equation in this paper we develop a new fourier pseudospectral method with a restrain operator for the modified korteweg de vries mkdv equation we prove the generalized stability of the scheme from which the convergence follows

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

a stabilized finite element method based on sgs models for compressible flows abstract we present an appropriate extension of the stabilized finite element formulation introduced in a corsini f rispoli a santoriello a variational multiscale high order finite element formulation for turbomachinery flow computations comput methods appl mech engrg 194 2005 4797 4823 for the prediction of incompressible flows aimed at compressible flows the stabilized formulation is the so called variable subgrid scale method v sgs based on an approximation of the class of subgrid scale models sgs derived from the hughes variational multiscale method hughes vms introduced in t j r hughes multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods comput methods appl mech engrg 127 1995 387 401 it is characterized by a variable stabilization parameter within the domain of element we also propose an innovative procedure for computing the stabilization parameter by using the one dimensional element green s functions corresponding to an advective diffusive differential operator the stabilization parameter is defined as the sum of two components the first providing its mean value and the second its element wise space dependency on this basis the present work proposes the stabilization matrix for the multidimensional advective diffusive system of equations governing compressible flows the procedure developed for computing the stabilization parameter can be applied to other differential operators as it has been shown in a corsini f rispoli a santoriello a variational multiscale high order finite element formulation for turbomachinery flow computations comput methods appl mech engrg 194 2005 4797 4823 the proposed stabilization device is validated by simulation of inviscid and viscous supersonic flow configurations

degrees of homogeneity of potential fields and structural indices of euler deconvolution homogeneity is a well known property of the potential fields of simple point sources used infield inversion wefind that the analytical expressions of potential fields created by sources of complicated shape and constant or variable density or magnetization also show this property this is true if all variables of length dimension are involved in the test of homogeneity the coordinates of observation points and the source coordinates and sizes form an extended set of variables in relation to which the field expression is homogeneous in this case the principal definition of homogeneity applied to a potential field can be treated as an operator of a space transform of similarity the ratio between the transformed and original fields determines the value and sign of thedegreeofhomogeneityn thelattermaytakeonpositive zero ornegativevalues thedegreeofhomogeneitydepends onthetypeoffieldandontheassumedphysicalparameterof the field source and can be nonunique for a given field element we analyze the potential field of one singular point as thesimplestcaseofhomogeneity thus wededuceresultsfor the structural index n n in euler deconvolution the structural index can also be positive zero or negative but it has a unique value analytical considerations as well as numerical tests on the gravity contact model confirm the proposed physical interpretation of n and lead to an extended versionofeuler sdifferentialequationforpotentialfields

on the estimation of the structural index from low pass filtered magnetic data abstractthe estimation of the structural index and of the depth to the source is the main task of many popular methods used to analyze potential field data such as euler deconvolution however these estimates are unstable even in the presence of a weak amount of noise and euler deconvolution of noisy data leads to an underestimation of structural index and depth we have studied how the structural index and depth estimates are affected by applying low pass filtering to the data physically based low pass filters such as upward continuation and integration have been shown to be the best choice over a range of altitudes upward continuation or orders integration filters mainly because their outputs have a well defined physical meaning in contrast mathematical low pass filters require that the filter parameters be tuned carefully by means of several trial tests to produce optimally smoothed fields the c norm criterion is a reliable strategy to produce a stabilized vertical derivative and we disc

euler deconvolution of gravity anomalies from thick contact fault structures with extended negative structural index the concept of extended euler homogeneity of potential fields is examined with respect to all variables of length dimension in their analytical expressions this reveals the possible existence of positive degrees of homogeneity or corresponding negative structural indices considered as extensions of thethompson s structural indices in euler deconvolution this approach is implemented for a contact gravity model represented by a 2d semi infinite slab with large thickness relative to its depth applying euler deconvolution on synthetic and field data indicates that the positive degree of homogeneity i e the extended negative structural index is the appropriate one for the inversion of gravity anomalies fromcontactstructures

a variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshes application to vortex shedding the variational multiscale formulation of large eddy simulations developed by hughes and his co workers for incompressible turbulent flows is first extended to the case of compressible turbulent flows next a finite volume element discretization on tetrahedral meshes of this method is proposed this discretization features an economical procedure based on agglomeration for separating a priori the scales a corresponding projector for eliminating the small scales from the system of equations and an effective control of the numerical dissipation induced by upwinding the resulting les method is validated with the three dimensional numerical simulation of a low speed flow past a square cylinder at m 0 1 and re 22 000 and various comparisons with experimental data

unification of euler and werner deconvolution in three dimensions via the generalized hilbert transform the extended euler deconvolution algorithm is shown to be a generalization and unification of 2 d euler deconvolution and werner deconvolution after recasting the extended euler algorithm in a way that suggests a natural generalization to three dimensions we show that the 3 d extension can be realized using generalized hilbert transforms the resulting algorithm is both a generalization of extended euler deconvolution to three dimensions and a 3 d extension of werner deconvolution at a practical level the new algorithm helps stabilize the euler algorithm by providing at each point three equations rather than one we illustrate the algorithm by explicit calculation for the potential of a vertical magnetic dipole

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

design criteria for local euler preconditioning euler preconditioning has remarkable benefits in removing stiffness making systems of equations behave as a scalar equation preserving accuracy and decoupling the euler equations design criteria for optimal euler preconditioning are discussed that retain the basic preconditioning benefits and remove the causes of instabilities due to the use of preconditioning new families of 1d and 2d optimal euler preconditioners are presented that may satisfy the design criteria in an optimal way in particular focusing on resolution of the stability problem associated with stagnation points a stagnation preconditioner and a suboptimal van leer lee roe preconditioner are studied these preconditioners are less sensitive to flow angle variation across cells and or produce a closer to orthogonal eigenvector system

a new finite element formulation for computational fluid dynamics x the compressible euler and navier stokes equations a space time element method is presented for solving the compressible euler and navier stokes equations the proposed formulation includes the variational equation predictor multi corrector algorithms and boundary conditions the variational equation is based on the time discontinuous galerkin method in which the physical entropy variables are employed a least squares operator and a discontinuity capturing operator are added resulting in a high order accurate and unconditionally stable method implicit explicit predictor multi corrector algorithms applicable to steady as well as unsteady problems are presented techniques are developed to enhance their efficiency implementation of boundary conditions is addressed in particular a technique is introduced to satisfy nonlinear essential boundary conditions and a consistent method is presented to calculate boundary fluxes numerical results are presented to demonstrate the performance of the method

a dynamic variational multiscale method for large eddy simulations on unstructured meshes abstract a dynamic version of the variational multiscale vms method for large eddy simulation les on unstructured meshes of turbulent compressible flows is presented its key features include two variational analogues of germano s algebraic identity and a computationally efficient agglomeration based numerical procedure agglomeration is used both for separating a priori the scales and computing dynamically a smagorinsky parameter and the turbulent prandtl number on unstructured finite volume or finite element meshes numerical predictions of low speed 20 angle of attack turbulent flows past a 6 1 prolate spheroid and a forward swept scaled wing are presented they are contrasted with counterpart results obtained using similar implementations on unstructured meshes of the vms les traditional les and traditional dynamic les methods they are also compared to experimental data when available

fourier spectral methods for pseudo parabolic equations fourier spectral methods of galerkin and collocation type for quasilinear pseudo parabolic equations are presented convergence with spectral accuracy is proved for both galerkin and collocation approximations unconditionally stable discretization in time by explicit one step methods is analyzed fast transform methods to compute the nonlinear terms are proposed

the structural index in gravity and magnetic interpretation errors uses and abuses abstractthe structural index si is based on the concept of euler homogeneity a description of scaling behavior it has found wide use in potential field depth estimation and is a constant integer for simple sources with single singularities points lines thin bed faults sheet edges infinite contacts for these cases the si is identical to the index of a simple power law field fall off with distance the simple euler formulation is only strictly correct for such simple sources and integer si values the widespread use of the simple euler method on more complex structures using fractional si values is likely to produce misleading results because the si is no longer a constant for any given source we examine a recently published example that used an arbitrary si to estimate depth to the base of the crust for africa and produced misleading results extension to more complex sources such as tabular bodies or thick steps requires one of several more generalized approaches which recognize all variabl

a spectral difference method for two dimensional viscous flow abstract we propose a spectral difference method for the 2 dimensional vorticity equation with a periodic boundary condition in one direction the solution satisfies a semidiscrete conservation law and thus better numerical results are obtained we also prove stability and convergence

preconditioned methods for solving the incompressible low speed compressible equations abstract acceleration methods are presented for solving the steady state incompressible equations these systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state we also consider the compressible equations in conservation form with slow flow two arbitrary functions and are introduced in the general preconditioning an analysis of this system is presented and an optimal value for is determined given a constant it is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed several generalizations to the compressible equations are presented which extend previous results

stabilization and shock capturing parameters in supg formulation of compressible flows abstract the streamline upwind petrov galerkin supg formulation is one of the most widely used stabilized methods in finite element computation of compressible flows it includes a stabilization parameter that is known as typically the supg formulation is used in combination with a shock capturing term that provides additional stability near the shock fronts the definition of the shock capturing term includes a shock capturing parameter in this paper we describe for the finite element formulation of compressible flows based on conservation variables new ways for determining the and the shock capturing parameter the new definitions for the shock capturing parameter are much simpler than the one based on the entropy variables involve less operations in calculating the shock capturing term and yield better shock quality in the test computations

a pseudospectral finite element method for solving two dimensional vorticity equations a fourier pseudospectral finite element method pfm with a control operator based on the bochner summation method is developed for solving two dimensional vorticity equations some approximation results in the nonisotropic sobolev spaces are given stability of the scheme and optimal rate of convergence are proved in the l 2 norm the methods in this paper are also applicable to pfms of other linear or nonlinear problems

a numerical scheme for periodic travelling wave simulations in some nonlinear dispersive wave models a numerical method for simulating periodic travelling wave solutions of some nonlinear dispersive wave equations is proposed the construction of the scheme is based on an efficient computation of the elements that characterize these solutions the initial profile and the velocity of the wave

numerical schemes for a model for nonlinear dispersive waves abstract a description is given of a number of numerical schemes to solve an evolution equation that arises when modelling the propagation of water waves in a channel the discussion also includes the results of numerical experiments made with each of the schemes it is suggested on the basis of these experiments that one of the schemes may have discrete solitary wave solutions

local preconditioning and variational multiscale stabilization for euler compressible steady flow abstract this paper introduces a preconditioned variational multiscale stabilization p vms method for compressible flows in this introductory paper we focus on inviscid flow and steady state problems the euler equations are solved on fully unstructured grids and discretized using the finite element method the p vms method can be decomposed into three parts first a local preconditioner is applied to the continuous equations to reduce the stiffness while covering a wide range of mach numbers then the resulting preconditioned system is discretized in space using finite elements and stabilized with a variational multiscale stabilization method adapted for the preconditioned equations in this paper the solution is advanced in time using a fully explicit time discretization although p vms is general and can be applied to fully implicit solvers the proposed method is assessed by comparing convergence and accuracy of the solutions between the non preconditioned and preconditioned cases in particular for van leer lee roe s 1991 and choi merkle s 1993 preconditioners in some selected examples covering a large range of mach numbers

spectral methods for a nonlinear initial value problem involving pseudo differential operators spectral methods fourier methods for approximating the solution of a nonlinear initial value problem involving pseudo differential operators are defined and analysed a semidiscrete approximation to the nonlinear equation based on an l 2 projection is described the semidiscrete l 2 approximation is shown to be a priori stable and convergent under sufficient decay and smoothness assumptions on the initial data it is shown that the semidiscrete method converges with infinite order that is higher order decay and smoothness assumptions imply higher order error bounds spectral schemes based on spacial collocation are also discussed

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

an entropy variable based vms gls method for the simulation of compressible flows on unstructured grids abstract the variational multiscale approach for large eddy simulation is investigated within a finite element framework a filtering analog of this method providing a model which shares basic structures with hyperviscosity models is used and evaluated on a freely decaying isotropic turbulence in the limit of infinite reynolds numbers the method used is based on a symmetric form of the navier stokes equations stabilized with a galerkin least squares approach the proposed procedure provides appealing results it is as satisfactory as the use of germano and lilly s dynamic algorithm without stabilization flaws and at a lower cost comparisons between numerical dissipation and the proper subgrid closure show that the variational multiscale models tend to compute a turbulent viscosity that accounts for the numerical sources of dissipation which reveals an important feature for an intensive and robust use of vms large eddy simulation

finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations a petrov galerkin finite element formulation is presented for first order hyperbolic systems of conservation laws with particular emphasis on the compressible euler equations applications of the methodology are made to one and two dimensional steady and unsteady flows with shocks results obtained suggest the potential of the type of methods developed

fully discrete galerkin methods for the korteweg de vries equation abstract fully discrete schemes for the numerical simulation of solutions of the periodic initial value problem for the korteweg de vries equation are introduced implemented and tested of special interest are stable schemes featuring rates of convergence of order higher than two in both the spatial and temporal variable a careful analysis of the relative and absolute efficiency of these schemes is carried out and one of the schemes is applied to demonstrate that solutions of certain generalized korteweg de vries equations apparently may develop singularities in finite time

accuracy and conservation properties in numerical integration the case of the korteweg de vries equation when numerically integrating time dependent differential equations it is often recommended to employ methods that preserve some of the invariant quantities mass energy etc of the problem being considered this recommendation is usually justified on the grounds that conservation of invariant quantities may ensure that the numerical solution possesses some important qualitative features however there are cases where schemes that preserve invariants are also advantageous in that they possess favourable error propagation mechanisms that render them superior from a quantitative point of view in the present paper we consider the korteweg de vries equation as a case study we show rigorously that for soliton problems and at leading order the error of conservative schemes consists of a phase error that grows linearly with time plus a complementary term that is bounded in the h 1 norm uniformly in time for general nonconservative schemes the error involves a linearly growing amplitude error a quadratically growing phase error and a complementary term that grows linearly in the h 1 norm numerical experiments are presented

magsounddst 3d automatic inversion of magnetic and gravity data based on the differential similarity transform we present an automatic procedure magnetic and gravity sounding differential similarity transformmagsounddst for inversion of regular or irregular magnetic and gravitygrid data measured on even or uneven surfaces it solves for horizontal position depth and structural index of simple sources and is independent of a linear background in addition it estimates the shape of sources consisting of several singular points and lines the method uses the property of the differential similarity transform dst of a magnetic or a gravity anomaly to become zero or linear at all observation points when the central point of similarity of the transform which we refer to as the probing point coincides with a source s singular point it uses a measured anomalous field and its calculated or measured gradiometry first order derivatives the method is independent of the magnetization vector direction in the magnetic data case and does not require reduction to the pole transformed data as input with magsounddst we provide an important alternative interpretation technique to the euler deconvolution procedures combining a moving window method whereby the solutions are linked to singular points of causative bodies with an approach in which the solutions are linked to the real sources the procedure involves calculating a 3d function that evaluates the linearity of the dst for different integer or noninteger structural indices using a moving window we sound the subsurface along a vertical line under each window center then we combine the 3d results for different structural indices and present them in three easy tointerpret maps avoiding the need for clustering techniques we deduce only one solution for location and type of simple sources which is a major advantage over euler deconvolution application to different cases of synthetic and real data shows the method s applicability to various types of magnetic and gravity field investigations

preconditioning applied to variable and constant density flows a time derivative preconditioning of the navier stokes equations suitable for both variable and constant density fluids is developed the ideas of low mach number preconditioning and artificial compressibility are combined into a unified approach designed to enhance convergence rates of density based time marching schemes for solving flows of incompressible and variable density fluids at all speeds the preconditioning is coupled with a dual time stepping scheme implemented within an explicit multistage algorithm for solving time accurate flows the resultant time integration scheme is used in conjunction with a finite volume discretization designed for unstructured solution adaptive mesh topologies this method is shown to provide accurate steady state solutions for transonic and low speed flow of variable density fluids the time accurate solution of unsteady incompressible flow is also demonstrated