an asymptotic preserving monte carlo method for the boltzmann equation abstract in this work we propose an asymptotic preserving monte carlo method for the boltzmann equation that is more efficient than the currently available monte carlo methods in the fluid dynamic regime this method is based on the successive penalty method 39 which is an improved bgk penalization method originally proposed by filbet and jin 16 here we introduce the monte carlo implementation of the method which despite its lower order accuracy is very efficient in higher dimensions or simulating some complicated chemical processes this method allows the time step independent of the mean free time which is prohibitively small in the fluid dynamic regime we study some basic properties of this method and compare it with some other asymptotic preserving monte carlo methods in terms of numerical performance in different regimes from rarefied to fluid dynamic regimes and their computational efficiency

a divide and conquer method for unitary and orthogonal eigenproblems leth n xn be a unitary upper hessenberg matrix whose eigenvalues and possibly also eigenvectors are to be determined we describe how this eigenproblem can be solved by a divide and conquer method in which the matrixh is split into two smaller unitary upper hessenberg matricesh 1 andh 2 by a rank one modification ofh the eigenproblems forh 1 andh 2 can be solved independently and the solutions of these smaller eigenproblems define a rational function whose zeros on the unit circle are the eigenvalues ofh the eigenvector ofh can be determined from the eigenvalues ofh and the eigenvectors ofh 1 andh 2 the outlined splitting of unitary upper hessenberg matrices into smaller such matrices is carried out recursively this gives rise to a divide and conquer method that is suitable for implementation on a parallel computer whenh n xn is orthogonal the divide and conquer scheme simplifies and is described separately our interest in the orthogonal eigenproblem stems from applications in signal processing numerical examples for the orthogonal eigenproblem conclude the paper

numerical methods for hyperbolic conservation laws with stiff relaxation ii higher order godunov methods a higher order godunov method is presented for hyperbolic systems of conservation laws with stiff relaxing source terms the goal is to develop a godunov method that produces higher order accurate

runge kutta methods for hyperbolic conservation laws with stiff relaxation terms underresolved numerical schemes for hyperbolic conservation laws with stiff relaxation terms may generate unphysical spurious numerical results or reduce to lower order if the small relaxation time is not temporally well resolved we design a second order runge kutta type splitting method that possesses the discrete analogue of the continuous asymptotic limit which thus is able to capture the correct physical behaviors with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved

a convergence proof for nanbu s simulation method for the full boltzmann equation it is shown that the discrete measures given by the nanbu simulation method converge with respect to the weak topology of measures to solutions of the boltzmann equation the main conditions for this result are that the cauchy problem for the boltzmann equation has a sufficiently smooth solution and that the discretization parameters cell size timestep and test particle number satisfy suitable constraints

iteration schemes for the divide and conquer eigenvalue solver we propose globally convergent iteration schemes for updating the eigenvalues of a symmetric matrix after a rank 1 modification such calculations are the core of the divide and conquer technique for the symmetric tridiagonal eigenvalue problem we prove the superlinear convergence right from the start of our schemes which allows us to improve the complexity bounds of 3 the effectiveness of our algorithms is confirmed by numerical results which are reported and discussed

an asymptotic preserving automatic domain decomposition method for the vlasov poisson bgk system with applications to plasmas in this work we present an efficient strategy to deal with plasma physics simulations in which localized departures from thermodynamical equilibrium are present the method is based on the introduction of intermediate regions which allows smooth transitions between kinetic and fluid zones in this paper we extend domain decomposition techniques obtained through dynamic coupling and buffer zones to the study of plasmas and moreover we combine them with asymptotic preserving and asymptotically accurate strategies for the time integration we use a hybrid scheme in which both kinetic and fluid descriptions are considered and coupled together while the kinetic model is solved by asymptotic preserving and accurate methods in order to guarantee high efficiency and accuracy in all regimes the numerical scheme is validated and its performances are analyzed by numerical simulations

a variant of van leer s method for multidimensional systems of conservation laws abstract we present a new variant of van leer s construction of upwind finite volume schemes for hyperbolic systems of conservation laws fluxes are computed with second order accuracy using an interpolation rather than a slope reconstruction a correction of the interpolated values is necessary and performed globally on each cell by a conservation argument it can be used on a rectangular or triangles based dual grid to obtain a genuinely multidimensional scheme one of our main concerns in this construction is to prove that the second order reconstruction combined with a boltzmann solver gives nonnegative values of the pressure and density for gas dynamics even on an unstructured mesh this allows us to derive a rigorous cfl condition thus our approach is very robust

hydrodynamics from kinetic models of conservative economies in this paper we introduce and discuss the passage to hydrodynamic equations for kinetic models of conservative economies in which the density of wealth depends on additional parameters like the propensity to invest as in kinetic theory of rarefied gases the closure depends on the knowledge of the homogeneous steady wealth distribution the maxwellian of the underlying kinetic model the collision operator used here is the fokker planck operator introduced by j p bouchaud and m mezard wealth condensation in a simple model of economy physica a 282 2000 536 545 which has been recently obtained in a suitable asymptotic of a boltzmann like model involving both exchanges between agents and speculative trading by s cordier l pareschi and one of the authors s cordier l pareschi g toscani on a kinetic model for a simple market economy j stat phys 120 2005 253 277 numerical simulations on the fluid equations are then proposed and analyzed for various laws of variation of the propensity

on the generalization of the boltzmann h theorem for a spatially homogeneous maxwell gas sufficient conditions are given for a convex and isotropic functional that are monotonically decreasing in time along the solution of the spatially homogeneous boltzmann equation for maxwellian molecules the conditions are valid for plane velocities and for axially symmetric solutions in any dimension explicit examples are given

on the time relaxed monte carlo computations for the lid driven micro cavity flow abstract in the present study the first second and third orders of the trmc scheme trmc1 trmc2 and trmc3 schemes are employed to numerically investigate the lid driven micro cavity flow with different lid velocities and knudsen numbers the simulations cover flow regimes from early slip to early transition regimes 0 005 k n 0 1 higher order terms in the wild sum expansion are considered to obtain higher order collisions the results are compared with those from the standard dsmc method comparisons show that among the studied schemes the results obtained from the trmc3 scheme have excellent agreement with the ones from the dsmc method on the other hand the results of trmc1 and trmc2 schemes show deviations compared to those from the dsmc method the deviations are more pronounced for the velocity distributions moreover the present investigation illustrates that as the knudsen number increases the accuracy of lower orders of the trmc scheme improves it is observed that truncating the wild sum expansion up to the third order approximation of the trmc scheme may be a proper alternative method for the dsmc method in simulating the lid driven micro cavity problem for knudsen numbers 0 005 k n 0 1 with reasonable accuracy and simplicity in mathematics

restricted rank modification of the symmetric eigenvalue problem theoretical considerations the problem is considered how to obtain the eigenvalues and vectors of a matrix a vvt where a is a symmetric matrix with known spectral decomposition and vvt is a positive semidefinite matrix of low rank it is shown that the eigenvalues of a vvt can easily be located to any desired accuracy by means of the inertia of the matrix i vt a 1v the problem of determining the eigenvalues of a restricted to r v can be treated likewise

numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a small relaxation limit governed by reduced systems of a parabolic or hyperbolic type in such systems the understanding of basic wave pattern is difficult to achieve and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved we develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit which are thus able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type

existence of natural frequencies of systems with artificial restraints and their convergence in asymptotic modelling a major limitation of the rayleigh ritz method for determining the natural frequencies of a system is the need to choose admissible functions that do not violate the geometric constraints of that system courant 1943 bulletin of the american mathematical society49 1 23 several researchers have attempted to overcome this problem by asymptotically modelling the rigid constraints with artificial imaginary restraints of very large stiffness courant 1943bulletin of the american mathematical society49 1 23 warburton and edney 1984 journal of sound and vibration95 537 552 gorman 1989 journal of applied mechanics56 893 899 kim et al 1990 journal of sound and vibration143 379 394 yuan and dickinson 1992 journal of sound and vibration153 203 216 yuan and dickinson 1992 journal of sound and vibration159 39 55 cheng and nicolas 1992 journal of sound and vibration155 231 247 yuan and dickinson 1994computers and structures53 327 334 lee and ng 1994 applied acoustics42 151 163 amabili and garziera 1999 journal of sound and vibration224 519 539 amabili and garziera 2000 journal of fluids and structures14 669 690 while the numerical results thus obtained for the systems considered in the literature were in close agreement with exact values for the natural frequencies corresponding to the first few modes sample calculations show that the error introduced by the asymptotic modelling increases with mode number and therefore to obtain accurate results for higher modes the magnitude of stiffness should also be increased in any event the error due to the asymptotic modelling would remain uncertain except when the correct frequency values are known however the use of artificial restraints with negative stiffness a new concept which was introduced in a recent publication ilanko and dickinson 1999 journal of sound and vibration219 370 378 paves the way for estimating the error due to asymptotic modelling this is possible since in this work the rayleigh ritz frequencies of the constrained system were found to be bracketed by the frequencies of the asymptotic models with positive and negative restraints however the use of artificial restraints with negative stiffness has raised some important questions would a system with a large negative restraint become unstable and if so what is the guarantee that the frequencies of the asymptotic model would converge to that of the constrained system this paper is the result of the author s attempt to answer these questions and gives a proof of existence of natural frequencies for systems with artificial restraints springs having positive or negative stiffness coefficients and their convergence towards constrained systems based on rayleigh s theorem of separation it has been shown that a vibratory system obtained by the addition of h restraints to an n degree of freedom d o f system where h n will have at least n h natural frequencies and modes and that as the magnitude of the stiffness of the added restraints becomes very large these n h natural frequencies will converge to the n h natural frequencies of a constrained system in which the displacements restrained by the springs are effectively constrained

asymptotic modelling theorems for the static analysis of linear elastic structures the work presented here justifies the use of restraints with large positive and negative stiffness values to asymptotically model geometric constraints of a structure in linear structural analysis replacing constraints by very stiff restraints improves the versatility of variational methods such as the rayleigh ritz method as the limitation on the choice of functions is removed based on recently published theorems on the existence of natural frequencies of systems with artificial restraints of positive and negative stiffness and their convergence towards the frequencies of the corresponding constrained systems a proof is given to show that the displacement of a constrained structure caused by any action along its direction is approached and bracketed by the displacement of systems with artificial restraints of positive and negative stiffness positive and negative penalty functions as the magnitude of stiffness is increased a procedure for the practical use of positive and negative penalty functions in static analysis of linear structures is also proposed

a class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources in this paper we propose a general time discrete framework to design asymptotic preserving schemes for initial value problem of the boltzmann kinetic and related equations numerically solving these equations are challenging due to the nonlinear stiff collision source terms induced by small mean free or relaxation time we propose to penalize the nonlinear collision term by a bgk type relaxation term which can be solved explicitly even if discretized implicitly in time moreover the bgk type relaxation operator helps to drive the density distribution toward the local maxwellian thus naturally imposes an asymptotic preserving scheme in the euler limit the scheme so designed does not need any nonlinear iterative solver or the use of wild sum it is uniformly stable in terms of the possibly small knudsen number and can capture the macroscopic fluid dynamic euler limit even if the small scale determined by the knudsen number is not numerically resolved it is also consistent to the compressible navier stokes equations if the viscosity and heat conductivity are numerically resolved the method is applicable to many other related problems such as hyperbolic systems with stiff relaxation and high order parabolic equations

eigenvalues of a symmetric tridiagonal matrix a divide and conquer approach the symmetric tridiagonal eigenproblem has been the topic of some recent work many methods have been advanced for the computation of the eigenvalues of such a matrix in this paper we present a divide and conquer approach to the computation of the eigenvalues of a symmetric tridiagonal matrix via the evaluation of the characteristic polynomial the problem of evaluation of the characteristic polynomial is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem we give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation furthermore this set of recursions can be implemented on a regulartree structure if the concurrency exhibited by this algorithm is exploited it can be shown that thetime for computation of all the eigenvalues becomeso nlogn instead ofo n2 as is the case for the approach where the order is increased by only one at every step we address the numerical problems associated with the use of the characteristic polynomial and present a numerically stable technique for the eigenvalue computation

the use of negative penalty functions in linear systems of equations contrary to what is commonly thought it is possible to obtain convergent results with negative rather than positive penalty functions this has been shown and proven on various occasions for vibration analysis but in this contribution it will also be shown and proven for systems of linear equations subjected to one or more constraints as a key ingredient in the developed arguments a pseudo force is identified as the derivative of the constrained degree of freedom with respect to the inverse of the penalty parameter since this pseudo force can be proven to be constant for large absolute values of the penalty parameter it follows that the exact solution is bounded by the results obtained with negative and positive penalty parameters the mathematical proofs are presented and two examples are shown to illustrate the principles

on the optimal choice of coefficients in a truncated wild sum and approximate solutions for the kac equation we study an approximate solution of the boltzmann problem for kac s caricature of a maxwellian gas by using a truncated and modified expansion of wild type we choose the coefficients in the wild sum approximation using a criterion based on exactly reproducing the behavior of the leading modes

fast algorithms for computing the boltzmann collision operator the development of accurate and fast numerical schemes for the five fold boltzmann collision integral represents a challenging problem in scientific computing for a particular class of interactions including the so called hard spheres model in dimension three we are able to derive spectral methods that can be evaluated through fast algorithms these algorithms are based on a suitable representation and approximation of the collision operator explicit expressions for the errors in the schemes are given and spectral accuracy is proved parallelization properties and adaptivity of the algorithms are also discussed

uniformly stable numerical schemes for the boltzmann equation preserving the compressible navier stokes asymptotics in this paper we develop a numerical method to solve boltzmann like equations of kinetic theory which is able to capture the compressible navier stokes dynamics at small knudsen numbers our approach is based on the micro macro decomposition technique which applies to general collision operators this decomposition is performed in all the phase space and leads to an equivalent formulation of the boltzmann or bgk equation that couples a kinetic equation with macroscopic ones this new formulation is then discretized with a semi implicit time scheme combined with a staggered grid space discretization finally several numerical tests are presented in order to illustrate the efficiency of our approach incidentally we also introduce in this paper a modification of a standard splitting method that allows to preserve the compressible navier stokes asymptotics in the case of the simplified bgk model up to our knowledge this property is not known for general collision operators

numerical solution of the boltzmann equation i spectrally accurate approximation of the collision operator in this paper we show that the use of spectral galerkin methods for the approximation of the boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost we prove that the spectral algorithm preserves the total mass and approximates with infinite order accuracy momentum and energy consistency of the method is also proved and a stability result for a smoothed positive scheme is given we demonstrate that the fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method

efficient computation of tridiagonal matrices largest eigenvalue abstract this paper proposes a method for a fast estimation of the largest eigenvalue of an asymmetric tridiagonal matrix the proposed method is based on the power method and the computation of the square of the original matrix the matrix square is computed through a proposed fast algorithm designed specifically for tridiagonal matrices implementations for compressed column ccs and compressed row storage crs formats are provided discussed and compared to a standard scientific library we investigate the roundoff numerical errors showing that the proposed method provides errors no greater than the usual power method we provide numerical results with simulations in c c implementation in order to demonstrate the effectiveness of the proposed method

numerical methods for hyperbolic conservation laws with stiff relaxation i spurious solutions the author considers the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing finite difference scheme on a fixed uniform spatial grid it is conjectured that certain a priori criteria ensure that the numerical method does not produce spurious solutions as the relaxation time vanishes one criterion is that the limits of vanishing relaxation time and vanishing viscosity commute for the viscous regularization of the hyperbolic system a second criterion is that a certain subcharacteristic condition be satisfied by the hyperbolic system this conjecture is supported with analytical and numerical results for a specific example the solution of generalized riemann problems of a model system of equations with a fractional step scheme in which godunov s method is coupled with the backward euler method similar ideas are applied to the numerical solution of stiff detonation problems

relaxation schemes for nonlinear kinetic equations a class of numerical schemes for nonlinear kinetic equations of boltzmann type is described following wild s approach the solution is represented as a power series with parameter depending exponentially on the knudsen number this permits us to derive accurate and stable time discretizations for all ranges of the mean free path these schemes preserve the main physical properties positivity conservation of mass momentum and energy moreover for some particular models the entropy property is also shown to hold

numerical passage from kinetic to fluid equations a numerical way to pass from the bhatnagar gross krook model of the boltzmann equation to compressible euler equations is presented in order to do so a stable discretization of the kinetic equati

the use of asymptotic modelling in vibration and stability analysis of structures abstract the use of springs with very large stiffness to model constraints in vibratory systems has been a popular approach to overcome the limitations on the choice of admissible functions in the rayleigh ritz method the maximum possible error resulting from this asymptotic modelling can be determined by using positive and negative stiffness values or in general terms using positive and negative penalty functions this paper illustrates how this method could be used to determine the critical loads of structures

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

bipenalty method for time domain computational dynamics penalty functions can be used to add constraints to systems of equations in computational mechanics stiffness type penalties are the common choice however in dynamic applications stiffness penalties have the disadvantage that they tend to decrease the critical time step in conditionally stable time integration schemes leading to increased cpu times for simulations in contrast inertia penalties increase the critical time step in this paper we suggest the simultaneous use of stiffness penalties and inertia penalties which is denoted as the bipenalty method we demonstrate that the accuracy of the bipenalty method is at least as good as and usually better than using either stiffness penalties or inertia penalties furthermore for a number of finite elements bar elements beam elements and square plane stress plane strain elements we have derived ratios of the two penalty parameters such that their combined effect on the critical time step is neutral the bipenalty method is thus superior to using stiffness penalties because the decrease in critical time step can be avoided the bipenalty method is also superior to using inertia penalties since the constraints are realized with higher accuracy

uniformly accurate schemes for hyperbolic systems with relaxation we develop high resolution shock capturing numerical schemes for hyperbolic systems with relaxation in such systems the relaxation time may vary from order 1 to much less than unity when the relaxation time is small the relaxation term becomes very strong and highly stiff and underresolved numerical schemes may produce spurious results usually one cannot decouple the problem into separate regimes and handle different regimes with different methods thus it is important to have a scheme that works uniformly with respect to the relaxation time using the broadwell model of the nonlinear boltzmann equation we develop a second order scheme that works effectively with a fixed spatial and temporal discretization for all ranges of the mean free path formal uniform consistency proof for a first order scheme and numerical convergence proof for the second order scheme are also presented we also make numerical comparisons of the new scheme with some other schemes this study is motivated by the reentry problem in hypersonic computations

asymptotic preserving implicit explicit runge kutta methods for nonlinear kinetic equations we discuss implicit explicit imex runge kutta methods which are particularly adapted to stiff kinetic equations of boltzmann type we consider both the case of easy invertible collision operators and the challenging case of boltzmann collision operators we give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate their monotonicity properties are also studied in the case of the boltzmann operator the methods are based on the introduction of a penalization technique for the collision integral this reformulation of the collision operator permits us to construct penalized imex schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator finally we show some numerical results which confirm the theoretical analysis

a parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix a parallel algorithm called polysection is presented for computing the eigenvalues of a symmetric tridiagonal matrix the method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros the signs of the polynomials at the interval endpoints are determined a priori and used to guarantee that all zeros are found the use of finite precision arithmetic may result in multiple zeros however in this case the intervals coalesce and their number determines exactly the multiplicity of the zero for an n x n matrix the eigenvalues can be determined in o log squared n time with n squared processors and o n time with n processors the method is compared with a parallel variant of bisection that requires o n squared time on a single processor o n time with n processors and o log n time with n squared processors

constraint relationships in linear and nonlinear finite element analyses finite element analysis of engineering structures commonly requires the use of inter nodal displacement constraints current algorithms for this facility elimination lagrange multipliers and penalty function methods are briefly reviewed an alternative approach is proposed which avoids some of the problems of these methods this technique uses solution of the unconstrained stiffness equation with an extended number of right hand sides the resulting solutions are then combined to satisfy the constraints this method has application in the synthesis of sub structures with incompatible boundary variables it is particularly efficient when the constraint affects many variables but the kinematics of the constraint can be expressed in terms of few independent variables e g rigid body motion of part of the structure in common with the lagrange method it produces data for use in a nonlinear analysis with constraints the formulation of rigid body constraints in geometric nonlinearity is detailed and demonstrated