on the σ2 invariants of the generalised r thompson groups of type f abstract the 2 invariants of the generalised thompson group f n x 0 x 1 x n x i x j x i n 1 for i j 0 are calculated for n 3 the case n 2 was solved in bieri et al 2010 5

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

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

blurred derivatives and meshless methods in this work we first introduce and describe the concept of blurred derivatives it is shown how they can be used both to approximate differential equations and to re express them in alternative ways in particular formulations in terms of functional integrals can be obtained using blurred derivatives and extended to non linear problems blurred derivatives are shown to provide higher flexibility for selection of approximation functions than strong and weak formulations some computational implementations of one dimensional problems are discussed and the relationship between several well known numerical methods is analysed finally a meshless numerical scheme for the poisson equation is described in detail its performance is compared with linear finite elements and generalized finite differences on unstructured meshes of points copyright 2002 john wiley sons ltd

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

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

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

variants of relaxed schemes and two dimensional gas dynamics we present a relaxed scheme with more precise information about local speeds of propagation and a multidimensional construction of the cell averages the physical domain of dependence is simulated correctly and high resolution is maintained relaxation schemes have advantages that include high resolution simplicity and explicitly no approximate riemann solvers and no characteristic decomposition is necessary performance of the scheme is illustrated by tests on two dimensional euler equations of gas dynamics

solving the boltzmann equation in n log 2 n in c mouhot and l pareschi fast algorithms for computing the boltzmann collision operator math comp to appear c mouhot and l pareschi c r math acad sci paris 339 2004 pp 71 76 fast deterministic algorithms based on spectral methods were derived for the boltzmann collision operator for a class of interactions including the hard spheres model in three dimensions these algorithms are implemented for the solution of the boltzmann equation in two and three dimensions first for homogeneous solutions then for general nonhomogeneous solutions the results are compared to explicit solutions when available and to monte carlo methods in particular the computational cost and accuracy are compared to those of monte carlo methods as well as to those of previous spectral methods finally for inhomogeneous solutions we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium which was suggested in the work l desvillettes and c villani invent math 159 2005 pp 245 316

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

implicit schemes for the fokker planck landau equation we propose time implicit schemes to solve the homogeneous fokker planck landau equation in both the isotropic and 3 dimensional geometries these schemes have properties of conservation and entropy moreover they allow for large time steps making them faster than the usual explicit schemes to solve the involved linear systems we prove that the use of krylov like solvers preserves the conservation properties we show in particular that the conjugate gradient method can be used numerical tests are performed for the isotropic case and demonstrate an important gain in terms of cpu time with the same accuracy as explicit schemes this work is a first step to the development of fast implicit schemes to solve a class of inhomogeneous kinetic equations

convergence and accuracy of the path integral approach for elastostatics abstract this paper addresses convergence rate and accuracy of a numerical technique for linear elastostatics based on a path integral formulation int j numer math eng 47 2000 1463 the computational implementation combines a simple polynomial approximation of the displacement field with an approximate statement of the exact evolution equations which is designated as functional integral method a convergence analysis is performed for some simple nodal arrays this is followed by two different estimations of the optimum parameter one is based on statistical arguments and the other on inspection of third order residuals when the eight closest neighbors to a node are used for polynomial approximation the optimum parameter is found to depend on poisson s ratio and lie in the range 0 5 0 while bilinear finite elements corresponds exactly to the choice 0 5 the use of the optimum parameter provides better precision than the other two methods with similar computational cost other nodal arrays are also studied both in two and three dimensions and the performance of the fim compared with the corresponding finite element and collocation schemes finally the implementation of fim on unstructured meshes is discussed and a numerical example solving laplace equation is analyzed it is shown that fim compares favorably with fem and offers a number of advantages

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

time relaxed monte carlo methods for the boltzmann equation a new family of monte carlo schemes is introduced for the numerical solution of the boltzmann equation of rarefied gas dynamics the schemes are inspired by the wild sum expansion of the solution of the boltzmann equation for maxwellian molecules and consist of a novel time discretization of the equation in particular high order terms in the expansion are replaced by the equilibrium maxwellian distribution the two main features of the schemes are high order accuracy in time and asymptotic preservation the first property allows to recover accurate solutions with time steps larger than those required by direct simulation monte carlo dsmc while the latter guarantees that for the vanishing knudsen number the numerical solution relaxes to the local maxwellian conservation of mass momentum and energy are preserved by the scheme numerical results on several space homogeneous problems show the improvement of the new schemes over standard dsmc applications to a one dimensional shock wave problem are also presented

the σ2 conjecture for metabelian groups the general case abstract the bieri neumann strebel invariant m g of a group g is a certain subset of a sphere that contains information about finiteness properties of subgroups of g in case of a metabelian group g the set 1 g completely characterizes finite presentability and it is conjectured that it also contains complete information about the higher finiteness properties fpm conjecture the m conjecture states how the higher invariants are obtained from 1 g in this paper we prove the 2 conjecture

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

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

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

functional integral formulation of classical wave equations it is shown in this paper that classical wave equations admit path integral formulations for this the evolution of the system is first set up in terms of a fundamental solution or propagator we choose this last name because it suggests a connection with functional integrals which are exploited in this work a functional integral in terms of non singular functions is then proposed and shown to converge to the propagator in the appropriate limit for the case of scalar wave equations one of the advantages of such formulation is that it provides an adequate framework for mesh free numerical methods this is demonstrated through a computational implementation that combines a simple second degree polynomial local approximation of the continuous field and an approximate statement of the exact evolution equations numerical simulations of modal analysis and transient dynamics indicate the feasibility of the technique

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

discrete velocity models and numerical schemes for the boltzmann bgk equation in plane and axisymmetric geometries we present new numerical models for computing transitional or rarefied gas flows as described by the boltzmann bgk and bgk es equations we first propose a new discrete velocity model based on the entropy minimization principle this model satisfies the conservation laws and the entropy dissipation moreover the problem of conservation and entropy for axisymmetric flows is investigated we find algebraic relations that must be satisfied by the discretization of the velocity derivative appearing in the transport operator then we propose some models that satisfy these constraints owing to these properties we obtain numerical schemes that are economic in terms of discretization and robust in particular we develop a linearized implicit scheme for computing stationary solutions of the discrete velocity bgk and bgk es models this scheme is the basis of a code which can compute high altitude hypersonic flows in 2d plane and axisymmetric geometries our results are analyzed and compared to other methods