a monte carlo method for series expansions the authors describe a simple and general algorithm to calculate series expansions in enumeration problems to large orders approximately by a monte carlo method it can be used to generate unbiased samples in cluster studies e g linear or branched polymers random surfaces etc in any dimension they calculate the number of site animals of size n on the square lattice for n or 50 and their average size to better than 1 accuracy

an implicit finite volume method for the solution of 3d low mach number viscous flows using a local preconditioning technique this paper presents a cell centered high order finite volume scheme for the solution of the three dimensional 3d navier stokes equations with low mach number the system of non linear equations is solved by means of a fully implicit pseudo transient scheme each pseudo time step is solved by a newton gmres procedure a local preconditioning technique is used to scale the speed of sound and to improve the system condition number for low mach number and low cell reynolds number this preconditioning is applied to the ausm up flux vector splitting function the method is tested on 2d and 3d low mach number laminar flows

persistency of two dimensional self avoiding walks the authors investigate the persistency of self avoiding walks on the square lattice by studying the distribution of endpoint displacements projected in the direction of the first step exact enumeration and the constant fugacity monte carlo data indicate that the mean displacement after n steps xn increases logarithmically with n while the higher odd moments x2k 1n vary as n2kv in n where v is the correlation length exponent this unusual behaviour for the moments is found to arise from a simple scaling behaviour of the asymmetric component of the displacement distribution which in the constant fugacity ensemble can be written in the form a x approximately n 2v in n exp bx nv where n is the average number of steps in the walk at a fixed value of the fugacity and b is a coefficient of order unity

tricriticality and persistency of trails and silhouettes trails and silhouettes polymers with loops are generated using exact enumeration on a face centred cubic lattice by assigning a tunable fugacity factor with each intersection it is observed that as the fugacity for intersections is increased the trail and silhouette configurations change from swollen to compact ones indicating the existence of tricritical points strong divergence of the specific heats for different path lengths is used to locate these tricritical points the dlog pade scheme is used to compute the corresponding tricritical exponents in the limit of large path lengths the tricritical exponents thus obtained show that these tricritical points are distinct from the usual theta point and that trails and silhouettes do not belong to the saw universality class the tricritical exponents further show that trails and silhouettes do not belong to the same universality class persistency arising from fixing the initial step is also investigated and it is found that odd moments of the persistency obey a power law that can be expressed in terms of the critical exponent nu and the path length l

nonperturbative tricritical exponents of trails i exact enumeration on a triangular lattice as the fugacity of intersections in trails intersecting but non overlapping lattice walks is increased the configurations change from swollen ones with the scaling exponents of self avoiding walks to compact ones separating the two regimes is a potentially new tricritical point with no perturbative renormalisation fixed point associated with it supporting evidence for the existence of a tricritical point its likely location and exponents are computed for the first time from exact enumeration of all trails up to length of fifteen lattice constants on the triangular lattice the divergence of the specific heat indicates the location of the tricritical point generalised ratio and pade methods are used to extract the scaling exponents for the number of configurations and their end to end distance

some recent variations on the expected number of distinct sites visited by an n step random walk abstract asymptotic forms for the expected number of distinct sites visited by an n step random walk being calculable for many random walks have been used in a number of analyses of physical models we describe three recent extensions of the problem the first replacing the single random walker by n random walkers the second to the study of a random walk in the presence of a trapping site and the third to a random walk in the presence of a trapping hyperplane

self avoiding polygons on the square l and manhattan lattices transfer matrix techniques are used to extend the self avoiding polygon generating function on the square lattice to terms in x46 corresponding to 46 step polygons these techniques are then extended to apply to directed square lattices such as the l and manhattan lattice and the self avoiding polygon generating function to x48 is found for these lattices series analysis confirms that the specific heat exponent alpha 1 2 for the self avoiding walk problem and gives the following estimates for the connective constants mu sq 2 638155 or 0 000025 mu l 1 5657 or 0 0019 and mu man 1 7328 or 0 0005 some evidence for a correction to scaling exponent delta approximately 0 84 is found from square lattice series

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

blended ausm method for all speeds and all grid aspect ratios an ausm based discretization method using an explicit third order discretization for the convective part and a line implicit central discretization for the acoustic part and for the diffusive part has been developed for incompressible and low speed compressible navier stokes equations the lines are chosen in the direction of the gridpoints with shortest connection the semi implicit line method is used in multistage form because of the explicit third order discretization of the convective part multigrid is used as acceleration technique because of the implicit treatment of the acoustic and the diffusive terms the stiffness otherwise caused by high aspect ratio cells is removed low mach number stiffness is treated by a preconditioning technique to ensure physical correct behavior of the discretization for vanishing mach number extreme care has been taken for vanishing mach number stabilization terms are added to the mass flux pressure and temperature stabilization terms are necessary the coefficients of these terms are chosen so that correct scaling with mach number is obtained a blend of the low speed algorithm with the original ausm algorithm developed for high speed applications has been constructed so that the resulting algorithm can be used at all speeds

on the survival probability of a random walk in a finite lattice with a single trap we consider the survival of a random walker in a finite lattice with periodic boundary conditions the initial position of the random walker is uniformly distributed on the lattice with respect to the trap we show that the survival of a random walker un can be exactly related to the expected number of distinct sites visted on a trap free lattice by un 1 sn nd wherend is the number of lattice points ind dimensions we then analyze the behavior of sn in any number of dimensions by using tauberian methods we find that at sufficiently long times sn decays exponentially withn in all numbers of dimensions ind 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid forn2 n 1 whend 1 andn lnn n 1 whend 2 no such crossover exists when z 3 the form of suggests that the single trap approximation is indeed a valid low concentration limit for survival on an infinite lattice with a finite concentration of traps

new exact exponents for two dimensional self avoiding walks the author establishes using standard coulomb gas methods the values of a new set of geometrical exponents that have been previously conjectured these exponents give for any p the number of configurations of p two dimensional self avoiding walks of the same length l which are attached by their ends

stability of parallel volterra runge kutta methods abstract in this paper we analyse parallel iteration of volterra runge kutta methods pivrk methods for solving second kind volterra integral equations on parallel computers we focus on the determination of the region of convergence c and on the stability region sm of the iterated method obtained after m iterations results are presented for the convolution test equation it turns out that the stability region sm does not necessarily converge to the stability region s of the corrector however for finite m sm need not to be contained in c or s and may be much larger than c

position space renormalisation group for isolated polymer chains we develop a cell position space renormalisation group psrg with which we study the scaling properties of isolated polymer chains we model a chain by a self avoiding walk constrained to a lattice for rescaling factors b s 6 we calculate recursion relations analytically on the square lattice with several different choices for the psrg weight function we also calculate implicit cell to cell transformations in which a cell of size b is rescaled to a cell of size b the results of these psrgs improve both as b increases and as b b 1 in addition we construct a true infinitesimal psrg transformation which appears to become exact as the dimensionality d approaches 1 we obtain the closed form expression for the correlation length exponent u d l d in d the flory formula deviates from this already at first order in d 1 we also develop a constant fugacity monte carlo method which enables us to simu late in an unbiased way within the grand canonical ensemble chains of up to lo3 bonds with this method we extend the psrg to larger cells b s 150 on the square lattice our numerical method provides high statistical accuracy for all cell sizes however in the range of b we study the asymptotic behaviour of our results appears to depend on the choice of weight function one weight function provides smooth behaviour as a function of b and with it we extrapolate to find u 0 756 0 004 further work is required to resolve the apparent anomalies in the results based on the other weight functions

backward differentiation type formulas for volterra integral equations of the second kind numerical integration formulas are discussed which are obtained by differentiation of the volterra integral equation and by applying backward differentiation formulas to the resulting integro differential equation in particular the stability of the method is investigated for a class of convolution kernels the accuracy and stability behaviour of the method proposed in this paper is compared with that of i a block implicit runge kutta scheme and ii the scheme obtained by applying directly a quadrature rule which is reducible to the backward differentiation formulas the present method is particularly advantageous in the case of stiff volterra integral equations

study of the θ point by enumeration of self avoiding walks on the triangular lattice the author reports series expansion analyses of self avoiding walks with nearest neighbour bond interactions the estimates 2 nu t 1 07 or 0 05 and phi 0 64 or 0 05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end to end distance data up to 16 steps on the triangular lattice

monte carlo test of a hyperscaling relation for the two dimensional self avoiding walk the authors simulated self avoiding walks on the square lattice with fixed endpoints by means of a dynamic monte carlo algorithm from these data they obtain an evaluation of the effective coordination number mu and the critical exponents alpha and v they can therefore test the hyperscaling relation 2 alpha dv with a careful estimate of systematic and statistical errors

polymer conformations through wiggling a new monte carlo method is proposed which allows for the efficient generation of equilibrium conformations of polymer chains in two and three dimensions the method treats each site monomer as a potential pivot around which a new conformation may be generated by rotating a portion of the chain the method does not suffer from the severe attrition associated with the simple sampling of self avoiding walks and may be extended to treat the interacting polymer chain the authors find in two dimensions that nu 0 748 or 0 005 exact 0 750 and in three dimensions nu 0 595 or 0 005 series expansion and renormalisation group predict nu approximately 0 588 the end end distances calculated for shorter chains are in good agreement with the exact values from enumeration techniques

a generating function for the second moment of the distinct number of sites visited by an n step lattice random walk we derive a generating function for the second moment of the distinct number of sites visited by an n step lattice random walk the formalism allows us to find asymptotic forms for the second moment as previously given in the mathematical literature using much more complicated techniques the general technique can in principle be utilized in a derivation of higher moments

on the mean radius of gyration of a polymer chain calculates the intrachain bead to bead square distance of a real polymer chain close to the critical dimensionality d 4 from this the authors obtain the mean square radius of gyration s2 of the polymer as a function of the molecular weight n of the chain and the excluded volume parameter mu a proportionality relationship between s2 and the mean square end to end distance r2 of the coil previously suggested by enumerations of self avoiding walks on lattice is shown to be true an estimate of the universal ratio s2 r2 is given

persistency studies of trails and silhouettes on square and triangular lattices the authors study the persistencies of trails and silhouettes on two dimensional square and triangular lattices these persistencies are induces by fixing the first step of the walk in the positive x direction trails are walks which are not allowed to overlap on themselves but may self intersect whereas silhouettes are just the shadows of trails associated with each self intersection is an attractive interaction energy epsilon mod epsilon mod thus a fugacity factor f i theta theta eitkbt ei theta where i is the number of intersections may be defined this has the advantage of providing a handle on the temperature and such dependence of persistencies on temperature is studied for the first time in this paper they are found to obey a law that may be expressed as a function of the critical exponents x2k 1l theta approximately lpk nu theta f l where l is the chain length nu is the correlation exponent f l approximately loge l or some weak function of l e g lw where w 1 0 p is a parameter and k 0 1 2

discrete time waveform relaxation volterra runge kutta methods convergence analysis the discrete time relaxation methods based on volterra runge kutta methods for solving large system of second kind volterra integral equations are proposed convergence of the discrete time iteration process with particular attention to parallel methods is investigated

persistency properties of models of polymers on simple cubic and face centred cubic lattices the first four odd moments of the persistence lengths of trails and silhouettes are studied on two different 3d lattices the simple cubic lattice and the face centred cubic lattice variations of the averaged persistence lengths with chain lengths l and inverse temperatures theta are systematically examined it is found that the averaged persistence lengths scale with a scaling law of the form xl2k 1 theta approximately lpk nu theta f l where nu is the correlation exponent p is a parameter k 0 1 2 and f l approximately constant in contrast to the results in 2d where f l approximately loge l

a note on lattice random walks with an excluded point we derive some asymptotic results for the moments of displacement of a random walk on a lattice with a single absorbing point in one dimension we are able to find an asymptotic expression for the probability distribution of random walks that have not been trapped

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

the persistence length of two dimensional self avoiding random walks the decay of directional correlations in self avoiding random walks on the square lattice is investigated analysis of exact enumerations and monte carlo data suggest that the correlation between the directions of the first step and the jth step of the walk decays faster than j 1 indicating that the persistence length of the walk is finite

a soluble seven vertex model for clusters with interfacial bending energy a model describing an ensemble of disjoint clusters on a square lattice is introduced in addition to the usual surface energy terms in x and in y the model includes a bending energy in c per corner in the interface it is formulated as a seven vertex model which is in general not solvable for beta in c ln 2 2 however this seven vertex model becomes free fermionic one and an exact expression for its free energy is derived in the extreme anisotropic hamiltonian limit the operator content coincides with that of the 1d ising model in a transverse field three critical lines with ising like behaviour are found

criticality of self avoiding walks with an excluded infinite needle the authors study the effect of repulsion for self avoiding walks and random walks from excluded sets they show in particular that the mean displacement away from an excluded infinite needle of self avoiding random walks in three dimensions has to diverge along the privileged axis as nsigma where n is the number of steps and sigma is a sub leading critical exponent for the two point function this exponent has been determined by using a high precision monte carlo simulation sigma 0 370 or 0 011 its knowledge is used to improve the measure of universal quantities like the exponent nu nu 0 5867 or 0 0025 in agreement with the in expansion estimate and with experimental data and amplitude ratios they verify also that for simple random walks the excluded needle introduces instead logarithmic violations to scaling

repulsion of random and self avoiding walks from excluded points and lines the authors investigate the repulsion of random walks rws and self avoiding walks saws induced by a excluding a single lattice point or b excluding all points on a half line from infinity to the origin for saws they use exact enumeration and monte carlo methods to study the asymptotic behaviour of the displacement away from the excluded set on three four and five dimensional hypercubic lattices when the saw begins one lattice site away from an excluded point along the x direction the mean displacement after n steps xn approaches a finite limit at a power law rate as n to infinity however the distribution of projected displacements exhibits a residual asymmetry in the asymptotic limit reflective of a long range influence of the excluded point this general pattern of behaviour also occurs for a purely random walk for a saw which starts one lattice spacing away from the axis of the excluded half line enumeration data suggest that the mean displacement along the axis diverges as n nu with v 3 4 in two dimensions but as nz with z approximately 0 35 in three dimensions this unexpected behaviour appears to be corroborated by constant fugacity monte carlo data however for the corresponding rw model both the solution to the diffusion equation and a heuristic argument indicate that xn diverges as n1 2 in two dimensions and as n1 2 ln n in three dimensions

random walk in a random multiplicative environment we investigate the dynamics of a random walk in a random multiplicative medium this results in a random but correlated multiplicative process for the spatial distribution of random walkers we show how the details of these correlations determine the asymptotic properties of the walk i e the central limit theorem does not apply to these multiplicative processes we also study a periodic source trap medium in which a unit cell contains one source followed byl 1 traps we calculate the asymptotic behavior of the number of particles and determine the conditions for which there is growth or decay in this average number finally we discuss the asymptotic behavior of a random walk in the presence of randomly distributed partially absoprbing traps for this case a temporal regime of purely exponential decay of the density can occur before the asymptotic stretched exponential decay exp at1 3 sets in

highly stable parallel volterra runge kutta methods a new family of parallel v stable methods for second kind volterra integral equations are proposed the methods belong to the class of volterra runge kutta methods and can be applied provided that some not very restrictive conditions on the integral equation kernel are satisfied

the winding angle of planar self avoiding walks the problem of understanding the asymptotic statistical behaviour of the winding angle on of a self avoiding walk of n steps on a planar lattice is posed exact series expansion data for the square lattice up to n 21 are reported these data and monte carlo estimates up to n g 170 steps are fitted well by a logarithmic growth law for 06 the ratio oh appears to approach a limit of 2 9 to 3 2 which is close to the gaussian value 3 heuristic scaling arguments are consistent with simple logarithmic growth and also illuminate the logarithmic behaviour known rigorously for free brownian motion

non perturbative tricritical exponents of trails ii exact enumerations on square and simple cubic lattices for pt i see ibid vol 21 p 773 1988 the authors present exact enumerations of trails self intersecting but non overlapping lattice walks tabulated according to their length l number of intersection i and end to end distance for the square lattice up to l 21 i 7 and for the simple cubic lattice up to l 15 i 5 they introduce a fugacity for intersection to explore the transition to a collapsed phase through a novel tricritical point which is unaccessible by the renormalisation group approach the existence of tricritical points in both lattices is manifested by the divergence of the specific heat the values of the tricritical couplings and exponents are extracted by a dlogpade analysis they find for the three dimensional lattice nu t approximately 0 48 and gamma t approximately 0 43 nu t approximately 0 52 and gamma t approximately 1 25 for the square lattice

end to end distance of linear polymers in two dimensions a reassessment recent exact enumeration studies of the two dimensional self avoiding walk have produced evidence in support of non analytic scaling corrections newly extended series for the triangular and square lattices to n 19 and 25 respectively are examined in the light of recent monte carlo results suggesting the absence of corrections of this type using a modified approach to series analysis it is concluded that the end to end distance can best be understood in terms of analytic corrections alone in view of the lack of theoretical support there is little reason to introduce additional kinds of correction in attempting to explain the data

parallel algorithm with spectral convergence for nonlinear integro differential equations we discuss a numerical algorithm for solving nonlinear integro differential equations and illustrate our findings for the particular case of volterra type equations the algorithm combines a perturbation approach meant to render a linearized version of the problem and a spectral method where unknown functions are expanded in terms of chebyshev polynomials el gendi s method this approach is shown to be suitable for the calculation of two point green functions required in next to leading order studies of time dependent quantum field theory

performance of compressible flow codes at low mach numbers the accuracy and the performance of three two dimensional compressible flow codes at freestream mach numbers as low as 0 001 are examined two of the codes employ a finite volume discretization scheme along with a multistage time stepping algorithm to solve the euler equations the two codes differ in their respective use of cell centered and node centered differencing schemes the third code uses an implicit finite difference procedure to solve the unsteady navier stokes equations computational test cases are the inviscid steady flow over a circular cylinder and the impulsively started viscous flow over a cylinder

a multigrid semi implicit line method for viscous incompressible and low mach number flows on high aspect ratio grids abstract discretization of viscous incompressible and viscous low mach number flows often leads to a system of equations which is very difficult to solve there are two reasons first the use of high aspect ratio grids results in a very numerically anisotropic behaviour of the diffusive and acoustic terms and second in low mach number flow the ratio of the convective and acoustic eigenvalues of the inviscid system becomes very high we implemented an ausm based discretization method using an explicit third order discretization for the convective part and a line implicit central discretization for the acoustic part and for the diffusive part the lines are chosen in the direction of the grid points with shortest connection the preconditioned semi implicit line method is used in multistage form because of the explicit third order discretization of the convective part multigrid is used as an acceleration technique it is shown that the convergence is very good independent of grid aspect ratio and mach number

new monte carlo method for the self avoiding walk we introduce a new monte carlo algorithm for the self avoiding walk saw and show that it is particularly efficient in the critical region long chains we also introduce new and more efficient statistical techniques we employ these methods to extract numerical estimates for the critical parameters of the saw on the square lattice we find 2 63820 0 00004 0 00030 1 352 0 006 0 025 v 0 7590 0 0062 0 0042 where the first error bar represents systematic error due to corrections to scaling subjective 95 confidence limits and the second bar represents statistical error classical 95 confidence limits these results are based on saws of average length 166 using 340 hours cpu time on a cdc cyber 170 730 we compare our results to previous work and indicate some directions for future research

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

the self avoiding walk on the honeycomb lattice the author analyses the chain generating function for the self avoiding walk problem on the honeycomb lattice she uses the exact critical point value of nienhuis and finds exponent values of gamma approximately 1 344 and delta 1 approximately 0 93 for the dominant and first confluent exponents the former value is in agreement with nienhuis exact result second and third confluent exponents delta 2 approximately 1 2 and delta 3 approximately 1 5 are also identified these exponents being consistent with monte carlo results and nienhuis exact irrelevant eigenvalue respectively

on solutions of the equation a ka ii the magnetic force free model for pt i see ibid vol 19 l101 4 1986 solutions of the vector differential equation del a ka k constant not 0 have been shown not to correspond to physical magnetic fields this is true even though they are solenoidal del a 0 which is a necessary but not sufficient condition for a vector field to be a magnetic field here the magnetic force free model is very carefully analysed and the inapplicability of the above equation is demonstrated using physical arguments

computation of flows with arbitrary equations of state the extension of time marching computations to fluids with arbitrary equations of state is demonstrated by means of stability analyses simplified problems and practical applications most of the examples use the properties of supercritical hydrogen for which the density varies by more than an order of magnitude for small changes in pressure and temperature but representative computations for incompressible fluids and perfect gases are also given to demonstrate the generality of the procedure because representative flow velocities in typical supercritical fluids applications are much lower than the speed of sound convergence enhancement through eigenvalue control is often necessary this is accomplished through a generalization of earlier preconditioning methods that enables efficient computation of arbitrary equation of state fluids perfect gases and incompressible fluids by a single procedure the present approach thus provides a single method that is uniformly applicable to all equations of state

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

random walks on lattices with points of two colours i abstract this paper is concerned with random walks on lattices with two kinds of points black and white the colours of the points are random variables with a translation invariant but otherwise arbitrary joint probability distribution the steps of the walk are independent of the colours we study the stochastic properties of the length of the subwalk from the starting point to a first black point and of subwalks between points visited in succession and establish a number of exact relations these relations can be applied to a trapping problem by identifying the black points with imperfect traps an example is discussed

a new flux splitting scheme a new flux splitting scheme is proposed the scheme is remarkably simple and yet its accuracy rivals and in some cases surpasses that of roe s solver in the euler and navier stokes solutions carried out in this study the scheme is robust and converges as fast as the roe splitting we propose an appropriately defined cell face advection mach number using values from the two straddling cells via associated characteristic speeds this interface mach number is then used to determine the upwind extrapolation for the convective quantities accordingly the name of the scheme is coined as the advection upstream splitting method ausm we also introduce a new pressure splitting which is shown to behave successfully yielding much smoother results than other existing pressure splittings of particular interest is the supersonic blunt body problem in which the roe scheme gives anomalous solutions the ausm produces correct solutions without difficulty for a wide range of flow conditions as well as grids

the diffusion of self avoiding random walk in high dimensions we use the brydges spencer lace expansion to prove that the mean square displacement of at step strictly self avoiding random walk in thed dimensional square lattice is asymptotically of the formdt ast approaches infinity ifd is sufficiently large the diffusion constantd is greater than one

two dimensional polymers universality and correction to scaling improved monte carlo methods have been used to study the asymptotic behaviour of self avoiding walks on two dimensional lattices the mean square end to end distance rn2 and radius of gyration sn2 are both found to scale as n2v with exponent values very close to the expected v 3 4 and an amplitude ratio that is universal small n deviations of rn2 from the asymptotic results indicate a correction to scaling exponent delta 1 a value that differs from previous estimates

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out

an exactly solvable self avoiding walks model ii triangular and honeycomb lattices a modified self avoiding walks model previously proposed for the square lattice is applied to the triangular and honeycomb lattices in the model the walker is restricted not to take any turn which will put the walker in a direction rotated by more than a certain angle phi max from any of the directions previously taken the generating functions are obtained and various quantities are evaluated exactly for all three square triangular and honeycomb lattices it is found that the model exhibits the characteristics reminiscent of one dimensional self avoiding walks in all directions while strongly retaining the anisotropic effect of the direction of the first step for the honeycomb lattice the authors have studied the model for phi max pi a model and phi max 3 2 pi b model unlike other walks the b model exhibits a special property the oscillations between even and odd steps in various quantities such as the number of n step walks do not decay as the number of steps increases

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