continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

structure and perimeters of percolation clusters site percolation clusters are simulated at the percolation threshold on the square lattice an algorithm for walks around each cluster is used to obtain information on its fractal geometry the fractal dimensionality of the external perimeter is found to depend on the size of adsorbent particles used to measure it if the vacant perimeter sites have only nearest neighbour connectivity then the perimeter has dimension de 1 37 or 0 03 instead of that of the hull dh 1 75 the authors also measured the mass of internal dangling sites surrounded by the backbone s blobs singly and doubly connected sites the number of blobs and the average linear distance between entry into the exit from a blob

dsc time domain solution of maxwell s equations a new computational algorithm the discrete singular convolution dsc is introduced for solving scattering and guided wave problems described by time domain maxwell s equations the dsc algorithm is utilized for the spatial discretization and the fourth order runge kutta scheme is used for the time advancing staggered meshes are used for electromagnetic fields four standard test problems a hollow air filled waveguide a dielectric slab loaded rectangular waveguide a shield microstrip line and a dielectric square are employed to illustrate the usefulness to test the accuracy and to explore the limitation of the dsc algorithm results are compared with those of finite difference scaling function multi resolution time domain and finite element based high frequency structure simulator numerical experiments indicate that the present algorithm is a promising approach for achieving high accuracy in electromagnetic wave computations

the fractal dimension of percolation cluster hulls large simulations of 2d site percolation on a square lattice provide several estimates of the fractal dimension dh of the hull or outer boundary of percolation clusters close to criticality statistical estimates based on an ensemble of more than 106 clusters with varying sizes yield dh 1 74 or 0 02 geometrical measurements of the hull two point correlation for individual clusters with more than 105 sites give dh 1 76 or 0 01 the observed scaling behaviour is the same both slightly above and below pc

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

crossover scaling functions and an extended minimal subtraction scheme abstract a field theoretic renormalization group method is presented which is capable of dealing with crossover problems associated with a change in the upper critical dimension the method leads to flow functions for the parameters and coupling constants of the model which depend on the set of parameters that characterize the fixed point landscape of the underlying problem similar to nelson s trajectory integral method any vertex function can be expressed as a line integral along a renormalization group trajectory which in the field theoretic formulation are given by the characteristics of the corresponding callan symanzik equation the field theoretic renormalization automatically leads to a separation of the regular and singular parts of all crossover scaling functions the method is exemplified for the crossover problem in magnetic phase transitions percolation problems and quantum phase transitions the broad applicability of the method is emphasized

crossover scaling a renormalization group approach we derive a theory of crossover scaling based on a scaling variable g g where g is the anisotropy parameter inducing the crossover and g is the correlation length in the presence of g our considerations are field theoretic and based on a renormalization group with a g dependent differential generator that interpolates between qualitatively different degrees of freedom g is a nonlinear scaling field for this renormalization group and interpolates between t t c g v 0 and t t c g v v 0 and v being the isotropic and anisotropic exponents respectively by expanding about a floating fixed point we can make corrections to scaling small throughout the crossover in this formulation effective scaling exponents obey standard scaling laws e g eff v eff 2 eff we discuss its advantages giving for various crossovers explicit supporting perturbative calculations of the susceptibility which is found to conform to the general form derived from the g dependent renormalization group

critical behavior of a colloid polymer mixture confined between walls we investigate the influence of confinement on phase separation in colloid polymer mixtures to describe the particle interactions the colloid polymer model of asakura and oosawa j chem phys 22 1255 1954 is used grand canonical monte carlo simulations are then applied to this model confined between two parallel hard walls separated by a distance d 5 colloid diameters we focus on the critical regime of the phase separation and look for signs of crossover from three dimensional 3d ising to two dimensional 2d ising universality to extract the critical behavior finite size scaling techniques are used including the recently proposed algorithm of kim et al phys rev lett 91 065701 2003 our results point to effective critical exponents that differ profoundly from 3d ising values and that are already very close to 2d ising values in particular we observe that the critical exponent of the order parameter in the confined system is smaller than in 3d bulk yielding a flatter binodal our results also show an increase in the critical colloid packing fraction in the confined system with respect to the bulk the latter seems consistent with theoretical expectations although subtleties due to singularities in the critical behavior of the coexistence diameter cannot be ruled out

series study of random percolation in three dimensions new high density series data for the mean number percolation probability and susceptibility of finite clusters are presented for bond and site percolation on four standard three dimensional lattices a pade approximant analysis of both high and low density series makes particular use of the rather precise unbiased estimates of the percolation threshold pc obtained recently by heermann and stauffer 1981 for the simple cubic lattice for this lattice the authors obtain the biased estimate gamma 1 73 or 0 03 for the site problem and a similar estimate but with larger uncertainties for the bond problem such a value is significantly larger than earlier series estimates assuming gamma to be universal they obtain precise although biased estimates of pc for both bond and site percolation on all four lattices using the bond estimates of pc they find an overall biased estimate of beta 0 454 or 0 008 for bond percolation on all three dimensional lattices the corresponding site problem requires further study scaling estimates of other critical exponents are alpha 0 64 or 0 05 delta 4 81 or 0 14 delta 2 18 or 0 04 nu 0 88 or 0 02 and nu 0 03 or 0 03

a low storage curvilinear discontinuous galerkin method for wave problems the low storage curvilinear discontinuous galerkin lsc dg method reduces the storage requirements for solving symmetric and linear linear symmetric conservation laws in curvilinear domains when compared to the standard discontinuous galerkin method we perform a semidiscrete a priori convergence analysis of lsc dg and determine sufficient conditions on sequences of meshes to guarantee the same rate of convergence as the original upwind dg method on curvilinear domains computational results confirm the optimal order convergence on sample mesh sequences that satisfy the sufficiency conditions additional results show that the sufficient conditions for optimal order convergence of lsc dg may also be necessary conditions

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

dimensional crossover and finite size scaling below t c using the formalism developed in earlier work dimensional crossover on ad dimensional layered ising type system satisfying periodic boundary conditions and of sizel is considered belowt c l t c l being the critical temperature of the finite size system effective critical exponents eff and eff are shown explicitly to crossover between theird and d 1 dimensional values for l in the limitsl l andl l 0 respectively l being the correlation length in the layers using anl dependent renormalization group the effective exponents are shown to satisfy natural generalizations of the standard scaling laws in addition l dependent global scaling fields which span the entire crossover are defined and a scaling form of the equation of state in terms of them derived all the above assertions are verified explicitly to one loop in perturbation theory in particular effective exponents and a universal crossover equation of state are obtained and shown in the above asymptotic limits to be in good agreement with known results

extrapolation of transfer matrix data for percolation and lattice animals by the romberg beleznay algorithm using beleznay s modified romberg algorithm the author re analyses the transfer matrix data for the two dimensional problems of derrida and stauffer 1985 he gets for percolation pc 0 59273 6 for the threshold of the square site problem nu 1 3330 7 for the correlation length exponent and omega 1 9 3 for the correction exponent in the case of lattice animals xc 0 246152 4 square lattice xc 0 19294 1 triangular lattice critical fugacity and nu 0 64062 10 radius exponent

a priori error analysis for hdg methods using extensions from subdomains to achieve boundary conformity we present the first a priori error analysis of a technique that allows us to numerically solve steady state diffusion problems defined on curved domains by using finite element methods defined in polyhedral subdomains dh for a wide variety of hybridizable discontinuous galerkin and mixed methods we prove that the order of convergence in the l2 norm of the approximate flux and scalar unknowns is optimal as long as the distance between the boundary of the original domain and that of the computational domain h is of order h we also prove that the l 2 norm of a projection of the error of the scalar variable superconverges with a full additional order when the distance between and h is of order h 5 4 but with only half an additional order when such a distance is of order h finally we present numerical experiments confirming the theoretical results and showing that even when the distance between and h is of order h the above mentioned projection of the error of the scalar variable can still superconverge with a full additional order

a simple and accurate discontinuous galerkin scheme for modeling scalar wave propagation in media with curved interfaces abstractconventional high order discontinuous galerkin dg schemes suffer from interface errors caused by the misalignment between straight sided elements and curved material interfaces we have developed a novel dg scheme to reduce those errors our new scheme uses the correct normal vectors to the curved interfaces whereas the conventional scheme uses the normal vectors to the element edge we modify the numerical fluxes to account for the curved interface our numerical modeling examples demonstrate that our new discontinuous galerkin scheme gives errors with much smaller magnitudes compared with the conventional scheme although both schemes have second order convergence moreover our method significantly suppresses the spurious diffractions seen in the results obtained using the conventional scheme the computational cost of our scheme is similar to that of the conventional scheme the new dg scheme we developed is thus particularly useful for large scale scalar wave modeling involving complex s

new heuristic percolation criterion for continuum systems a heuristic criterion is introduced and applied for derivation of percolation thresholds in continuum systems the criterion is comprehensive in the sense that it applies to systems with diverse bonding criteria as well as to systems with a correlated distribution of sites its usefulness is manifested by the excellent quantitative agreement between the values obtained and available monte carlo data

monte carlo experiments on percolation the influence of boundary conditions the slow convergence of the cluster size distribution observed in recent monte carlo simulations of two dimensional site percolation is shown to be due to the use of free boundary conditions when periodic boundaries are employed the convergence is improved considerably to the extent that the integrated size distribution for large typically 1010 site lattices is essentially flat over a range of cluster sizes spanning three orders of magnitude the existence of this plateau originally predicted by scaling theory leads to an improved estimate of the critical probability for the square lattice the observed correction to scaling is not readily characterised by a single additive term a fact which helps one to understand the wide range of published exponent estimates

regular article a discontinuous galerkin ale method for compressible viscous flows in moving domains we present a matrix free discontinuous galerkin method for simulating compressible viscous flows in two and three dimensional moving domains to this end we solve the navier stokes equations in an arbitrary lagrangian eulerian ale framework spatial discretization is based on standard structured and unstructured grids but using an orthogonal hierarchical spectral basis the method is third order accurate in time and converges exponentially fast in space for smooth solutions a novelty of the method is the use of a force directed algorithm from graph theory that requires no matrix inversion to efficiently update the grid while minimizing distortions we present several simulations using the new method including validation with published results from a pitching airfoil and new results for flow past a three dimensional wing subject to large flapping insect like motion

illumination a new method for studying 3d percolation fronts in a concentration gradient a new method of studying percolation fronts is introduced in situations where the concentration is not constant gradient percolation approach it consists in an efficient way of determining subsets of the percolation cluster which are exactly at the percolation threshold the method is applied to both lattice and continuum percolation in order to determine the percolation threshold in 3d more accurately two variations of this method illumination and projection seem to give a similar scaling behaviour at low gradients for the cubic lattice case leading to a new estimate for the percolation threshold pc 0 31173 or 0 00007 for continuum percolation of overlapping spheres the new estimate for the critical volume fraction is 0 291 or 0 002 and the value of the critical exponent alpha sigma for the width of the front scaled with the concentration gradient is very close to the one for the lattice case

convergent cartesian grid methods for maxwell s equations in complex geometries a convergent second order cartesian grid finite difference scheme for the solution of maxwell s equations is presented the scheme employs a staggered grid in space and represents the physical location of the material and metallic boundaries correctly hence eliminating problems caused by staircasing and contrary to the popular yee scheme enforces the correct jump conditions on the field components across material interfaces a detailed analysis of the accuracy of the new embedding scheme is presented confirming its second order global accuracy furthermore the scheme is proven to be a bounded error scheme and thus convergent conditions for fully discrete stability is furthermore established this enables the derivation of bounds for fully discrete stability with cfl restrictions being almost identical to those of the much simpler yee scheme the analysis exposes that the effects of staircasing as well as a lack of properly enforced jump conditions on the field components have significant consequences for the global accuracy it is among other things shown that for cases in which a field component is discontinuous along a grid line as happens at general two and three dimensional material interfaces the yee scheme may exhibit local divergence and loss of global convergence to validate the analysis several one and two dimensional test cases are presented showing an improvement of typically 1 to 2 orders of accuracy at little or no additional computational cost over the yee scheme which in most cases exhibits first order accuracy c 2001 academic press

simplified model for fractal dimension of clouds abstract projections of three dimensional random percolation clusters onto a two dimensional plane give roughly the same fractal dimension 1 3 for the perimeter versus area relation as lovejoy s observation of real clouds and cellular automata simulations of nagel and raschke

concentration gradient approach to continuum percolation in two dimensions the author has investigated percolation properties of a system of overlapping discs randomly distributed with a gradient of concentration fractal properties and critical exponents of this system appear to be identical to their counterparts on a lattice this is in agreement with the universality of critical exponents of percolation the concentration gradient approach permits a precise calculation of the percolation threshold corresponding to a critical area fraction of 0 6766 or 0 0005 he has investigated the part of the external perimeter of the percolation cluster which is accessible to discs of various sizes the fractal dimension of this accessible perimeter is found to decrease abruptly as a function of the radius of invading discs from 1 75 or 0 02 to 1 35 or 0 02

a curved boundary treatment for discontinuous galerkin schemes solving time dependent problems for problems defined in a two dimensional domain with boundary conditions specified on a curve we consider discontinuous galerkin dg schemes with high order polynomial basis functions on a geometry fitting triangular mesh it is well known that directly imposing the given boundary conditions on a piecewise segment approximation boundary h will render any finite element method to be at most second order accurate unless the boundary conditions can be accurately transferred from to h in general curvilinear element method should be used to obtain high order accuracy we discuss a simple boundary treatment which can be implemented as a modified dg scheme defined on triangles adjacent to h even though integration along the curve is still necessary integrals over any curved element are avoided if the domain is convex or if is nonconvex and the true solutions can be smoothly extended to the exterior of the modified dg scheme is high order accurate in these cases numerical tests on first order and second order partial differential equations including hyperbolic systems and the scalar wave equation suggest that it is as accurate as the full curvilinear dg scheme

phase separation of symmetrical polymer mixtures in thin film geometry monte carlo simulations of the bond fluctuation model of symmetrical polymer blends confined between two neutral repulsive walls are presented for chain lengthna nb 32 and a wide range of film thicknessd fromd 8 tod 48 in units of the lattice spacing the critical temperaturestc d of unmixing are located by finite size scaling methods and it is shown that t c infty t c d propto d 1 mathord left vphantom 1 v 3 right kern nulldelimiterspace v 3 wherev3 0 63 is the correlation length exponent of the three dimensional ising model universality class contrary to this result it is argued that the critical behavior of the films is ruled by two dimensional exponents e g the coexistence curve difference in volume fraction of a rich and a poor phases scales as phi cocx 2 phi cocx 1 hat b d left 1 t mathord left vphantom t t right kern nulldelimiterspace t c d right beta 2 where 2 is the critical exponent of the two dimensional ising universality class 2 1 8 since for larged this asymptotic critical behavior is confined to an extremely narrow vicinity oftc d one observes in practice effective exponents which gradually cross over from 2 to 3 with increasing film thickness this anomalous flattening of the coexistence curve should be observable experimentally

optimal energy conserving local discontinuous galerkin methods for second order wave equation in heterogeneous media abstract solving wave propagation problems within heterogeneous media has been of great interest and has a wide range of applications in physics and engineering the design of numerical methods for such general wave propagation problems is challenging because the energy conserving property has to be incorporated in the numerical algorithms in order to minimize the phase or shape errors after long time integration in this paper we focus on multi dimensional wave problems and consider linear second order wave equation in heterogeneous media we develop and analyze an ldg method in which numerical fluxes are carefully designed to maintain the energy conserving property and accuracy compatible high order energy conserving time integrators are also proposed the optimal error estimates and the energy conserving property are proved for the semi discrete methods our numerical experiments demonstrate optimal rates of convergence and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property

monte carlo study of invasion percolation clusters in two and three dimensions the authors consider the process of growing invasion percolation clusters from a point into an infinite medium the form considered is the simplest one in which the cluster is permitted to grow into regions it has previously surrounded it is shown that this process can yield extremely good monte carlo estimates of the percolation threshold pc for the square triangular and match square lattices they obtain pc values of 0 5925 or 0 0003 0 5000 or 0 0003 and 0 4072 or 0 0002 and for the simple cubic lattice 0 31158 or 0 00006 the errors quoted are purely statistical and represent one standard deviation two critical exponents are obtained which they suggest should be identified in terms of the fractal dimension d and gap exponent delta of ordinary percolation based on these identifications they obtain values for 1 d and 1 delta of 0 527 or 0 002 and 0 393 or 0 004 in two dimensions and 0 402 or 0 003 and 0 454 or 0 005 in three dimensions these results are consistent with known exact results and best series and monte carlo estimates suggesting that the form of invasion percolation considered is probably in the same universality class as ordinary percolation

site percolation threshold for honeycomb and square lattices a new method of estimating the critical percolation threshold is proposed based on stauffer s cluster number scaling hypothesis and the universality with respect to lattice structure of the corresponding stauffer scaling function this method is illustrated by obtaining estimates of the site percolation threshold for the honeycomb lattice pc 0 6962 or 0 0006 and for the square lattice pc 0 5923 or 0 0007 the error bars or confidence limits of our estimates are substantially smaller than previous series estimates and for the honeycomb lattice would have to be multiplied by the factor 18 to include the value pc 2 12 recently proposed by kondor 1980 to be exact an additional result is r 4 95 or 0 15 where r lims to infinity rs and rs is the ratio of the cluster number scaling function at its maximum to its value at pc for clusters of size s

application of discontinuous galerkin method in supersonic and hypersonic gas flows abstract most high order methods often suffer from two and three dimensional unstructured grids particularly for supersonic and hypersonic gas flows we applied a high order mixed modal discontinuous galerkin method with a coupled slope limiter to investigate multidimensional cases of supersonic and hypersonic gas flows furthermore we provide detailed implementations and validations of two and three dimensional problems examples include double mach reflection problems and supersonic and hypersonic gas flows around a cylinder in addition we provide case studies of a 3d hypersonic vehicle to evaluate its computational performance on complex hypersonic gas flows this study can be regarded as an attempt to apply the discontinuous galerkin method to engineering problems