simulation of confined magnetohydrodynamic flows with dirichlet boundary conditions using a pseudo spectral method with volume penalization a volume penalization approach to simulate magnetohydrodynamic mhd flows in confined domains is presented here the incompressible visco resistive mhd equations are solved using parallel pseudo spectral solvers in cartesian geometries the volume penalization technique is an immersed boundary method which is characterized by a high flexibility for the geometry of the considered flow in the present case it allows to use other than periodic boundary conditions in a fourier pseudo spectral approach the numerical method is validated and its convergence is assessed for two and three dimensional hydrodynamic hd and mhd flows by comparing the numerical results with results from literature and analytical solutions the test cases considered are two dimensional taylor couette flow the z pinch configuration three dimensional orszag tang flow ohmic decay in a periodic cylinder three dimensional taylor couette flow with and without axial magnetic field and three dimensional hartmann instabilities in a cylinder with an imposed helical magnetic field finally we present a magnetohydrodynamic flow simulation in toroidal geometry with non symmetric cross section and imposing a helical magnetic field to illustrate the potential of the method

a multidimensional flux function with applications to the euler and navier stokes equations abstract a grid independent approximate riemann solver has been developed for use in both two and three dimensional flows governed by the euler or navier stokes equations fluxes on grid faces are obtained via wave decomposition by assuming that information propagates in the velocity difference directions rather than in the grids normal directions as in a standard grid aligned solver this flux function more appropriately interprets and hence more accurately resolves shock and shear waves when these lie oblique to the grid the model which describes the difference in states at each grid interface by the action of five waves produces a significant increase in accuracy for both supersonic and subsonic first order spatially accurate computations second order computations with the grid independent flux function are generally only worth the additional effort for flowfields whose primary structures include shear waves that lie oblique to the grid included in this category is the viscous flow over an airfoil in which a separated shear layer emanates from the airfoil surface at an angle to the grid pressure distortions which can result from misinterpretation of the oblique waves by a grid aligned solver are essentially eliminated by the grid independent flux function

on the numerical solution of two point boundary value problems abstract this paper presents a new numerical method for the solution of linear two point boundary value problems of ordinary differential equations after reducing the differential equation to a second kind integral equation the latter are discretized via a high order nystrom scheme a somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using o n dot p2 operations where n is the number of nodes on the interval and p is the desired order of convergence thus the advantages of the integral equation formulation small condition number insensitivity to boundary layers insensitivity to end point singularities etc are retained while achieving a computational efficiency previously available only to finite difference or finite element methods keywords two point boundary value problems integral equations chebyshev polynomials approximation theory

interior penalty discontinuous galerkin method for magnetic induction equation with resistivity abstract we design and analyze the interior penalty discontinuous galerkin discretization of the magnetic induction equation with resistivity the resulting semi discrete scheme is shown to be energy stable and consistent numerical experiments are performed in order to demonstrate the accuracy and convergence of the dg scheme through the l 2 error and divergence error analysis by incorporating several time discretization schemes

finite difference approximation of a nonlinear integro differential system finite difference approximation of the nonlinear integro differential system associated with the penetration of a magnetic field into a substance is studied the convergence of the finite difference scheme is proved the rate of convergence of the discrete scheme is given the decay of the numerical solution is compared with the analytical results proven earlier

stretched exponential fixation in stochastic ising models at zero temperature we study a class of continuous time markov processes which describes 1 spin flip dynamics on the hypercubic lattice d d 2 with initial spin configurations chosen according to the bernoulli product measure with density p of spins 1 during the evolution the spin at each site flips at rate c 0 or 0 0 q t exp t 1 d for large t in d 2 we obtain the complementary bound for arbitrary 0 q t exp t 1 2 for large t

phase ordering after a deep quench the stochastic ising and hard core gas models on a tree consider a low temperature stochastic ising model in the phase coexistence regime with markov semigroup p t a fundamental and still largely open problem is the understanding of the long time behavior of p t when the initial configuration is sampled from a highly disordered state e g a product bernoulli measure or a high temperature gibbs measure exploiting recent progresses in the analysis of the mixing time of monte carlo markov chains for discrete spin models on a regular b ary tree open image in new window we study the above problem for the ising and hard core gas independent sets models on open image in new window if is a biased product bernoulli law then under various assumptions on the bias and on the thermodynamic parameters we prove almost sure weak convergence of p t to an extremal gibbs measure pure phase and show that the limit is approached at least as fast as a stretched exponential of the time t in the context of randomized algorithms and if one considers the glauber dynamics on a large finite tree our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution

an integral equation approach to the unsteady convection diffusion equations an integral equation approach is proposed to solve the unsteady convection diffusion equations in this approach the second order adams moulton method is firstly utilized for the time discretization then by using the green s function of the laplace equation in the series form the convection diffusion equation is transformed into an integral equation that is further converted into an algebraic equation system the accuracy convergence and the stability of this integral equation approach are examined by four examples in comparison with the characteristic variational multiscale method and the finite volume element method the integral equation approach shows a higher accuracy compared with the finite volume element method the integral equation approach has a better convergence in solving the convection dominated convection diffusion problems the integral equation approach demonstrates a good stability

integral equation method for coupled schrodinger equations a new integral equation method for the numerical solution of the radial schrodinger equation in one dimension developed by the authors 1997 j comput phys 134 134 is extended to systems of coupled schrodinger equations with both positive and negative channel energies the method carried out in configuration space is based on the conversion of differential equations into a system of integral equations together with the application of a spectral type clenshaw curtis quadrature an accompanying general multichannel fortran code is available upon request

glauber dynamics on trees and hyperbolic graphs we study continuous time glauber dynamics for random configurations with local constraints e g proper coloring ising and potts models on finite graphs with n vertices and of bounded degree we show that the relaxation time defined as the reciprocal of the spectral gap 1 2 for the dynamics on trees and on planar hyperbolic graphs is polynomial in n for these hyperbolic graphs this yields a general polynomial sampling algorithm for random configurations we then show that for general graphs if the relaxation time 2 satisfies 2 o 1 then the correlation coefficient and the mutual information between any local function which depends only on the configuration in a fixed window and the boundary conditions decays exponentially in the distance between the window and the boundary for the ising model on a regular tree this condition is sharp

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

an interior penalty galerkin method for the mhd equations in heterogeneous domains the maxwell equations in the magnetohydrodynamic mhd limit in heterogeneous domains composed of conducting and non conducting regions are solved by using lagrange finite elements and by enforcing continuities across interfaces using an interior penalty technique a la baker finite element methods for elliptic equations using non conforming elements math comp 31 137 1977 45 59 the method is shown to be stable and convergent and is validated by convergence tests it is used to compute ohmic decay in various compact conducting domains and to simulate the kinematic dynamo action in two different geometries

discontinuous galerkin methods for hamiltonian odes and pdes in this article we present a unified framework of discontinuous galerkin dg discretizations for hamiltonian odes and pdes we show that with appropriate numerical fluxes the numerical algorithms deduced from dg discretizations can be combined with the symplectic methods in time to derive the multi symplectic prk schemes the resulting numerical discretizations are applied to the linear and nonlinear schrodinger equations some conservative properties of the numerical schemes are investigated and confirmed in the numerical experiments

glauber dynamics on trees boundary conditions and mixing time we give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the glauber dynamics in the so called bethe approximation specifically we show that the spectral gap and the log sobolev constant of the glauber dynamics for the ising model on an n vertex regular tree with boundary are bounded below by a constant independent of n at all temperatures and all external fields this implies that the mixing time is o logn in contrast to the free boundary case where it is not bounded by any fixed polynomial at low temperatures in addition our methods yield simpler proofs and stronger results for the spectral gap and log sobolev constant in the regime where the mixing time is insensitive to the boundary condition our techniques also apply to a much wider class of models including those with hard core constraints like the antiferromagnetic potts model at zero temperature proper colorings and the hard core lattice gas independent sets

zero temperature glauber dynamics on the 3 regular tree and the median process in zero temperature glauber dynamics vertices of a graph are given i i d initial spins sigma x 0 from 1 1 with mathbb p p sigma x 0 1 p and they update their spins at the arrival times of i i d poisson processes to agree with a majority of their neighbors we study this process on the 3 regular tree mathbb t 3 where it is known that the critical threshold p c below which mathbb p p a s all spins fixate to 1 is strictly less than 1 2 defining theta p to be the mathbb p p probability that a vertex fixates to 1 we show that theta is a continuous function on 0 1 so that in particular theta p c 0 to do this we introduce a new continuous spin process we call the median process which gives a coupling of all the measures mathbb p p along the way we study the time infinity agreement clusters of the median process show that they are a s finite and deduce that all continuous spins flip finitely often in the second half of the paper we show a correlation decay statement for the discrete spins under mathbb p p for a e value of p the proof relies on finiteness of a vertex s trace in the median process to derive a stability of discrete spins under finite resampling last we use our methods to answer a question of howard j appl probab 37 736 747 2000 on the emergence of spin chains in mathbb t 3 in finite time

functionally fitted energy preserving methods for solving oscillatory nonlinear hamiltonian systems in the last few decades numerical simulation for nonlinear oscillators has received a great deal of attention and manyresearchers have been concerned with the design and analysis ofnumerical methods for solving oscillatory problems in this paper from the perspective of the continuous finite element method we propose and analyze new energy preserving functionally fitted methods in particular trigonometrically fitted methods of an arbitrarily high order for solving oscillatory nonlinear hamiltonian systems with a fixed frequency to implement these new methods in a widespread way they are transformed into a class of continuous stage runge kutta methods this paper is accompanied by numerical experiments on oscillatory hamiltonian systems such as the fpu problem and nonlinear schrodinger equation the numerical results demonstrate the remarkable accuracy and efficiency of our new methods compared with the existing high order energy preserving methods in the literature

a high order scheme for unsteady heat conduction equations abstract in this work a high order scheme is proposed for solving the heat conduction problems in this scheme the governing equation of heat conduction is firstly written into a system of first order partial differential equations which are integrated over the control volumes around each node point and a finite time step the resulting time integrals are approximated by numerical quadrature with an undetermined weighting parameter the error analysis allows us to determine the parameter and the ratio of the time step to the square of grid spacing by making the error of the scheme as small as possible both theoretical and numerical results show that the proposed high order scheme can achieve at least sixth order accuracy at the expense of the same computing cost as the traditional finite volume method and the high order scheme demonstrates a better performance than the high order compact finite difference method and the runge kutta discontinuous galerkin method

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

self organized segregation on the grid we consider an agent based model with exponentially distributed waiting times in which two types of agents interact locally over a graph and based on this interaction and on the value of a common intolerance threshold tau decide whether to change their types this is equivalent to a zero temperature ising model with glauber dynamics an asynchronous cellular automaton with extended moore neighborhoods or a schelling model of self organized segregation in an open system and has applications in the analysis of social and biological networks and spin glasses systems some rigorous results were recently obtained in the theoretical computer science literature and this work provides several extensions we enlarge the intolerance interval leading to the expected formation of large segregated regions of agents of a single type from the known size epsilon 0 to size approx 0 134 namely we show that for 0 433 tau 1 2 and by symmetry 1 2 tau 0 567 the expected size of the largest segregated region containing an arbitrary agent is exponential in the size of the neighborhood we further extend the interval leading to expected large segregated regions to size approx 0 312 considering almost segregated regions namely regions where the ratio of the number of agents of one type and the number of agents of the other type vanishes quickly as the size of the neighborhood grows in this case we show that for 0 344 tau le 0 433 and by symmetry for 0 567 le tau 0 656 the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood this behavior is reminiscent of supercritical percolation where small clusters of empty sites can be observed within any sufficiently large region of the occupied percolation cluster the exponential bounds that we provide also imply that complete segregation where agents of a single type cover the whole grid does not occur with high probability for p 1 2 and the range of intolerance considered

extension of the piecewise parabolic method to multidimensional ideal magnetohydrodynamics an extension of the piecewise parabolic method to treat multidimensional ideal magnetohydrodynamical equations is presented in this paper the multidimensional scheme is constructed from a one dimensional functioning code based on the dimensional splitting method originally suggested by strang the functioning code is built upon a nonlinear riemann solver for ideal mhd equations recently developed by the authors the correctness of the scheme is tested in the steepening of waves in both one and two dimensional situations and in various mhd shock tube problems which involve all the discontinuities in ideal mhd the robust character of the scheme is demonstrated in the shock tube problems and in the interaction between mhd shocks and a cloud the results of these problems show that the scheme keeps the principal advantages of a high order godunov scheme robust operation in the presence of very strong waves thin shock fronts with little attendant noise generation and thin contact discontinuity

exponential integrators with quadratic energy preservation for linear poisson systems abstract motivated by the recent paper wang and wu 2018 12 where the authors proposed a class of functionally fitted energy preserving integrators for poisson systems in this paper we aim at developing exponential integrators with quadratic energy preservation for the important class of linear poisson systems by rewriting the linear poisson system in a new formulation we apply the symplectic exponential runge kutta methods and thus obtain a novel class of exponential integrators for the linear poisson system it is proved that the obtained exponential integrator can exactly preserve the quadratic energy in comparison with the functionally fitted energy preserving methods whose numerical performance depends on the fitted frequency the new exponential integrators are independent of the fitted frequency and convenient to use furthermore we investigate the convergence of the fixed point iteration used in the implementation of the proposed exponential integrator and show that the convergence region of the stepsize is independent of the norm of the skew symmetric matrix finally numerical results with the euler equations of a free rigid body soundly support the remarkable performance of the exponential integrator in both the global errors and the energy preservation

wulff droplets and the metastable relaxation of kinetic ising models we consider the kinetic ising models glauber dynamics corresponding to the infinite volume ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h minimal conditions on the flip rates are assumed so that all the common choices are being considered we study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature t and the evolution is started from a distribution which is stochastically lower than the phase we show that as h 0 the relaxation time blows up as exp c t h with lgr c t w t 2 12 t m t here m t is the spontaneous magnetization and w t is the integrated surface tension of the wulff body of unit volume moreover for 0 c the state of the process at time exp h is shown to be close when h is small to the phase the difference between this state and the phase can be described in terms of an asymptotic expansion in powers of the external field this expansion can be interpreted as describing a set of continuations in h of the family of gibbs distributions with the negative magnetic fields into the region of positive fields

symmetric integrators based on continuous stage runge kutta nystrom methods for reversible systems abstract in this paper we study symmetric integrators for solving second order ordinary differential equations on the basis of the notion of continuous stage runge kutta nystrom methods the construction of such methods heavily relies on the legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions new families of symmetric integrators as illustrative examples are presented for comparing the numerical behaviors of the presented methods some numerical experiments are also reported

poloidal toroidal decomposition in a finite cylinder i influence matrices for the magnetohydrodynamic equations the navier stokes equations and magnetohydrodynamics equations are written in terms of poloidal and toroidal potentials in a finite cylinder this formulation insures that the velocity and magnetic fields are divergence free by construction but leads to systems of partial differential equations of higher order whose boundary conditions are coupled the influence matrix technique is used to transform these systems into decoupled parabolic and elliptic problems the magnetic field in the induction equation is matched to that in an exterior vacuum by means of the dirichlet to neumann mapping thus eliminating the need to discretize the exterior the influence matrix is scaled in order to attain an acceptable condition number

a hybrid finite difference boundary element procedure for the simulation of turbulent mhd duct flow at finite magnetic reynolds number a conservative coupled finite difference boundary element computational procedure for the simulation of turbulent magnetohydrodynamic flow in a straight rectangular duct at finite magnetic reynolds number is presented the flow is assumed to be periodic in the streamwise direction and is driven by a mean pressure gradient the duct walls are considered to be electrically insulated the co evolution of the velocity and magnetic fields as described respectively by the navier stokes and the magnetic induction equations together with the coupling of the magnetic field between the conducting domain and the non conducting exterior is solved using the magnetic field formulation the aim is to simulate localized magnetic fields interacting with turbulent duct flow detailed verification of the implementation of the numerical scheme is conducted in the limiting case of low magnetic reynolds number by comparing with the results obtained using a quasistatic approach that has no coupling with the exterior the rigorous procedure with non local magnetic boundary conditions is compared with simplified pseudo vacuum boundary conditions and the differences are quantified our first direct numerical simulations of turbulent hartmann duct flow at moderate magnetic reynolds numbers and a low flow reynolds number show significant differences in the duct flow turbulence even at low interaction level between the flow and magnetic field

a staggered mesh algorithm using high order godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations the equations of magnetohydrodynamics mhd have been formulated as a hyperbolic system of conservation laws in that form it becomes possible to use higher order godunov schemes for their solution this results in a robust and accurate solution strategy however the magnetic field also satisfies a constraint that requires its divergence to be zero at all times this is a property that cannot be guaranteed in the zone centered discretizations that are favored in godunov schemes without involving a divergence cleaning step in this paper we present a staggered mesh strategy which directly uses the properly upwinded fluxes that are provided by a godunov scheme the process of directly using the upwinded fluxes relies on a duality that exists between the fluxes obtained from a higher order godunov scheme and the electric fields in a plasma by exploiting this duality we have been able to construct a higher order godunov scheme that ensures that the magnetic field remains divergence free up to the computer s round off error we have even presented a variant of the basic algorithm that uses multidimensional features in the flow to design an upwinded strategy that aligns itself with the predominant upwinded direction in the flow we have devised several stringent test problems to show that the scheme works robustly and accurately in all situations in doing so we have shown that a scheme that involves a collocation of magnetic field variables that is different from the one traditionally favored in the design of higher order godunov schemes can nevertheless offer the same robust and accurate performance of higher order godunov schemes provided the properly upwinded fluxes from the godunov methodology are used in the scheme s construction

phase transitions on nonamenable graphs we survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph most attention is devoted to transitive graphs and trees

exponential collocation methods for conservative or dissipative systems abstract in this paper we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods it is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or lyapunov functions we also consider order estimates of the new methods furthermore we explore and discuss the application of our methods in important stiff gradient systems and it turns out that our methods are unconditionally energy diminishing and strongly damped even for very stiff gradient systems practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear schrodinger equation as a byproduct of this paper arbitrary order trigonometric rkn collocation methods are also presented and analysed for second order highly oscillatory general systems the paper is accompanied by numerical results that demonstrate the great potential of this work

the integral equation method for a steady kinematic dynamo problem with only a few exceptions the numerical simulation of cosmic and laboratory hydromagnetic dynamos has been carried out in the framework of the differential equation method however the integral equation method is known to provide robust and accurate tools for the numerical solution of many problems in other fields of physics the paper is intended to facilitate the use of integral equation solvers in dynamo theory in concrete the integral equation method is employed to solve the eigenvalue problem for a hydromagnetic dynamo model with an isotropic helical turbulence parameter for the case of spherical geometry three examples of the function r with steady and oscillatory solutions are considered a convergence rate proportional to the inverse squared of the number of grid points is achieved based on this method a convergence accelerating strategy is developed and the convergence rate is improved remarkably typically quite accurate results can be obtained with a few tens of grid points in order to demonstrate its suitability for the treatment of dynamos in other than spherical domains the method is also applied to x2 dynamos in rectangular boxes the magnetic fields and the electric potentials for the first eigenvalues are visualized

integral equations a tool to solve the schrodinger equation from the integral equation equivalent to the one dimensional schrodinger equation with the boundary conditions appropriate for a bound state problem an exact three point integration rule is derived the approximate evaluation of this rule by means of the euler mclaurin sum rule gives rise to various o h2 and o h4 integration methods associated with the different ways of splitting the starting equation into a free part and an interaction term the method is particularly useful to deal with the singularities of the potential

an integro differential formulation for magnetic induction in bounded domains boundary element finite volume method simulations of magnetohydrodynamic mhd flows in bounded domains using spectral methods suffer from a number of serious limitations alternative methods based on local discretization raise the problem of how to implement non local boundary conditions for the magnetic field we have developed a new strategy for the numerical solution of mhd problems in bounded domains which combines the flexibility of a local discretization with a rigorous formulation of magnetic boundary conditions next to an insulator in arbitrary geometries in accordance with the character of underlying equations we apply a global integral approach at the boundary and a differential approach inside the conducting domain the formulation of the boundary problem in terms of primitive variables allows us to combine these approaches and propose a mixed finite volume and boundary element method we illustrate its efficiency on magnetic diffusion problems in a sphere and in a finite cylinder

analysis and numerical approximation of a stationary mhd flow problem with nonideal boundary we are concerned with the steady flow of a conducting fluid confined to a bounded region of space and driven by a combination of body forces externally generated magnetic fields and currents entering and leaving the fluid through electrodes attached to the surface the flow is governed by the navier stokes equations in the fluid region and maxwell s equations in all of space coupled via ohm s law and the lorentz force by means of the biot savart law we reduce the problem to a system of integro differential equations in the fluid region derive a mixed variational formulation and prove its well posedness under a small data assumption we then study the finite element approximation of solutions in the case of unique solvability and establish optimal order error estimates finally an implementation of the method is described and illustrated with the results of some numerical experiments

zero temperature glauber dynamics on mathbb z d we study zero temperature glauber dynamics on mathbb z d which is a dynamic version of the ising model of ferromagnetism spins are initially chosen according to a bernoulli distribution with density p and then the states are continuously and randomly updated according to the majority rule this corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field to one at zero temperature with no external field define p c mathbb z d to be the infimum over p such that the system fixates at with probability 1 it is a folklore conjecture that p c mathbb z d 1 2 for every 2 le d in mathbb n we prove that p c mathbb z d to 1 2 as d

spectral method for matching exterior and interior elliptic problems a spectral method is described for solving coupled elliptic problems on an interior and an exterior domain the method is formulated and tested on the two dimensional interior poisson and exterior laplace problems whose solutions and their normal derivatives are required to be continuous across the interface a complete basis of homogeneous solutions for the interior and exterior regions corresponding to all possible dirichlet boundary values at the interface are calculated in a preprocessing step this basis is used to construct the influence matrix which serves to transform the coupled boundary conditions into conditions on the interior problem chebyshev approximations are used to represent both the interior solutions and the boundary values a standard chebyshev spectral method is used to calculate the interior solutions the exterior harmonic solutions are calculated as the convolution of the free space green s function with a surface density this surface density is itself the solution to an integral equation which has an analytic solution when the boundary values are given as a chebyshev expansion properties of chebyshev approximations insure that the basis of exterior harmonic functions represents the external near boundary solutions uniformly the method is tested by calculating the electrostatic potential resulting from charge distributions in a rectangle the resulting influence matrix is well conditioned and solutions converge exponentially as the resolution is increased the generalization of this approach to three dimensional problems is discussed in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum

the percolation transition for the zero temperature stochastic ising model on the hexagonal lattice on the planar hexagonal lattice mathbb h we analyze the markov process whose state t in 1 1 mathbb h updates each site v asynchronously in continuous time t 0 so that v t agrees with a majority of its three neighbors the initial v 0 s are i i d with p v 0 1 p 0 1 we study both rigorously and by monte carlo simulation the existence and nature of the percolation transition as t and p 1 2 denoting by t p the expected size of the plus cluster containing the origin we 1 prove that 1 2 and 2 study numerically critical exponents associated with the divergence of p as p 1 2 a detailed finite size scaling analysis suggests that the exponents and of this t dependent percolation model have the same values 4 3 and 43 18 as standard two dimensional independent percolation we also present numerical evidence that the rate at which t as t is exponential

integral equation method for the continuous spectrum radial schrodinger equation a new approach to the numerical solution of boundary value problems for differential equations which originated in recent papers by greengard and rokhlin is improved and adapted to the numerical solution of the radial schrodinger equation the approach is based on the conversion of the differential equation into an integral equation together with the application of a spectral type clenshaw curtis quadrature method through numerical examples the integral equation method is shown to be superior to finite difference methods