a second order operator splitting method for allen cahn type equations with nonlinear source terms abstract allen cahn ac type equations with nonlinear source terms have been applied to a wide range of problems for example the vector valued ac equation for phase separation and the phase field equation for dendritic crystal growth in contrast to the well developed first and second order methods for the ac equation not many second order methods are suggested for the ac type equations with nonlinear source terms due to the difficulties in dealing with the nonlinear source term numerically in this paper we propose a simple and stable second order operator splitting method a core idea of the method is to decompose the original equation into three subequations with the free energy evolution term the heat evolution term and a nonlinear source term respectively it is important to combine these three subequations in proper order to achieve the second order accuracy and stability we propose a method with a half time free energy evolution solver a half time heat evolution solver a full time midpoint solver for the nonlinear source term and a half time heat evolution solver followed by a final half time free energy evolution solver we numerically demonstrate the second order accuracy of the new numerical method through the simulations of the phase separation and the dendritic crystal growth

first and second order operator splitting methods for the phase field crystal equation in this paper we present operator splitting methods for solving the phase field crystal equation which is a model for the microstructural evolution of two phase systems on atomic length and diffusive time scales a core idea of the methods is to decompose the original equation into linear and nonlinear subequations in which the linear subequation has a closed form solution in the fourier space we apply a nonlinear newton type iterative method to solve the nonlinear subequation at the implicit time level and thus a considerably large time step can be used by combining these subequations we achieve the first and second order accuracy in time we present numerical experiments to show the accuracy and efficiency of the proposed methods

adaptive finite elements with high aspect ratio for the computation of coalescence using a phase field model a multiphase field model for the description of coalescence in a binary alloy is solved numerically using adaptive finite elements with high aspect ratio the unknown of the multiphase field model are the three phase fields solid phase 1 solid phase 2 and liquid phase a lagrange multiplier and the concentration field an euler implicit scheme is used for time discretization together with continuous piecewise linear finite elements at each time step a linear system corresponding to the three phases plus the lagrange multiplier has to be solved then the linear system pertaining to concentration is solved an adaptive finite element algorithm is proposed in order to reduce the number of mesh vertices the generated meshes contain elements with high aspect ratio the refinement and coarsening criteria are based on an error indicator which has already been justified theoretically for simpler problems numerical results on two test cases show the efficiency of the method

unconditionally stable second order schemes for gradient regularized non convex finite strain elasticity modeling martensitic phase transformations abstract in the setting of continuum elasticity martensitic phase transformations are characterized by a non convex free energy density function that possesses multiple wells in strain space and includes higher order gradient terms for regularization metastable martensitic microstructures defined as solutions that are local minimizers of the total free energy are of interest and are obtained as steady state solutions to the resulting transient formulation of toupin s gradient elasticity at finite strain this type of problem poses several numerical challenges including stiffness the need for fine discretization to resolve microstructures and following solution branches accurate time integration schemes are essential to obtain meaningful solutions at reasonable computational cost in this work we introduce two classes of unconditionally stable second order time integration schemes for gradient elasticity each having relative advantages over the other numerical examples are shown highlighting these features

discontinuous galerkin finite element approximation of the cahn hilliard equation with convection the paper is concerned with the construction and convergence analysis of a discontinuous galerkin finite element method for the cahn hilliard equation with convection using discontinuous piecewise polynomials of degree p geq1 and backward euler discretization in time we show that the order parameter c is approximated in the broken rm l infty rm h 1 norm with optimal order cal o h p tau the associated chemical potential w phi c gamma 2 delta c is shown to be approximated with optimal order cal o h p tau in the broken rm l 2 rm h 1 norm here phi c frac 1 4 1 c 2 2 is a quartic free energy function and gamma 0 is an interface parameter numerical results are presented with polynomials of degree p 1 2 3

a stable and conservative finite difference scheme for the cahn hilliard equation we propose a stable and conservative finite difference scheme to solve numerically the cahn hilliard equation which describes a phase separation phenomenon numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill posed problem we design a new difference scheme based on a general strategy proposed recently by furihata and mori the new scheme inherits characteristic properties the conservation of mass and the decrease of the total energy from the equation the decrease of the total energy implies boundedness of discretized sobolev norm of the solution this in turn implies by discretized sobolev s lemma boundedness of max norm of the solution and hence the stability of the solution an error estimate for the solution is obtained and the order is o delta x 2 delta t 2 numerical examples demonstrate the effectiveness of the proposed scheme

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

second order convex splitting schemes for periodic nonlocal cahn hilliard and allen cahn equations we devise second order accurate unconditionally uniquely solvable and unconditionally energy stable schemes for the nonlocal cahn hilliard nch and nonlocal allen cahn nac equations for a large class of interaction kernels we present numerical evidence that both schemes are convergent we solve the nonlinear equations resulting from discretization using an efficient nonlinear multigrid method and demonstrate the performance of our algorithms by simulating nucleation and crystal growth for several different choices of interaction kernels

stable and efficient finite difference nonlinear multigrid schemes for the phase field crystal equation in this paper we present and compare two unconditionally energy stable finite difference schemes for the phase field crystal equation the first is a one step scheme based on a convex splitting of a discrete energy by wise et al s m wise c wang j s lowengrub an energy stable and convergent finite difference scheme for the phase field crystal equation siam j numer anal in press in this scheme which is first order in time and second order in space the discrete energy is non increasing for any time step the second scheme we consider is a new fully second order two step algorithm in the new scheme the discrete energy is bounded by its initial value for any time step in both methods the equations at the implicit time level are nonlinear but represent the gradients of strictly convex functions and are thus uniquely solvable regardless of time step size we solve the nonlinear equations using an efficient nonlinear multigrid method numerical simulations are presented and confirm the stability efficiency and accuracy of the schemes

upper semicontinuity of attractors for approximations of semigroups and partial differential equations abstract suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite dimensional system conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined applications are given to parabolic and hyperbolic partial differential equations

convergence of a fast explicit operator splitting method for the epitaxial growth model with slope selection a fast explicit operator splitting method for the epitaxial growth model with slope selection has been presented in cheng et al j comput phys 303 2015 pp 45 65 the original problem is split into linear and nonlinear subproblems for the linear part the pseudospectral method is adopted for the nonlinear part a 33 point difference scheme is constructed here we give a compact center difference scheme involving fewer points for the nonlinear subproblem in addition we analyze the convergence rate of the algorithm the global error order mathcal o tau 2 h 4 in discrete l 2 norm is proved theoretically and verified numerically some numerical experiments show the robustness of the algorithm for small coefficients of the fourth order term for the one dimensional case in addition coarsening dynamics are simulated in large domains and the 1 3 power laws are observed for the two dimensional case

robustness of design against autocorrelation in time ii optimally theoretical and numerical results for the first order autoregressive process abstract the optimum design and its efficiency relative to the uniform design for estimating the mean of a stationary first order autoregressive process plus an independent error are characterized completely this is related to the work of jones 1948 optimum designs and variances of least squares estimates are calculated numerically for the problem of estimating the slope in a simple linear regression when the errors follow the above structure for a range of values of the sample size and the parameters of the process numerical results in both cases are compared with asymptotic values obtained in bickel and herzberg 1979 the asymptotic optimality of the uniform design is borne out

efficient local energy dissipation preserving algorithms for the cahn hilliard equation abstract in this paper we show that the cahn hilliard equation possesses a local energy dissipation law which is independent of boundary conditions and produces much more information of the original problem to inherit the intrinsic property we derive three novel local structure preserving algorithms for the 2d cahn hilliard equation by the concatenating method in particular when the nonlinear bulk potential f in the equation is chosen as the ginzburg landau double well potential the method discussed by zhang and qiao 2012 50 is a special case of our scheme ii thanks to the leibnitz rules and properties of operators the three schemes are rigorously proven to conserve the discrete local energy dissipation law in any local time space region under periodic boundary conditions the schemes are proven to possess the discrete mass conservation and total energy dissipation laws numerical experiments are conducted to show the performance of the proposed schemes

a new conservative swift hohenberg equation and its mass conservative method abstract the swift hohenberg sh energy functional has been widely used to study pattern formation the l 2 and h 1 gradient flows for the sh energy functional are the sh and phase field crystal pfc equations respectively the sh equation is of lower order in space than the pfc equation but does not conserve the total mass furthermore when the sh energy functional does not have the cubic nonlinearity the sh equation unlike the pfc equation only shows striped patterns even for various parameter values in this study we introduce a new mass conservative sh equation by using a lagrange multiplier in order to solve the conservative sh equation that is an integro partial differential equation we propose operator splitting methods that are shown analytically to inherit the mass conservation numerical examples including standard tests in the pfc equation are presented to show the applicability of the proposed framework

an unconditionally energy stable second order finite element method for solving the allen cahn equation abstract in this paper we design analyze and numerically validate an unconditionally energy stable second order numerical method for solving the allen cahn equation which represents a model for anti phase domain coarsening in a binary alloy the proposed scheme inherits the property of the decrease of the total energy from the allen cahn equation an error estimate for the fully discretized scheme is also established numerical examples are presented to confirm the accuracy efficiency and stability of the proposed method in particular the method is shown to be unconditionally energy stable and second order accurate in both time and space discretizations

a lyapunov type theorem for dissipative numerical integrators with adaptive time stepping the asymptotic behavior of continuous dissipative systems and dissipative numerical integrators with fixed time stepping can be fully investigated by a lyapunov type theorem on continuous and discrete dynamical systems respectively however once adaptive time stepping is involved such theories cease to work and usually the dynamics should be investigated from the past instead of the standard forward way such as in terms of pullback attractors in this paper we present a different approach we stick to a forward definition of limit sets and show that still we can establish a lyapunov type theorem which reveals the precise asymptotic behavior of adaptive time stepping integrators in the presence of a discrete lyapunov functional

cosine law at the atomic scale toward realistic simulations of knudsen diffusion we propose to reconsider the diffusion of atoms in the knudsen regime in terms of a complex dynamical reflection process by means of molecular dynamics simulations we emphasize the asymptotic nature of the cosine law of reflection at the atomic scale and carefully analyze the resulting strong correlations in the reflection events a dynamical interpretation of the accommodation coefficient associated with the slip at the wall interface is also proposed finally we show that the first two moments of the stochastic process of reflection depend nonuniformly on the incident angle

a new approach to optimal design for linear models with correlated observations we consider the problem of designing experiments for regression in the presence of correlated observations with the location model as the main example for a fixed correlation structure approximate optimal designs are determined explicitly and it is demonstrated that under the model assumptions made by bickel and herzberg 1979 for the determination of asymptotic optimal design the designs derived in this article converge weakly to the measures obtained by these authors we also compare the asymptotic optimal design concepts of sacks and ylvisaker 1966 1968 and bickel and herzberg 1979 and point out some inconsistencies of the latter finally we combine the best features of both concepts to develop a new approach for the design of experiments for correlated observations and it is demonstrated that the resulting design problems are related to the logarithmic potential theory

the spectrum of the billiard laplacian of a family of random billiards random billiards are billiard dynamical systems for which the reflection law giving the post collision direction of a billiard particle as a function of the pre collision direction is specified by a markov scattering operator p billiards with microstructure are random billiards whose markov operator is derived from a microscopic surface structure on the boundary of the billiard table the microstructure in turn is defined in terms of what we call a billiard cellq the shape of which completely determines the operator p this operator defined on an appropriate hilbert space is bounded self adjoint and for the examples considered here a hilbert schmidt operator a central problem in the statistical theory of such random billiards is to relate the geometric characteristics of q and the spectrum of p we show for a particular family of billiard cell shapes parametrized by a scale invariant curvature k fig 2 that the billiard laplacian p i is closely related to the ordinary spherical laplacian and indicate by partly analytical and partly numerical means how this provides asymptotic information about the spectrum of p for small values of k it is shown in particular that the second moment of scattering about the incidence angle closely approximates the spectral gap of p

computation of multiphase systems with phase field models phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near critical interfacial phenomena phase separation under shear and microstructure evolution during solidification however because interfaces are replaced by thin transition regions diffuse interfaces phase field simulations require resolution of very thin layers to capture the physics of the problems studied this demands robust numerical methods that can efficiently achieve high resolution and accuracy especially in three dimensions we present here an accurate and efficient numerical method to solve the coupled cahn hilliard navier stokes system known as model h that constitutes a phase field model for density matched binary fluids with variable mobility and viscosity the numerical method is a time split scheme that combines a novel semi implicit discretization for the convective cahn hilliard equation with an innovative application of high resolution schemes employed for direct numerical simulations of turbulence this new semi implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the cahn hilliard equation at the same cost as that of an explicit scheme it is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity moreover we solve the navier stokes equations with a robust time discretization of the projection method that guarantees better stability properties than those for crank nicolson based projection methods for channel geometries the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction the capabilities of the method are demonstrated with several examples including phase separation with and without shear in two and three dimensions the method effectively resolves interfacial layers of as few as three mesh points the numerical examples show agreement with analytical solutions and scaling laws where available and the 3d simulations in the presence of shear reveal rich and complex structures including strings

convex splitting rungekutta methods for phase field models in this paper we present the convex splitting rungekutta csrk methods which provide a simple unified framework to solve phase field models such as the allencahn cahnhilliard and phase field crystal equations the core idea of the csrk methods is the combination of convex splitting methods and multi stage implicitexplicit rungekutta methods our csrk methods are high order accurate in time and we investigate the energy stability numerically we present numerical experiments to show the accuracy and efficiency of the proposed methods up to the third order accuracy

a simple and efficient finite difference method for the phase field crystal equation on curved surfaces abstract we present a simple and efficient finite difference method for the phase field crystal pfc equation on curved surfaces embedded in r 3 we employ a narrow band neighborhood of a curved surface that is defined as a zero level set of a signed distance function the pfc equation on the surface is extended to the three dimensional narrow band domain by using the closest point method and applying a pseudo neumann boundary condition we can use the standard seven point discrete laplacian operator instead of the discrete laplace beltrami operator on the surface the pfc equation on the narrow band domain is discretized using an unconditionally stable scheme and the resulting implicit discrete system of equations is solved by using the jacobi iterative method computational results are presented to demonstrate the efficiency and usefulness of the proposed method

first and second order numerical methods based on a new convex splitting for phase field crystal equation the phase field crystal equation derived from the swifthohenberg energy functional is a sixth order nonlinear equation we propose numerical methods based on a new convex splitting for the phase field crystal equation the first order convex splitting method based on the proposed splitting is unconditionally gradient stable which means that the discrete energy is non increasing for any time step the second order scheme is unconditionally weakly energy stable which means that the discrete energy is bounded by its initial value for any time step we prove mass conservation unique solvability energy stability and the order of truncation error for the proposed methods numerical experiments are presented to show the accuracy and stability of the proposed splitting methods compared to the existing other splitting methods numerical tests indicate that the proposed convex splitting is a good choice for numerical methods of the phase field crystal equation

limit theorems in the stadium billiard we prove that the birkhoff sums for almost every relevant observable in the stadium billiard obey a non standard limit law more precisely the usual central limit theorem holds for an observable if and only if its integral along a one codimensional invariant set vanishes otherwise a open image in new window normalization is needed as one of the two key steps in the argument we obtain a limit theorem that holds in young towers with exponential return time statistics in general an abstract result that seems to be applicable to many other situations

an efficient and stable compact fourth order finite difference scheme for the phase field crystal equation abstract in this paper we present a high order accurate compact scheme for the phase field crystal model in two and three dimensional spaces the proposed scheme is derived by combining a fourth order compact finite difference formula in space and a backward differentiation for the time derivative term which is second order accurate in time furthermore a nonlinearly stabilized splitting scheme is used and thus a larger time step can be allowed since the equations at the implicit time level are nonlinear we introduce a newton type iterative method and employ a fast and efficient nonlinear multigrid solver to solve the resulting discrete system in particular we implement the compact scheme in the adaptive mesh refinement framework an adaptive time step method for the phase field crystal model is also proposed various numerical experiments are presented and confirm the accuracy stability and efficiency of our proposed method

second order schemes and time step adaptivity for allen cahn and cahn hilliard models abstract in this paper we focus on efficient second order in time approximations of the allen cahn and cahn hilliard equations first of all we present the equations generic second order schemes based on a mid point approximation of the diffusion term and some schemes already introduced in the literature then we propose new ways of deriving second order in time approximations of the potential term starting from the main schemes introduced in guillen gonzalez and tierra 2013 yielding to new second order schemes for these schemes and other second order schemes previously introduced in the literature we study the constraints on the physical and discrete parameters that can appear to assure the energy stability unique solvability and in the case of nonlinear schemes the convergence of newton s method to the nonlinear schemes moreover in order to save computational cost we have developed a new adaptive time stepping algorithm based on the numerical dissipation introduced in the discrete energy law in each time step finally we compare the behaviour of the schemes and the effectiveness of the adaptive time stepping algorithm through several computational experiments

error estimates for time discretizations of cahn hilliard and allen cahn phase field models for two phase incompressible flows we carry out rigorous error analysis for some energy stable time discretization schemes developed in shen and yang siam j sci comput 32 3 1159 1179 2010 for a cahn hilliard phase field model and in shen and yang chin ann math ser b 31 5 743 758 2010 for an allen cahn phase field model

billiards in a general domain with random reflections we study stochastic billiards on general tables a particle moves according to its constant velocity inside some domain fancyscript d subset mathbb r d until it hits the boundary and bounces randomly inside according to some reflection law we assume that the boundary of the domain is locally lipschitz and almost everywhere continuously differentiable the angle of the outgoing velocity with the inner normal vector has a specified absolutely continuous density we construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location velocity we mainly focus on the case of bounded domains then we prove exponential ergodicity of these two markov processes we study their invariant distribution and their normal gaussian fluctuations of particular interest is the case of the cosine reflection law the stationary distributions for the two processes are uniform in this case the discrete time chain is reversible though the continuous time process is quasi reversible also in this case we give a natural construction of a chord picked at random in fancyscript d and we study the angle of intersection of the process with a d 1 dimensional manifold contained in fancyscript d

unconditionally stable methods for gradient flow using convex splitting runge kutta scheme abstract we propose a convex splitting runge kutta csrk scheme which provides a simple unified framework to solve a gradient flow in an unconditionally gradient stable manner the key feature of the scheme is a combination of a convex splitting method and a specially designed multi stage two additive runge kutta method our methods are high order accurate in time and assure the gradient energy stability for any time step size we provide detailed proof of the unconditional energy stability and present issues on the practical implementations we demonstrate the accuracy and stability of the proposed methods using numerical experiments of the cahn hilliard equation

the numerical computation of heteroclinic connections in systems of gradient partial differential equations the numerical computation of heteroclinic connections in partial differential equations pdes with a gradient structure such as those arising in the modeling of phase transitions is considered initially a scalar reaction diffusion equation is studied structural assumptions are made on the problem to ensure the existence of absorbing sets and consequently a global attractor as a result of the gradient structure it is known that if all equilibria are hyperbolic the global attractor comprises the set of equilibria and heteroclinic orbits connecting equilibria to one another thus it is natural to consider direct approximation of the set of equilibria and the connecting orbits results are proved about the fourier spanning basis for branches of equilibria and also for certain heteroclinic connections these results exploit the oddness of the nonlinearity the reaction diffusion equation is then approximated by a galerkin spectral discretization to produce a system of ordinary differential equations

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints abstract we present a set of linear second order unconditionally energy stable schemes for the allen cahn model with nonlocal constraints for crystal growth that conserves the mass of each phase solvability conditions are established for the linear systems resulting from the schemes convergence rates are verified numerically dynamics obtained using the allen cahn model with nonlocal constraints are compared with the one obtained using the classic allen cahn model as well as the cahn hilliard model respectively demonstrating slower dynamics than that of the allen cahn model but faster dynamics than that of the cahn hilliard model thus the allen cahn model with nonlocal constraints can serve as an alternative to the cahn hilliard model in simulating crystal growth while conserving the mass of each phase two benchmark examples are presented to contrast the predictions made with the four models highlighting the accuracy and effectiveness of the allen cahn model with nonlocal constraints

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

the global dynamics of discrete semilinear parabolic equations a class of scalar semilinear parabolic equations possessing absorbing sets a lyapunov functional and a global attractor are considered the gradient structure of the problem implies that provided all steady states are isolated solutions approach a steady state as t to infty the dynamical properties of various finite difference and finite element schemes for the equations are analysed the existence of absorbing sets bounded independently of the mesh size is proved for the numerical methods discrete lyapunov functions are constructed to show that under appropriate conditions on the mesh parameters numerical orbits approach steady state solutions as discrete time increases however it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented furthermore it is also shown that the explicit euler scheme introduces spurious solutions with period 2 in the timestep as a result the absorbing set is destroyed and there is initial data leading to blow up of the scheme however small the mesh parameters are taken to obtain stabilization to a steady state for this scheme it is necessary to restrict the timestep in terms of the initial data and the space step implicit schemes are constructed for which absorbing sets and lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step both one step and multistep bdf methods are studied

a new approach to optimal designs for correlated observations this paper presents a new and efficient method for the construction of optimal designs for regression models with dependent error processes in contrast to most of the work in this field which starts with a model for a finite number of observations and considers the asymptotic properties of estimators and designs as the sample size converges to infinity our approach is based on a continuous time model we use results from stochastic anal ysis to identify the best linear unbiased estimator blue in this model based on the blue we construct an efficient linear estimator and corresponding optimal designs in the model for finite sample size by minimizing the mean squared error between the opti mal solution in the continuous time model and its discrete approximation with respect to the weights of the linear estimator and the optimal design points in particular in the multi parameter case in contrast to previous work on the subject the resulting estimators and corresponding optimal designs are very efficient and easy to implement this means that they are practi cally not distinguishable from the weighted least squares estimator and the corresponding optimal designs which have to be found numerically by non convex discrete optimization the advantages of the new approach are illustrated in several numerical examples

optimal design for linear models with correlated observations in the common linear regression model the problem of determining optimal designs for least squares estimation is considered in the case where the observations are correlated a necessary condition for the optimality of a given design is provided which extends the classical equivalence theory for optimal designs in models with uncorrelated errors to the case of dependent data if the regression functions are eigenfunctions of an integral operator defined by the covariance kernel it is shown that the corresponding measure defines a universally optimal design for several models universally optimal designs can be identified explicitly in particular it is proved that the uniform distribution is universally optimal for a class of trigonometric regression models with a broad class of covariance kernels and that the arcsine distribution is universally optimal for the polynomial regression model with correlation structure defined by the logarithmic potential to the best knowledge of the authors these findings provide the first explicit results on optimal designs for regression models with correlated observations which are not restricted to the location scale model

spectral gap for a class of random billiards a random billiard is a random dynamical system similar to an ordinary billiard system except that the standard specular reflection law is replaced with a more general stochastic operator specifying the post collision distribution of velocities for any given pre collision velocity we consider such collision operators for certain random billiards that we call billiards with microstructure collisions modeled by these operators can still be thought of as elastic and time reversible the operators are canonically determined by a second deterministic billiard system that models microscopic roughness on the billiard table boundary our main purpose here is to develop some general tools for the analysis of the collision operator of such random billiards among the main results we give geometric conditions for these operators to be hilbert schmidt and relate their spectrum and speed of convergence to stationary markov chains with geometric features of the microscopic billiard structure the relationship between spectral gap and the shape of the microstructure is illustrated with several simple examples

on knudsen flows within thin tubes the dynamics of a test particle in an infinite tube is investigated it is proven that the evolution tends to that of a diffusion process as the radius of the tube decreases to zero this justifies the hypotheses underlying an experiment of clausing 1930

energy decaying extrapolated rk sav methods for the allen cahn and cahn hilliard equations we construct and analyze a class of extrapolated and linearized runge kutta rk methods which can be of arbitrarily high order for the time discretization of the allen cahn and cahn hilliard

measures for designs in experiments with correlated errors summary in this paper we consider optimal design of experiments in the case of correlated obser vations we use and further develop the concept of design measures introduced by paizman mtiller 1998 for the construction of a simple quick and elegant design algorithm we support the construction of this algorithm for a general correlation structure by an interpretation in terms of norms examples demonstrate that our results are useful for generating exact designs by sampling from the obtained design measures most of the literature on experimental design operates under the assumption of uncorre lated errors and by exploiting the concept of a design measure introduced by kiefer 1959 in this setting an optimal design measure is usually computed by an iterative algorithm and then an exact design with independent replications approximately proportional to that measure is employed in the correlated case when replications are not allowed the implementation of design measures is not so straightforward recently however by adding virtual design dependent noise to the process paizman muiller 1998 have introduced a way of using this popular concept albeit requiring a very different interpretation furthermore paizman muller 2000 showed that solving minimisation problems in terms of these measures corresponds to computation of exact designs these measures serve as building blocks for the methods presented in this paper which provides a general numerical tool for screening designs by a different type of smoothing of some important nondifferentiable terms in the algorithm and without imposing restric tions on the number of support points we obtain a design measure the relative magnitude of this measure for each support point reflects the importance of its inclusion in any exact design this algorithm can also be interpreted in terms of certain information norms as pre

efficient energy stable numerical schemes for a phase field moving contact line model in this paper we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line the model is a coupled system that consists of incompressible navier stokes equations with a generalized navier boundary condition and cahn hilliard equation in conserved form in both schemes the projection method is used to deal with the navier stokes equations and stabilization approach is used for the non convex ginzburg landau bulk potential by some subtle explicit implicit treatments we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions an efficient spectral galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes numerical results show that the proposed schemes are very efficient and accurate we proposed two linear energy stable numerical schemes for a phase field moving contact line model discrete energy laws that are consistent to the continuous ones are proved the efficiency and fast convergence of the proposed schemes with spectral spatial approximations are numerically verified

stable second order schemes for the space fractional cahn hilliard and allen cahn equations abstract in this paper we propose stable second order numerical schemes for the fractional cahn hilliard and allen cahn equations which are based on the convex splitting in time and the fourier spectral method in space it is shown that the scheme for the fractional cahn hilliard equation preserves mass meanwhile the unique solvability and energy stability of the numerical schemes for the fractional cahn hilliard and allen cahn equations are proved finally we present some numerical experiments to confirm the accuracy and the effectiveness of the proposed methods

a three level finite element approximation of a pattern formation model in developmental biology this paper concerns a second order three level piecewise linear finite element scheme 2 sbdf ruuth in j math biol 34 148 176 1995 for approximating the stationary turing patterns of a well known experimental substrate inhibition reaction diffusion thomas system thomas in analysis and control of immobilized enzyme systems pp 115 150 1975 a numerical analysis of the semi discrete in time approximations leads to semi discrete a priori bounds and an optimal error estimate the analysis highlights the technical challenges in undertaking the numerical analysis of multi level ge 3 3 schemes we illustrate the effectiveness of the numerical method by repeating an important classical experiment in mathematical biology namely to approximate the turing patterns of the thomas system over a schematic mammal skin domain with fixed geometry at various scales we also make some comments on the correct procedure for simulating turing patterns in general reaction diffusion systems

a semi analytical fourier spectral method for the allen cahn equation in recent years fourier spectral methods have been widely used as a powerful tool for solving phase field equations to improve its effectiveness many researchers have employed stabilized semi implicit fourier spectral sifs methods which allow a much larger time step than a usual explicit scheme our mathematical analysis and numerical experiments however suggest that an effective time step is smaller than a time step specified in the sifs schemes in consequence the sifs scheme is inaccurate for a considerably large time step and may lead to incorrect morphologies in phase separation processes in order to remove the time step constraint and guarantee the accuracy in time for a sufficiently large time step we present a first and a second order semi analytical fourier spectral safs methods for solving the allen cahn equation the core idea of the methods is to decompose the original equation into linear and nonlinear subequations which have closed form solutions in the fourier and physical spaces respectively both the first and the second order methods are unconditionally stable and numerical experiments demonstrate that our proposed methods are more accurate than the stabilized semi implicit fourier spectral method

error analysis of full discrete invariant energy quadratization schemes for the cahn hilliard type equation abstract in this paper we present the error analysis for a fully discrete scheme of the cahn hilliard type equation along with numerical verifications the numerical schedule is developed by first transforming the cahn hilliard type equation into an equivalent form using the invariant energy quadratization ieq technique then the equivalent form is discretized by using the linear implicit crank nicolson method for the time variable and the fourier pseudo spectral method for the spatial variables the resulted full discrete scheme is linear and unconditionally energy stable which makes it easy to implement by constructing an appropriate interpolation equation the uniform boundedness of the numerical solution is obtained theoretically then we prove that the numerical solutions converge with the order o t 2 h m where t is the temporal step and h is the spatial step with m the regularity of the exact solution several numerical examples are presented to confirm the theoretical results and demonstrate the effectiveness of the presented linear scheme the numerical strategies and analytical tools in this paper could be readily applied to study other phase field models or gradient flow problems

attractors and inertial manifolds for finite difference approximations of the complex ginzburg landau equation a semidiscrete spatial finite difference approximation and two fully discrete finite difference approximations to the complex ginzburg landau equation are considered in this paper the existence of an inertial manifold is proved inside a discrete h1 absorbing ball for the semi discrete approximation by showing that certain spectral properties hold for the linear operator and certain lipschitz properties hold for the nonlinear term convergence of the true and discrete inertial manifolds is then shown by considering the contraction mappings used to construct the inertial manifolds a fully implicit scheme is shown to form a continuous semigroup in a discrete l2 space and absorbing balls of radii independent of the spatial and temporal mesh sizes in discrete l2 and h1 spaces are constructed existence of a global attractor follows furthermore the existence of an inertial manifold is proved by showing that certain spectral and lipschitz properties hold for a semi implicit scheme existence of absorbing balls in the discrete space l2 of radii independent of the spatial and temporal mesh is proved existence of a global attractor follows

the numerical solutions for the energy dissipative and mass conservative allen cahn equation abstract there are three criteria when we model with phase field equations the first one is the evolution of phase separation the second one is energy dissipation law the last one is whether we have the law of conservation of mass or not if we minimize the ginzburg landau free energy functional the kinetics of phase separation inherently follows however the other properties such as energy dissipation and mass conservation are more difficult to model hitherto there have been a few proposals to conserve mass using the allen cahn ac equation but we do not have the energy dissipation law in these models when considered in light of local effects of mass correction for the reasons certain researchers still prefer to use the cahn hilliard equation e g biharmonic equation instead of using the allen cahn equation when we consider a mass conserving case this paper is concerned with numerical solutions of the ac equation in the mass conserving space where the solution is corrected at each iteration for mass conservation we introduce two different methods for solving the conservative ac equation with periodic boundary conditions in this work we have the ac equation in common but use two different mass correction operators note that both of mass correction operators satisfy the discrete energy dissipation property first we project the solution of the ac equation to the mass conserving space and then recover the linear convex splitting scheme for the conventional mass conserving equation the linear convex splitting scheme is well known but the convergence of numerical solutions is elaborately studied in terms of gradient projection next we propose the other energy minimizing method on the mass conserving space it translates the solution of the ac equation to the mass conserving space via an energy dissipation operator the dissipation of discrete energy functional for the proposed method is also established as far as the author knows no report has been presented for the numerical scheme having discrete energy dissipation property of the mass conserving ac equation without using the nonlocal and constant multiplier to verify the common properties of mass projection and proposed methods numerically we simulate the evolutions for phase separation dissipation of energy and mass conservation we moreover consider the short and long time stages of evolutions to see the difference between two methods these several tests are provided for its applications of the proposed method for modeling natural phenomena

long time numerical simulations for phase field problems using p adaptive spectral deferred correction methods a high order and energy stable scheme is developed to simulate phase field models by combining the semi implicit spectral deferred correction sdc method and the energy stable convex splitting technique the convex splitting scheme we use here is a linear unconditionally stable method but is only of first order accuracy so the sdc method can be used to iteratively improve the rate of convergence however it is found that the accuracy improvement may affect the overall energy stability which is intrinsic to the phase field models to compromise the accuracy and stability a local p adaptive strategy is proposed to enhance the accuracy by sacrificing some local energy stability in an acceptable level the proposed strategy is found very useful for producing accurate numerical solutions at small time dynamics as well as long time steady state with reasonably large time stepsizes numerical experiments are carried out to demonstrate the high effectiveness of the proposed numerical strategy

maximally fast coarsening algorithms we present maximally fast numerical algorithms for conserved coarsening systems that are stable and accurate with a growing natural time step deltat at2 3s we compare the scaling structure obtained from our maximally fast conserved systems directly against the standard fixed time step euler algorithm and find that the error scales as square root of a so arbitrary accuracy can be achieved for nonconserved systems only effectively finite time steps are accessible for similar unconditionally stable algorithms

n th order operator splitting schemes and nonreversible systems this paper is concerned with partitioned nth order accurate split operator schemes built using m distinct solution operators for semidiscrete equations of the form frac d dt u j f j 1 u k cdots f j m u k which arise among others from constant coefficient parabolic partial differential equations we prove that for every n geq 3 and m geqslant 2 each solution operator must be applied for at least one backward fractional time step during each complete time step this result has important consequences for applications to the complex ginzburg landau equation with periodic boundary conditions and other partial differential equations with both reversible and nonreversible components furthermore for the special case of n 3 and m 2 we analytically determine all possible schemes

runge kutta methods for dissipative and gradient dynamical systems the numerical approximation of dissipative initial value problems by fixed time stepping runge kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared a general class of ordinary differential equations for which dissipativity is induced through an inner product is studied throughout this class arises naturally in many finite dimensional applications such as the lorenz equations and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics it is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step size h 0 the numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor a h exists upper semicontinuity of a h at h 0 is established which shows that for h small every point on the numerical attractor is close to a point on the true global attractor a under the additional assumption that the problem is globally lipschitz it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of a h at h 0 is again established for gradient systems with globally lipschitz vector fields it is shown that any runge kutta method preserves the gradient structure for h sufficiently small for general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small convergence of the numerical attractor is studied and for a dissipative gradient system with hyperbolic equilibria lower semicontinuity at h 0 is established thus for such a system a h converges to a in the hausdorff metric as h 0

multiple scattering in random mechanical systems and diffusion approximation this paper is concerned with stochastic processes that model multiple or iterated scattering in classical mechanical systems of billiard type defined below from a given deterministic system of billiard type a random process with transition probabilities operator p is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions of particular interest are systems with weak scattering which are associated to parametric families of operators ph depending on a geometric or mechanical parameter h that approaches the identity as h goes to 0 it is shown that ph i h converges for small h to a second order elliptic differential operator mathcal l on compactly supported functions and that the markov chain process associated to ph converges to a diffusion with infinitesimal generator mathcal l both ph and mathcal l are self adjoint densely defined on the space l 2 mathbb h eta of square integrable functions over the lower half space mathbb h in mathbb r m where is a stationary measure this measure s density is either post collision maxwell boltzmann distribution or knudsen cosine law and the random processes with infinitesimal generator mathcal l respectively correspond to what we call mb diffusion and generalized legendre diffusion concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes

on the solution of convection diffusion boundary value problems using equidistributed grids the effect of using grid adaptation on the numerical solution of model convection diffusion equations with a conservation form is studied the grid adaptation technique studied is based on moving a fixed number of mesh points to equidistribute a generalization of the arc length of the solution in particular a parameter dependent monitor function is introduced which incorporates fixed meshes approximate arc length equidistribution and equidistribution of the absolute value of the solution in a single framework thus the resulting numerical method is a coupled nonlinear system of equations for the mesh spacings and the nodal values a class of singularly perturbed problems including burgers s equation in the limit of small viscosity is studied singular perturbation and bifurcation techniques are used to analyze the solution of the discretized equations and numerical results are compared with the results from the analysis computation of the bifurcation diagram of the system is performed numerically using a continuation method and the results are used to illustrate the theory it is shown that equidistribution does not remove spurious solutions present on a fixed mesh and that furthermore the spurious solutions can be stable for an appropriate moving mesh method

an efficient algorithm for solving the phase field crystal model we present and discuss the development of an unconditionally stable algorithm used to solve the evolution equations of the phase field crystal pfc model this algorithm allows for an arbitrarily large algorithmic time step as the basis for our analysis of the accuracy of this algorithm we determine an effective time step in fourier space we then compare our calculations with a set of representative numerical results and demonstrate that this algorithm is an effective approach for the study of the pfc models yielding a time step effectively 180 times larger than the euler algorithm for a representative set of material parameters as the pfc model is just a simple example of a wide class of density functional theories we expect this method will have wide applicability to modeling systems of considerable interest to the materials modeling communities

an energy stable runge kutta method for convex gradient problems abstract we propose a class of runge kutta methods which provide a simple unified framework to solve the gradient flow of a convex functional in an unconditionally energy stable manner stiffly accurate runge kutta methods are high order accurate in terms of time and also assure the energy stability for any time step size when they satisfy the positive definite condition we provide a detailed proof of the unconditional energy stability as well as unique solvability of the proposed scheme we demonstrate the accuracy and stability of the proposed methods using numerical experiments for a specific example

optimal designs in regression with correlated errors this paper discusses the problem of determining optimal designs for regression models when the observations are dependent and taken on an interval a complete solution of this challenging optimal design problem is given for a broad class of regression models and covariance kernels we propose a class of estimators which are only slightly more complicated than the ordinary least squares estimators we then demonstrate that we can design the experiments such that asymptotically the new estimators achieve the same precision as the best linear unbiased estimator computed for the whole trajectory of the process as a by product we derive explicit expressions for the blue in the continuous time model and analytic expressions for the optimal designs in a wide class of regression models we also demonstrate that for a finite number of observations the precision of the proposed procedure which includes the estimator and design is very close to the best achievable the results are illustrated on a few numerical examples

a time adaptive finite volume method for the cahn hilliard and kuramoto sivashinsky equations this paper presents a complete finite volume method for the cahn hilliard and kuramoto sivashinsky type of equations the spatial discretization is high order accurate and suitable for general unstructured grids the time integration is addressed by means of implicit an implicit explicit fourth order runge kutta schemes with error control and adaptive time stepping the outcome is a practical accurate and efficient simulation tool which has been successfully applied to accuracy tests and representative simulations the use of adaptive time stepping is of paramount importance in problems governed by the cahn hilliard model an adaptive method may be several orders of magnitude more efficient than schemes using constant or heuristic time steps in addition to driving the simulations efficiently the time adaptive procedure provides a quantitative not just qualitative characterization of the rich temporal scales present in phase separation processes governed by the cahn hilliard phase field model

a novel linear unconditional energy stable scheme for the incompressible cahn hilliard navier stokes phase field model abstract in this paper we propose a linear unconditional energy stable time discretization scheme for cahn hilliard navier stokes model which is a phase field model for two phase incompressible flow based on a lagrange multiplier approach our proposed scheme is linearized by using implicit explicit treatments and the computation of the velocity field u the pressure p the phase function are decoupled using the mathematical induction the bound n l is obtained rigorous error analysis for the variable u p and are carried out in semi discrete form finally some numerical experiments are done to demonstrate the effectiveness for the proposed method