computationally efficient adaptive time step method for the cahnhilliard equation in this work we propose a fast and efficient adaptive time step procedure for the cahnhilliard equation the temporal evolution of the cahnhilliard equation has multiple time scales for spinodal decomposition simulation an initial random perturbation evolves on a fast time scale and later coarsening evolves on a very slow time scale therefore if a small time step is used to capture the fast dynamics the computation is quite costly on the other hand if a large time step is used fast time evolutions may be missed hence it is essential to use an adaptive time step method to simulate phenomena with multiple time scales the proposed time adaptivity algorithm is based on the discrete maximum norm of the difference between two consecutive time step numerical solutions numerical experiments in one two and three dimensions are presented to demonstrate the performance and effectiveness of the adaptive time stepping algorithm

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

isogeometric analysis of the advective cahn hilliard equation spinodal decomposition under shear flow we present a numerical study of the spinodal decomposition of a binary fluid undergoing shear flow using the advective cahn hilliard equation a stiff nonlinear parabolic equation characterized by the presence of fourth order spatial derivatives our numerical solution procedure is based on isogeometric analysis an approximation technique for which basis functions of high order continuity are employed these basis functions allow us to directly discretize the advective cahn hilliard equation without resorting to a mixed formulation we present steady state solutions for rectangular domains in two dimensions and for the first time in three dimensions we also present steady state solutions for the two dimensional taylor couette cell to enforce periodic boundary conditions in this curved domain we derive and utilize a new periodic bezier extraction operator we present an extensive numerical study showing the effects of shear rate surface tension and the geometry of the domain on the phase evolution of the binary fluid theoretical and experimental results are compared with our simulations

resonance tongues and instability pockets in the quasi periodic hill schrodinger equation this paper concerns hill s equation with a parametric forcing that is real analytic and quasi periodic with frequency vector d and a frequency or energy parameter a and a small parameter b the 1 dimensional schrodinger equation with quasi periodic potential occurs as a particular case in the parameter plane 2 a b for small values of b we show the following the resonance tongues with rotation number frac 1 2 langle bf k omega rangle bf k in mathbb z d have c boundary curves our arguments are based on reducibility and certain properties of the schrodinger operator with quasi periodic potential analogous to the case of hill s equation with periodic forcing i e d 1 several further results are obtained with respect to the geometry of the tongues one result regards transversality of the boundaries at b 0 another result concerns the generic occurrence of instability pockets in the tongues in a reversible near mathieu case that may depend on several deformation parameters these pockets describe the generic opening and closing behaviour of spectral gaps of the schrodinger operator in dependence of the parameter b this result uses a refined averaging technique also consequences are given for the behaviour of the lyapunov exponent and rotation number in dependence of a for fixed b

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

nonoscillation of mathieu equations with two frequencies abstract as is well known mathieu s equation is a representative of mathematical models describing parametric excitation phenomena this paper deals with the oscillation problem for mathieu s equation with two frequencies the ratio of these two frequencies is not necessarily a rational number when the ratio is an irrational number the coefficient of mathieu s equation is quasi periodic but not periodic for this reason the basic knowledge for linear periodic systems such as floquet theory is not useful whether all solutions of mathieu s equation oscillate or not is determined by parameters and frequencies our results provide parametric conditions to guarantee that all solutions are nonoscillatory the advantage of the obtained parametric conditions is that it can be easily checked parametric nonoscillation region is drawn to understand these results easily finally several simulations are carried out to clarify the remaining problems

a second order accurate linearized difference scheme for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is of fourth order in space in this paper a linearized finite difference scheme is derived by the method of reduction of order it is proved that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete l2 norm the coefficient matrix of the difference system is symmetric and positive definite so many well known iterative methods e g gauss seidel sor can be used to solve the system

a practically unconditionally gradient stable scheme for the n component cahn hilliard system we present a practically unconditionally gradient stable conservative nonlinear numerical scheme for the n component cahn hilliard system modeling the phase separation of an n component mixture the scheme is based on a nonlinear splitting method and is solved by an efficient and accurate nonlinear multigrid method and the scheme allows us to convert the n component cahn hilliard system into a system of n 1 binary cahn hilliard equations and significantly reduces the required computer memory and cpu time we observe that our numerical solutions are consistent with the linear stability analysis results we also demonstrate the efficiency of the proposed scheme with various numerical experiments

fast and accurate coarsening simulation with an unconditionally stable time step we present cahn hilliard and allen cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy controlled simulation our stability analysis is based on eyre s theorem and unconditional von neumann stability analysis both of which we present numerical tests confirm the accuracy of the von neumann approach which is straightforward and should be widely applicable in phase field modeling for the cahn hilliard case we show that accuracy can be controlled with an unbounded time step dt that grows with time t as dt t a we develop a classification scheme for the step exponenta and demonstrate that a class of simple linear algorithms gives a51 3 for this class the speedup relative to a fixed time step grows with n the linear size of the system as n lnn with conservative choices for the parameters controlling accuracy and finite size effects we find that an 8192 2 lattice can be integrated 300 times faster than with the euler method

transition curves for the quasi periodic mathieu equation in this work we investigate an extension of mathieu s equation the quasi periodic qp mathieu equation given by ddot psi delta eps cos t cos omega t psi 0 for small eps and irrational omega of interest is the generation of stability diagrams that identify the points or regions in the delta omega parameter plane for fixed eps for which all solutions of the qp mathieu equation are bounded numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots oflyapunov exponents in addition we derive approximate analytic expressions for transition curves using two distinct techniques 1 a regular perturbation method under which transition curves delta delta omega eps are each expanded in powers of eps and 2 the method of harmonic balance utilizing hill s determinants both analytic methods deliver results in good agreement with those generated numerically in the sense that pre

interval oscillation theorems for asecond order linear differential equation interval oscillation criteria are given for the forced second order linear differential equation ly t p t y q t y t email protected 0 where p q are locally integrable functions and p t 0 for t 0 no restriction is imposed on t to be the second derivative of an oscillatory function as assumed by kartsatos 1 our results also allow both q and f to change sign in the neighborhood at infinity in particular we show that all solutions of y c sin t y t b cos t with b 0 are oscillatory for c 1 3448 this improves an estimate given by nasr 2 for the linear equation

a gradient stable scheme for a phase field model for the moving contact line problem in this paper an efficient numerical scheme is designed for a phase field model for the moving contact line problem which consists of a coupled system of the cahn hilliard and navier stokes equations with the generalized navier boundary condition 1 2 4 the nonlinear version of the scheme is semi implicit in time and is based on a convex splitting of the cahn hilliard free energy including the boundary energy together with a projection method for the navier stokes equations we show under certain conditions the scheme has the total energy decaying property and is unconditionally stable the linearized scheme is easy to implement and introduces only mild cfl time constraint numerical tests are carried out to verify the accuracy and stability of the scheme the behavior of the solution near the contact line is examined it is verified that when the interface intersects with the boundary the consistent splitting scheme 21 22 for the navier stokes equations has the better accuracy for pressure

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

symmetry properties for nonnegative solutions of non uniformly elliptic equations in the hyperbolic space abstract we are interested in monotonicity and symmetry properties for nonnegative solutions of elliptic equations defined in geodesic balls of the hyperbolic space h n which is the simplest example of manifold with negative curvature more precisely let b be a geodesic ball in h n and let u w 1 p b l b be a sufficiently regular solution of p u f u 0 in b with boundary condition u 0 where p is the p laplace beltrami operator with p 2 then we prove local or global symmetry results for nonnegative solutions according to the assumptions about the zeros of the nonlinearity f s which is merely continuous

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

coarsening kinetics from a variable mobility cahn hilliard equation application of a semi implicit fourier spectral method an efficient semi implicit fourier spectral method is implemented to solve the cahn hilliard equation with a variable mobility the method is orders of magnitude more efficient than the conventional forward euler finite difference method thus allowing us to simulate large systems for longer times we studied the coarsening kinetics of interconnected two phase mixtures using a cahn hilliard equation with its mobility depending on local compositions in particular we compared the kinetics of bulk diffusion dominated and interface diffusion dominated coarsening in two phase systems results are compared with existing theories and previous computer simulations

oscillation and nonoscillation of hill s equation with periodic damping we prove new results on the oscillation and nonoscillation of the hill s equation with periodic damping y p t y q t y 0 t 0 where p t and q t are continuous and periodic the results show that the equation y sint y cost y 0 is nonoscillatory whilst the equation y cost y sint y 0 is oscillatory

a nonconforming finite element method for the two dimensional cahn hilliard equation the cahn hilliard equation is a nonlinear evolutionary equation that is fourth order in space in this paper a continuous in time finite element galerkin approximation is considered we use the nonconforming morley element and derive optimal order error bounds in l 2

a class of stable spectral methods for the cahn hilliard equation in this work the initial boundary value problem of two dimensional cahn hilliard equation is considered a class of fully discrete dissipative fourier spectral schemes are proposed moreover semi implicit prediction correction schemes are presented the numerical simulations are performed to demonstrate the effectiveness of the proposed schemes

accurate efficient and iso geometrically flexible collocation methods for phase field models we propose new collocation methods for phase field models our algorithms are based on isogeometric analysis a new technology that makes use of functions from computational geometry such as for example non uniform rational b splines nurbs nurbs exhibit excellent approximability and controllable global smoothness and can represent exactly most geometries encapsulated in computer aided design cad models these attributes permitted us to derive accurate efficient and geometrically flexible collocation methods for phase field models the performance of our method is demonstrated by several numerical examples of phase separation modeled by the cahn hilliard equation we feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo spectral collocation methods and is a viable alternative to classical collocation methods

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

on the two phase stefan problem with interfacial energy and entropy abstract the classical stefan theory for the melting of a solid or the freezing of a liquid is too simplistic to describe phenomena such as supercooling in which a liquid supports temperatures below its freezing point or superheating the analog for solids or dendritic growth in which simple shapes evolve to complicated tree like structures this paper develops a general theory for two phase phenomena of this type it develops partial differential equations satisfied in the phase regions and free boundary conditions satisfied on the interface between phases and gives arguments which indicate that the resulting boundary value problems predict the formation of dendrites keywords equilibrium liapunoo functions mathematical physics

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 nonoscillation theorem for half linear differential equations with periodic coefficients abstract the half linear differential equation p x a t p x b t p x 0 is considered under the assumption that the coefficient a t and an indefinite integral b t of b t are periodic functions with period t 0 it is proved that p 1 p b t a t b t 0 0 t t is sufficient for all nontrivial solutions to be nonoscillatory here p 1 and q y y q 2 y for q p or q p p p 1 the proof is given by means of riccati technique the condition is shown to be sharp sufficient conditions are also presented for all nontrivial solutions are oscillatory in the linear case p 2 some examples and simulations are included to illustrate our results

numerical study of the cahn hilliard equation in one two and three dimensions abstract the cahn hilliard ch equation is related with a number of interesting physical phenomena like the spinodal decomposition phase separation and phase ordering dynamics on the other hand this equation is very stiff and the difficulty to solve it numerically increases with the dimensionality and therefore there are several published numerical studies in one dimension 1d dealing with different approaches and much fewer in two dimensions 2d in three dimensions 3d there are very few publications usually concentrate in some specific result without the details of the used numerical scheme we present here a stable and fast conservative finite difference scheme to solve the ch with two improvements a splitting potential into an implicit and explicit in time part and the use of free boundary conditions we show that gradient stability is achieved in one two and three dimensions with large time marching steps than normal methods

simple conditions for parametrically excited oscillations of generalized mathieu equations abstract the following equation is considered in this paper x cos t x 0 where and are real parameters and 0 this equation is referred to as mathieu s equation when 2 the parameters determine whether all solutions of this equation are oscillatory or nonoscillatory our results provide parametric conditions for oscillation and nonoscillation there is a feature in which it is very easy to check whether these conditions are satisfied or not parametric oscillation and nonoscillation regions are drawn to help understand the obtained results

chebyshev collocation method for the constant mobility cahn hilliard equation in a square domain abstract chebyshev collocation method was developed for constant mobility cahn hilliard equation the accuracy of the method was indirectly checked with the aid of two separate calculations of temporal derivative of total free energy we found that the derivatives of total free energy by two different ways of calculation coincided exactly when the number of grid points in the transitional layer within mixing region was 16 or more when the coefficient of gradient energy 2 was chosen correspondingly in the given domain we also investigated the application of kosloff and tal ezer mapping for the chebyshev collocation grid points in space and showed the possibility of the use of much larger timestep size than that of the chebyshev collocation method without mapping

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

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29