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

membrane bound turing patterns motivated by recent observations in biological cells we study turing patterns in bounded regions where the nonlinear chemical reactions occur on the boundary and where reagent transport occurs in the bulk within a generic model we formulate the stability problem and discuss the conditions for the occurrence of a turing instability by choosing other model parameters to be unequal we find that turing patterns exist even in the case of equal diffusion constants finally a recently introduced computation technique is utilized to follow the nascent pattern into the highly nonlinear regime

a moving mesh method for the solution of the one dimensional phase field equations a moving mesh method is developed for the numerical solution of one dimensional phase change problems modelled by the phase field equations the computational mesh is obtained by equidistribution of a monitor function tailored for the functional variation of the phase field in the interfacial region existence and uniqueness of the discretised equations using a moving mesh are also established numerical results are given for classical and modified stefan test problems the numerical algorithm is relatively simple and is shown to be far more efficient than fixed grid methods

turing type instabilities in bulk surface reaction diffusion systems in this contribution we consider a coupled bulk surface reaction diffusion system and for infinitely fast bulk diffusion its formal reduction to a non local surface pde model thereby we review results of linear stability analyses for both models showing that turing type instabilities can occur for equal lateral diffusion coefficients the stability results are confirmed by new numerical results as a specific application we study a model for a spatial and reaction cycle of signalling molecules in a cell

quantitative phase field modeling of dendritic growth in two and three dimensions we report the results of quantitative phase field simulations of the dendritic crystallization of a pure melt in two and three dimensions these simulations exploit a recently developed thin interface limit of the phase field model a karma and w j rappel phys rev e 53 r3017 1996 which is given here a detailed exposition this limit makes it possible to perform efficient computations with a smaller ratio of capillary length to interface thickness and with an arbitrary interface kinetic coefficient simulations in one and two dimensions are first carried out to test the accuracy of phase field computations performed within this limit dendrite tip velocities and tip shapes are found to be in excellent quantitative agreement with exact numerical benchmarks of solvability theory obtained by a boundary integral method both with and without interface kinetics simulations in three dimensions exploit in addition to the asymptotics a methodology to calculate grid corrections due to the surface tension and kinetic anisotropies they are used to test basic aspects of dendritic growth theory that pertain to the selection of the operating state of the tip and to the three dimensional morphology of needle crystals without sidebranches for small crystalline anisotropy simulated values of ensuremath sigma are slightly larger than solvability theory predictions computed by the boundary integral method assuming an axisymmetric shape and agree relatively well with experiments for succinonitrile given the uncertainty in the measured anisotropy in contrast for large anisotropy simulated ensuremath sigma values are significantly larger than the predicted values this disagreement however does not signal a breakdown of solvability theory it is consistent with the finding that the amplitude of the mathrm cos 4 ensuremath varphi mode which measures the departure of the tip morphology from a shape of revolution increases with anisotropy this departure can therefore influence the tip selection in a way that is not accurately captured by the axisymmetric approximation for large anisotropy finally the tail shape at a distance behind the tip that is large compared to the diffusion length is described by a linear law r ensuremath sim z with a slope dr dz that is nearly equal to the ratio of the two dimensional and three dimensional steady state tip velocities furthermore the evolution of the cross section of a three dimensional needle crystal with increasing distance behind the tip is nearly identical to the evolution of a two dimensional growth shape in time in accord with the current theory of the three dimensional needle crystal shape

large order perturbation expansions for the charged harmonic oscillator the charged oscillator defined by the hamiltonian h d2 dr2 r2 lambda r in the domain 0 infinity is a particular case of the family of spike oscillators which does not behave as a supersingular hamiltonian this problem is analysed around the three regions lambda to infinity lambda to 0 and lambda to infinity by using rayleigh ritz large order perturbative expansions a path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda finally the riccati pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod

oscillatory dynamics for a coupled membrane bulk diffusion model with fitzhugh nagumo membrane kinetics oscillatory dynamics associated with the coupled membrane bulk pde ode model of gomez marin garcia ojalvon and sancho phys rev lett 98 2007 168303 in one spatial dimension is analyzed using a combination of asymptotic analysis linear stability theory and numerical bifurcation software the mathematical model consists of two dynamically active membranes with fitzhugh nagumo kinetics separated spatially by a distance l that are coupled together through a diffusion field that occupies the bulk region 0

comment on the letter a new efficient method for calculating perturbation energies using functions which are not quadratically integrable by l skala and j cizek we suggest that the recent numerical integration perturbative approach to bound states as proposed by skala and cizek may be generalized via its renormalizations

the reaction diffusion problem with dynamical boundary condition abstract the aim of this paper is to prove the existence and uniqueness of classical solutions to a coupled system of three parabolic and one ordinary differential equations with the latter determined on the boundary this system describes the receptor toxin antibody interaction problem taking into account diffusion of reacting species

a second order accurate symmetric discretization of the poisson equation on irregular domains in this paper we consider the variable coefficient poisson equation with dirichlet boundary conditions on an irregular domain and show that one can obtain second order accuracy with a rather simple discretization moreover since our discretization matrix is symmetric it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second order accurate discretizations of this problem multidimensional computational results are presented to demonstrate the second order accuracy of this numerical method in addition we use our approach to formulate a second order accurate symmetric implicit time discretization of the heat equation on irregular domains then we briefly consider stefan problems

a hybrid finite volume finite element method for bulk surface coupled problems abstract the paper develops a hybrid method for solving a system of advection diffusion equations in a bulk domain coupled to advection diffusion equations on an embedded surface a monotone nonlinear finite volume method for equations posed in the bulk is combined with a trace finite element method for equations posed on the surface in our approach the surface is not fitted by the mesh and is allowed to cut through the background mesh in an arbitrary way moreover a triangulation of the surface into regular shaped elements is not required the background mesh is an octree grid with cubic cells as an example of an application we consider the modeling of contaminant transport in fractured porous media one standard model leads to a coupled system of advection diffusion equations in a bulk matrix and along a surface fracture a series of numerical experiments with both steady and unsteady problems and different embedded geometries illustrate the numerical properties of the hybrid approach the method demonstrates great flexibility in handling curvilinear or branching lower dimensional embedded structures

thin interface asymptotics for an energy entropy approach to phase field models with unequal conductivities karma and rappel phys rev e 57 1998 4342 recently developed a new sharp interface asymptotic analysis of the phase field equations that is especially appropriate for modeling dendritic growth at low undercoolings their approach relieves a stringent restriction on the interface thickness that applies in the conventional asymptotic analysis and has the added advantage that interfacial kinetic effects can also be eliminated however their analysis focussed on the case of equal thermal conductivities in the solid and liquid phases when applied to a standard phase field model with unequal conductivities anomalous terms arise in the limiting forms of the boundary conditions for the interfacial temperature that are not present in conventional sharp interface solidification models as discussed further by almgren siam j appl math 59 1999 2086 in this paper we apply their asymptotic methodology to a generalized phase field model which is derived using a thermodynamically consistent approach that is based on independent entropy and internal energy gradient functionals that include double wells in both the entropy and internal energy densities the additional degrees of freedom associated with the generalized phase field equations can be used to eliminate the anomalous terms that arise for unequal conductivities

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

the summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic sextic and octic anharmonic oscillators using nonlinear sequence transformations the strongly divergent rayleigh schrodinger perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator and a corresponding renormalized perturbation expansion f vinette and j ci ek j math phys 32 3392 1991 are summed by pade approximants by levin s sequence transformation d levin int j comput math b 3 371 1973 and by a closely related sequence transformation which was suggested recently e j weniger comput phys rep 10 189 1989 it is shown that the renormalized perturbation expansion can be summed much more easily than the original perturbation expansion from which it was derived and that levin s sequence transformation diverges and is not able to sum the perturbation expansions the pade summation of the renormalized perturbation expansions gives relatively good results in the quartic and sextic case in the case of the octic anharmonic oscillator even the renormalized perturbation expansion is not pade summable the best results are clearly obtained by the new sequence transformation which for instance is able to sum the renormalized perturbation expansions for the infinite coupling limits of the quartic sextic and octic anharmonic oscillator and which produces at least in the quartic and sextic case extremely accurate results

well posedness for solid liquid phase transitions with a fourth order nonlinearity abstract a phase field system which describes the evolution of both the absolute temperature and the phase variable f during first order transitions in thermal insulators is considered a thermodynamic approach is developed by regarding the order parameter as a phase field and its evolution equation as a balance law by virtue of the special form of the internal energy a third order nonlinearity g 2 f appears in the energy balance in place of the customarily constant latent heat as a consequence the bounds 0 f 1 hold whenever is positive valued in addition a nonlinear fourier law with conductivity proportional to temperature is assumed well posedness for the resulting initial and boundary value problem are then established in a suitable setting

global existence and blow up phenomena for nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions abstract in this paper we consider nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions we establish respectively the conditions on nonlinearities to guarantee that u x t exists globally or blows up at some finite time if blow up occurs we obtain upper and lower bounds of the blow up time

rd oscillators with arbitrary d 0 and perturbation expansions with sturmians in contrast to widespread belief the current rayleigh schrodinger perturbation theory may provide an easy description of double well oscillators and or of the strongly anharmonic forces with an arbitrary power law asymptotical growth one has only to work in a suitable sturmian basis the feasibility and numerical efficiency of the construction is illustrated on a few one dimensional one body examples

a comparative study of penalization and phase field methods for the solution of the diffusion equation in complex geometries we present a comparative study of penalization and phase field methods for the solution of the diffusion equation in complex geometries embedded using simple cartesian meshes the two methods have been widely employed to solve partial differential equations in complex and moving geometries for applications ranging from solid and fluid mechanics to biology and geophysics their popularity is largely due to their discretization on cartesian meshes thus avoiding the need to create body fitted grids at the same time there are questions regarding their accuracy and it appears that the use of each one is confined by disciplinary boundaries here we compare penalization and phase field methods to handle problems with neumann and robin boundary conditions we discuss extensions for dirichlet boundary conditions and in turn compare with methods that have been explicitly designed to handle dirichlet boundary conditions the accuracy of all methods is analyzed using one and two dimensional benchmark problems such as the flow induced by an oscillating wall and by a cylinder performing rotary oscillations this comparative study provides information to decide which methods to consider for a given application and their incorporation in broader computational frameworks we demonstrate that phase field methods are more accurate than penalization methods on problems with neumann boundary conditions and we present an error analysis explaining this result

effective characteristic polynomials and two point pade approximants as summation techniques for the strongly divergent perturbation expansions of the ground state energies of anharmonic oscillators pad e approximants are able to sum effectively the rayleigh schr odinger perturbation series for the ground state energy of the quartic anharmonic oscillator as well as the corresponding renormalized perturbation expansion e j weniger j ifmmode check c else v c fi i ifmmode check z else v z fi ek and f vinette j math phys 34 571 1993 in the sextic case pad e approximants are still able to sum these perturbation series but convergence is so slow that they are computationally useless in the octic case pad e approximants are not powerful enough and fail on the other hand the inclusion of only a few additional data from the strong coupling domain e j weniger ann phys ny to be published greatly enhances the power of summation methods the summation techniques which we consider are two point pad e approximants and effective characteristic polynomials it is shown that these summation methods give good results for the quartic and sextic anharmonic oscillators and even in the case of the octic anharmonic oscillator which represents an extremely challenging summation problem two point pad e approximants give relatively good results textcopyright 1996 the american physical society

comparison of asymptotic solutions of a phase field model to a sharp interface model a one dimensional directional solidification problem is considered for the purpose of analyzing the relationship between the solution resulting from a phase field model to that from a sharp interface model the solidification problem is posed within a finite domain rather than an infinite extent as in classical stefan problems an asymptotic analysis based upon a small stefan number is performed on the sharp interface model in the phase field case the small stefan number expansion is coupled with a small interface thickness boundary layer expansion this approach enables us to develop analytical solutions to the phase field model the results show agreement at leading order between the two models for the location of the solidification front and the temperature profiles in the solid and liquid phases however due to the nonzero interface thickness in the phase field model corrections to the sharp interface location and temperature profiles develop these corrections result from the conduction of late

efficient thermal field computation in phase field models we solve the phase field equations in two dimensions to simulate crystal growth in the low undercooling regime the novelty is the use of a fast solver for the free space heat equation to compute the thermal field this solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient free space heat equation with a smooth source term the computational cost and memory requirements of the new solver are reasonable and no artificial boundary conditions are needed this allows one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field which can result in significant computational savings in the low undercooling regime

a level set simulation of dendritic solidification with combined features of front tracking and fixed domain methods a method combining features of front tracking methods and fixed domain methods is presented to model dendritic solidification of pure materials to explicitly track the interface growth and shape of the solidifying crystals a front tracking approach based on the level set method is implemented to easily model the heat and momentum transport a fixed domain method is implemented assuming a diffused freezing front where the liquid fraction is defined in terms of the level set function the fixed domain approach by avoiding the explicit application of essential boundary conditions on the freezing front leads to an energy conserving methodology that is not sensitive to the mesh size to compute the freezing front morphology an extended stefan condition is considered applications to several classical stefan problems and two and three dimensional crystal growth of pure materials in an undercooled melt including the effects of melt flow are considered the computed results agree very well with available analytical solutions as well as with results obtained using front tracking techniques and the phase field method

sharp interface cartesian grid method iii solidification of pure materials and binary solutions a numerical technique is presented for computing dendritic growth of crystals from pure melts and binary solutions the governing equations are solved on a fixed cartesian mesh and the immersed phase boundary is treated as a sharp solid fluid interface the interface is tracked using a level set field a finite difference scheme is presented that incorporates the immersed phase boundary with only a small change to a standard cartesian grid poisson solver the scheme is simple to implement in three dimensions the results from our calculations show excellent agreement with two dimensional microscopic solvability theory for pure material solidification it is shown that the method predicts dendrite tip characteristics in excellent agreement with the theory the sharp interface treatment allows discontinuous material property variation at the solid liquid interface this facilitates sharp interface simulations of dendritic solidification of binary solutions

short note a volume of fluid approach for crystal growth simulation a new approach to simulating the dendritic growth of pure metals based on a recent volume of fluid vof method with plic piecewise linear interface calculation reconstruction of the interface is presented the energy equation is solved using a diffuse interface method which avoids the need to apply the thermal boundary conditions directly at the solid front the thermal gradients at both sides of the interface which are needed to obtain the front velocity are calculated with the aid of a distance function to the reconstructed interface the advection equation of a discretized solid fraction function is solved using the unsplit vof advection method proposed by lopez et al j comput phys 195 2004 718 742 extended to three dimensions by hernandez et al int j numer methods fluids 58 2008 897 921 and the interface curvature is computed using an improved height function technique which provides second order accuracy the proposed methodology is assessed by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions

the blow up behavior of the heat equation with neumann boundary conditions abstract we establish the blow up rate for the solution of the heat equation u t u xx 0 x t 0 subject to neumann boundary conditions u x 0 t 0 u x 1 t e u 1 t with convex initial data

global and blow up solutions for nonlinear parabolic equations with robin boundary conditions abstract in this paper we discuss the blow up for classical solutions to the following class of parabolic equations with robin boundary condition b u t g u u f u in 0 t u n u 0 on 0 t u x 0 h x 0 in where is a bounded domain of r n n 2 with smooth boundary by constructing some appropriate auxiliary functions and using a first order differential inequality technique we derive conditions on the data which guarantee the blow up or the global existence of the solution for the blow up solution a lower bound on blow up time is also obtained moreover some examples are presented to illustrate the applications

phase field modelling of solidification microstructure formation using the discontinuous finite element method this paper presents a discontinuous galerkin finite element computational methodology for solving the coupled phase field and heat diffusion equations to predict the microstructure evolution during solidification the phase field modelling of microstructure formation is briefly discussed the discontinuous galerkin finite element formulation for the phase field model systems for solidification microstructure formation is described in detail numerical stability is performed using the neumann method the accuracy of the discontinuous model is verified by the analytic solution of a simple 1 d solidification problem numerical calculations using the discontinuous finite element phase field model have been performed for simulating the complex 2 d and 3 d dendrite structures formed in supercooled melts and the results are compared well with those in literature using the finite difference methods parallel computing algorithm is presented and results show that the minimization of the intercommunication between microprocessors is the key to increase the effectiveness in parallel computing with the discontinuous finite element phase field model for solidification microstructure formation copyright 2006 john wiley sons ltd

numerical approximation of a thermally driven interface using finite elements a two dimensional finite element model for dendritic solidification has been developed that is based on the direct solution of the energy equation over a fixed mesh the model tracks the position of the sharp solid liquid interface using a set of marker points placed on the interface the simulations require calculation of the temperature gradients on both sides of the interface in the direction normal to it at the interface the heat flux is discontinuous due to the release of latent heat during the solidification melting process two ways to calculate the temperature gradients at the interface evaluating their interpolants at gauss points were proposed using known one and two dimensional solutions to stable solidification problems the stefan problem it was shown that the method converges with second order accuracy when applied to the unstable solidification of a crystal into an undercooled liquid it was found that the numerical solution is extremely sensitive to the mesh size and the type of approximation used to calculate the temperature gradients at the interface i e different approximations and different meshes can yield different solutions the cause of these difficulties is examined the effect of different types of interpolation on the simulations is investigated and the necessary criteria to ensure converged solutions are established copyright 2003 john wiley sons ltd

numerical simulation of crystal growth in three dimensions using a sharp interface finite element method a sharp interface numerical model is presented to simulate thermally driven crystal growth in three dimensional space the model is formulated using the finite element method and works directly with primitive variables it solves the energy equation in a fixed volume mesh while explicitly tracking the motion of the solid liquid interface the three dimensional interface is represented by connected planar triangles that form a surface mesh to accurately capture the morphology of the growing dendrite the surface mesh is updated every few time steps so that the quality of the triangles is maintained and the size of the triangles is always kept in a range associated with the element size of the fixed volume mesh the interface curvature is calculated by a least squares paraboloid fitting to neighbouring nodes the model is validated through a comparison with an exact solution of a three dimensional stefan problem a mesh refinement study a mesh orientation test and a comparison with solvability theory it is shown that the interface position is tracked to second order accuracy simulations are performed under different combinations of the undercooling and surface energy the effects of these parameters on the growth and morphology of the dendrites are studied copyright 2006 john wiley sons ltd

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

surface evolution of elastically stressed films under deposition by a diffuse interface model we consider the heteroepitaxial growth of thin films by numerical simulations within a diffuse interface model the model is applicable to describe the self organization of nanostructures the influence of strain surface energies and kinetics on the surface evolution is considered a matched asymptotic analysis shows the formal convergence of an anisotropic viscous cahn hilliard model to a general surface evolution equation the system is solved by adaptive finite elements in three dimensions and in special cases compared with sharp interface models

multiscale finite difference diffusion monte carlo method for simulating dendritic solidification abstract we present a novel hybrid computational method to simulate accurately dendritic solidification in the low undercooling limit where the dendrite tip radius is one or more orders of magnitude smaller than the characteristic spatial scale of variation of the surrounding thermal or solutal diffusion field the first key feature of this method is an efficient multiscale diffusion monte carlo dmc algorithm which allows off lattice random walkers to take longer and concomitantly rarer steps with increasing distance away from the solid liquid interface as a result the computational cost of evolving the large scale diffusion field becomes insignificant when compared to that of calculating the interface evolution the second key feature is that random walks are only permitted outside of a thin liquid layer surrounding the interface inside this layer and in the solid the diffusion equation is solved using a standard finite difference algorithm that is interfaced with the dmc algorithm using the local conservation law for the diffusing quantity here we combine this algorithm with a previously developed phase field formulation of the interface dynamics and demonstrate that it can accurately simulate three dimensional dendritic growth in a previously unreachable range of low undercoolings that is of direct experimental relevance

blow up solutions and global solutions for nonlinear parabolic problems abstract this paper deals with the blow up of positive solutions for a nonlinear parabolic equation subject to nonlinear boundary conditions we obtain the conditions under which the solutions may exist globally or blow up in a finite time by a new approach moreover upper estimates of the blow up time blow up rate and global solutions are obtained also

the general structure of eigenvalues in nonlinear oscillators hilbert spaces of bounded one dimensional nonlinear oscillators are studied it is shown that the eigenvalue structure of all such oscillators have the same general form they depend only on the ground state energy of the system and a single function h of the hamiltonian operator h it is also found that the hilbert space of the nonlinear oscillator is unitarily inequivalent to the hilbert space of the simple harmonic oscillator providing an explicit example of haag s theorem a number operator for the nonlinear oscillator is constructed and the general form of the partition function and average energy of a nonlinear oscillator in contact with a heat bath is determined connection with the wkb result in the semiclassical limit is made the analysis is then applied to the case of the x4 anharmonic oscillator as an explicit example

robust second order accurate discretizations of the multi dimensional heaviside and dirac delta functions we present a robust second order accurate method for discretizing the multi dimensional heaviside and the dirac delta functions on irregular domains the method is robust in the following ways 1 integrations of source terms on a co dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain s boundary 2 the method depends only on the function value of a level function not on its derivatives we present the discretizations in tabulated form to make their implementations straightforward we present numerical results in two and three spatial dimensions to demonstrate the second order accuracy in the l 1 norm in the case of the solution of pdes with singular source terms in the case of evaluating the contribution of singular source terms on interfaces the method is also second order accurate in the l norm

a conservative numerical method for the cahn hilliard equation with dirichlet boundary conditions in complex domains in this paper we present a conservative numerical method for the cahn hilliard equation with dirichlet boundary conditions in complex domains the method uses an unconditionally gradient stable nonlinear splitting numerical scheme to remove the high order time step stability constraints the continuous problem has the conservation of mass and we prove the conservative property of the proposed discrete scheme in complex domains we describe the implementation of the proposed numerical scheme in detail the resulting system of discrete equations is solved by a nonlinear multigrid method we demonstrate the accuracy and robustness of the proposed dirichlet boundary formulation using various numerical experiments we numerically show the total energy decrease and the unconditionally gradient stability in particular the numerical results indicate the potential usefulness of the proposed method for accurately calculating biological membrane dynamics in confined domains

anharmonic oscillator constructing the strong coupling expansions two novel approaches to construction of the strong coupling expansion for the anharmonic oscillator with the potential v x 1 2 x2 g 4 x4 are proposed the first one is simply a straightforward solution of the schrodinger equation via the nonlinearization technique resulting in the rapidly convergent perturbation series the second one is a version of the path integral perturbation theory but with an unconventional choice of the zeroth approximation action nine leading coefficients of the strong coupling expansion are computed they decrease rapidly the ninth one being of the order of 10 9 three leading corrections of the nonlinearization approach provide the ground state energy within a relative accuracy of 10 7 10 9 at an arbitrary coupling g the explicit formulas for corrections enable one to study the analyticity properties of the energy as a function of the coupling g in the second approach the strong coupling expansion coefficients are computed as an infinite linear series of the weak

an implicit parallel multigrid computing scheme to solve coupled thermal solute phase field equations for dendrite evolution an implicit second order space and time discretization scheme together with a parallel multigrid method involving a strip grid domain partitioning has been developed to solve fully coupled nonlinear phase field equations involving solute and heat transport for multiple solidifying dendrites the computational algorithm has been shown to be stable and monotonously convergent and allowed time marching steps that were 3 4 orders of magnitude larger than those employed in similar explicit approaches resulting in an increase of 3 4 orders of magnitude in computing efficiency full solute and thermal coupling was achieved for metallic alloys with a realistic high lewis number of 10 4 the parallel multigrid computing scheme is shown to provide a scalable methodology that allowed the efficient use of distributed supercomputing resource to simulate the evolution of tens of complex shaped 2d dendrites in a computational domain containing tens or even hundreds of millions of grid points the simulations have provided insight into the dynamic interplay of many growing dendrites in a more realistic fully coupled thermal solute condition capturing for the first time fine scale features such as dendrite splitting

efficient symmetric discretization for the poisson heat and stefan type problems with robin boundary conditions we present a novel and efficient method for solving the poisson equation the heat equation and stefan type problems with robin boundary conditions over potentially moving arbitrarily shaped domains the method utilizes a level set framework thus it has all of the benefits of a sharp implicitly represented interface such as the ease of handling complex topological changes this method is straightforward to implement and leads to a linear system that is symmetric and positive definite which can be inverted efficiently with standard iterative methods this approach is second order accurate for both the poisson and heat equations and first order accurate for the stefan problem we demonstrate the accuracy in the l 1 and l norms

an adaptive mesh redistribution method for the incompressible mixture flows using phase field model a phase field model which describes the motion of mixtures of two incompressible fluids is presented by liu and shen c liu j shen a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method phys d 179 2003 211 228 the model is based on an energetic variational formulation in this work we develop an efficient adaptive mesh method for solving a phase field model for the mixture flow of two incompressible fluids it is a coupled nonlinear system of navier stokes equations and allen cahn phase equation phase field equation through an extra stress term and the transport term the numerical strategy is based on the approach proposed by li et al r li t tang p w zhang moving mesh methods in multiple dimensions based on harmonic maps j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution in the pde evolution part the phase field equation is numerically solved by a conservative scheme with a lagrange multiplier and the coupled incompressible navier stokes equations are solved by the incremental pressure correction projection scheme based on the semi staggered grid method in the mesh moving part the mesh points are iteratively redistributed by solving the euler lagrange equations with a parameter free monitor function in each iteration the pressure and the phase are updated on the resulting new grid by a conservative interpolation formula while the velocity is re mapped in a non conservative approach a simple method for preserving divergence free is obtained by projecting the velocity onto the divergence free space after generating the new mesh at the last iterative step numerical experiments are presented to demonstrate the effectiveness of the proposed method for solving the incompressible mixture flows

phase field model for hele shaw flows with arbitrary viscosity contrast i theoretical approach we present a phase field model for the dynamics of the interface between two inmiscible fluids with arbitrary viscosity contrast in a rectangular hele shaw cell with asymptotic matching techniques we check the model to yield the right hele shaw equations in the sharp interface limit and compute the corrections to these equations to first order in the interface thickness we also compute the effect of such corrections on the linear dispersion relation of the planar interface we discuss in detail the conditions on the interface thickness to control the accuracy and convergence of the phase field model to the limiting hele shaw dynamics in particular the convergence appears to be slower for high viscosity contrasts

a necessary and sufficient condition for global existence for a degenerate parabolic boundary value problem consider the degenerate parabolic boundary value problemut u f u on 0 in which is a bounded domain inrnand thec 0 functionsfand are nonnegative and nondecreasing with s f s 0 ifs 0 and 0 0 assume homogeneous neumann boundary conditions and an initial condition that is nonnegative nontrivial and continuous on because the function is not sufficiently nice to allow this problem to have a classical solution we consider generalized solutions in a manner similar to that of benilan crandall and sacks appl math optim 17 1988 203 224 we show that this initial boundary value problem has such a nonnegative generalized solution if and only ifi 0 ds 1 f s

blow up and global existence for nonlinear parabolic equations with neumann boundary conditions in this paper we study the global and blow up solutions of the following problem h u t a u t b x u g t f u in dx 0 t u n 0on dx 0 t u x 0 u 0 x 0in d where d r n is a bounded domain with smooth boundary d by constructing auxiliary functions and using maximum principles and comparison principles the sufficient conditions for the existence of global solution an upper estimate of the global solution the sufficient conditions for the existence of the blow up solution an upper bound for the blow up time and an upper estimate of the blow up rate are specified under some appropriate assumptions on the functions a b f g and h

modeling the growth and interaction of multiple dendrites in solidification using a level set method abstract a level set method is presented to study the growth and interaction of multiple dendrites in solidification the method couples thermal and solute diffusion with propagation of multiple interfaces a single signed distance function is used to track the solid liquid interface with the aid of markers the value of which is the orientation angle for identification of different crystals the problem of evolving multiple crystal interfaces is reduced to two tasks 1 tracking one level set variable signed distance function and 2 determination of the marker for a newly solidified finite element nodal point tracking a single level set variable is implemented by solving the level set equation with interface velocity computed from an extended stefan equation using the marker information crystal orientation determination of the marker for a newly solidified finite element nodal point is implemented by using an algorithm modified from the fast marching technique both of these two steps are computationally efficient and the approach is suitable for incorporating effects of multiple crystals convergence and accuracy of this approach are demonstrated by using different grid spacings and comparing with results obtained from the multi phase level set method a parametric study is performed to investigate the effects of solidification speed and thermal gradient on the resulting solidification microstructure pattern numerical results of columnar to equiaxed transition cet qualitatively agree with an analytical estimation and are similar to previous numerical results obtained using a phase field method a convergence study is performed to determine the appropriate grid spacing for numerical simulation at lower surface tension cet occurs at a lower thermal gradient for a giving solidification speed secondary dendrite formation is more apparent with lower surface tension the differences and similarities between the three dimensional and two dimensional growth results are analyzed randomness in crystal orientation and required undercooling for nucleation are modeled and found to have a great effect on the microstructure pattern the efficiency of the present approach is finally demonstrated with an example that includes the growth of hundreds of crystals with consideration of randomness effects

blow up problems for a semilinear heat equation with large diffusion abstract we consider the blow up problem of a semilinear heat equation u t d u u p in 0 t d u x t 0 on 0 t d u x 0 x 0 in where is a bounded smooth domain in r n t d 0 d 0 and p 1 we study the blow up time the location of the blow up set and the blow up profile of the blow up solution for sufficiently large d in particular we prove that for almost all initial data if d is sufficiently large then the solution blows up only near the maximum points of the orthogonal projection of the initial data from l 2 onto the second neumann eigenspace

a diffuse interface method for two phase flows with soluble surfactants a method is presented to solve two phase problems involving soluble surfactants the incompressible navier stokes equations are solved along with equations for the bulk and interfacial surfactant concentrations a nonlinear equation of state is used to relate the surface tension to the interfacial surfactant concentration the method is based on the use of a diffuse interface which allows a simple implementation using standard finite difference or finite element techniques here finite difference methods on a block structured adaptive grid are used and the resulting equations are solved using a nonlinear multigrid method results are presented for a drop in shear flow in both 2d and 3d and the effect of solubility is discussed

front tracking finite element method for dendritic solidification 765 abstract a two dimensional finite element method capable of tracking sharp interfaces is developed the method is based on a fixed mesh of bilinear isoparametric elements and is extremely simple and easy to use the interfaces are tracked with a set of marker points that define their position at all times several different approaches to finding the interface position and velocity are discussed and their effect on the convergence rate is examined it is shown through numerical examples that the approximation to the interface position converges with second order accuracy in two dimensional problems as opposed to previously developed finite difference algorithms that are only first order accurate the approximation to the interface velocity is shown to be of first order a number of examples are examined including several that provide a careful comparison with previously published results

adaptive mesh computation of polycrystalline pattern formation using a renormalization group reduction of the phase field crystal model we implement an adaptive mesh algorithm for calculating the space and time dependence of the atomic density field in microscopic material processes our numerical approach uses the systematic renormalization group formulation of a phase field crystal model of a pure material to provide the underlying equations for the complex amplitude of the atomic density field a quantity that is spatially uniform except near topological defects grain boundaries and other lattice imperfections our algorithm employs a hybrid formulation of the amplitude equations combining cartesian and polar decompositions of the complex amplitude we show that this approach leads to an acceleration by three orders of magnitude in model calculations of polycrystalline grain growth in two dimensions

a simple moving mesh method for one and two dimensional phase field equations a simple moving mesh method is proposed for solving phase field equations the numerical strategy is based on the approach proposed in li et al j comput phys 170 2001 562 588 to separate the mesh moving and pde evolution the phase field equations are discretized by a finite volume method and the mesh moving part is realized by solving the conventional euler lagrange equations with the standard gradient based monitors numerical results demonstrate the accuracy and effectiveness of the proposed algorithm

stochastic reaction and diffusion on growing domains understanding the breakdown of robust pattern formation many biological patterns from population densities to animal coat markings can be thought of as heterogeneous spatiotemporal distributions of mobile agents many mathematical models have been proposed to account for the emergence of this complexity but in general they have consisted of deterministic systems of differential equations which do not take into account the stochastic nature of population interactions one particular pertinent criticism of these deterministic systems is that the exhibited patterns can often be highly sensitive to changes in initial conditions domain geometry parameter values etc due to this sensitivity we seek to understand the effects of stochasticity and growth on paradigm biological patterning models in this paper we extend spatial fourier analysis and growing domain mapping techniques to encompass stochastic turing systems through this we find that the stochastic systems are able to realize much richer dynamics than their deterministic counterparts in that patterns are able to exist outside the standard turing parameter range further it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth when considering which wave modes are excited finally although growth is able to generate robust pattern sequences in the deterministic case we see that stochastic effects destroy this mechanism for conferring robustness however through fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it

the strong coupling expansion for anharmonic oscillators we accurately calculate the strong coupling expansion for anharmonic oscillators by means of a robust and stable numerical algorithm the method applies to any state and to any anharmonicity degree by means of the perturbation coefficients we estimate the location of the branch points that determine the convergence radius of the strong coupling expansion

summation of asymptotic expansions of multiple valued functions using algebraic approximants application to anharmonic oscillators the divergent rayleigh schrodinger perturbation expansions for energy eigenvalues of cubic quartic sextic and octic oscillators are summed using algebraic approximants these approximants are generalized pade approximants that are obtained from an algebraic equation of arbitrary degree numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree different branches of the approximants converge to different branches of the function the success of the high degree approximants is attributed to their ability to model the function on multiple sheets of the riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter an efficient recursive algorithm for computing the diagonal approximant sequence is presented

diffuse interface approximations of osmosis free boundary problems free boundary problems based on mass conservation and surface tension with application in osmotic swelling are the topic of this contribution phase field approximations of such models are introduced in order to numerically investigate properties of the solutions formal justification of the proposed approximations is provided by matched asymptotic expansions supported by numerical tests reproducing the convergence for shrinking interface thickness

reply to comments on a new efficient method for calculating perturbation energies using functions which are not quadratically integrable in a previous paper sk la and c zek 1996 a new approach to calculating perturbation energies of bound states based on the use of functions which are not quadratically integrable was suggested this method has been analysed in a comment by guardiola and ros 1996 the present comment is the reply to this previous comment the aim is to give a simple and straightforward proof of the method and clarify some other points important for its application

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

spectral methods for partial differential equations in irregular domains the spectral smoothed boundary method in this paper we propose a numerical method to approximate the solution of partial differential equations in irregular domains with no flux boundary conditions the idea is to embed the domain into a box and use a smoothing term to encode the boundary conditions into a modified equation that can be approached by standard spectral methods the main features of this method are its capability to deal with domains of arbitrary shape and its easy implementation via fast fourier transform routines we discuss several examples of practical interest and test the results against standard numerical methods

fast convergent resummation algorithm and critical exponents of φ4 theory in three dimensions we develop an efficient algorithm for evaluating divergent perturbation expansions of field theories in the bare coupling constant gb for which we possess a finite number l of expansion coefficients plus two more information the knowledge of the large order behavior proportional to kk k gbk with a known growth parameter and the knowledge of the approach to scaling of the type c c gb with constants c c and a critical exponent of approach the latter information leads to an increase in the speed of convergence and a high accuracy of the results the algorithm is applied to the six and seven loop expansions for the critical exponents of o n symmetric 4 theories and the result for the critical exponent is compared with a recent satellite experiment

large order behaviour of the strong coupling perturbation expansion for anharmonic oscillators a new formula describing the large order behaviour of the strong coupling perturbation coefficients for the anharmonic oscillators with the hamiltonian is suggested a new method for the accurate calculation of the square root branch points of the energy from the numerical values of the coefficients is also suggested the branch points and the related minimal values of the coupling constant for which the expansion converges are calculated for the ground state of the quartic sextic octic and decadic oscillators

asymptotic response of observables from divergent weak coupling expansions a fractional calculus assisted pade technique appropriate constructions of pade approximants are believed to provide reasonable estimates of the asymptotic large coupling amplitude and exponent of an observable given its weak coupling expansion to some desired order in many instances however sequences of such approximants are seen to converge very poorly we outline here a strategy that exploits the idea of fractional calculus to considerably improve the convergence behavior pilot calculations on the ground state perturbative energy series of quartic sextic and octic anharmonic oscillators reveal clearly the worth of our endeavor

spectral implementation of an adaptive moving mesh method for phase field equations phase field simulations have been extensively applied to modeling microstructure evolution during various materials processes however large scale simulations of three dimensional 3d microstructures are still computationally expensive among recent efforts to develop advanced numerical algorithms the semi implicit fourier spectral method is found to be particularly efficient for systems involving long range interactions as it is able to utilize the fast fourier transforms fft on uniform grids in this paper we report our recent progress in making grid points spatially adaptive in the physical domain via a moving mesh strategy while maintaining a uniform grid in the computational domain for the spectral implementation this approach not only provides more accurate treatment at the interfaces requiring higher resolution but also retains the numerical efficiency of the semi implicit fourier spectral method numerical examples using the new adaptive moving mesh semi implicit fourier spectral method are presented for both two and three space dimensional microstructure simulations and they are compared with those obtained by other methods by maintaining a similar accuracy the proposed method is shown to be far more efficient than the existing methods for microstructures with small ratios of interfacial widths to the domain size

blow up and global solutions for nonlinear reaction diffusion equations with neumann boundary conditions abstract the type of problem under consideration is 1 u ln 1 u t ln 1 u u 1 u ln 1 u in d 0 t u n 0 on d 0 t u x 0 u 0 x 0 in d where d r n is a bounded domain with smooth boundary d n 2 it is proved that if 1 0 the positive solution u x t blows up globally in d whereas if 0 1 the positive solution u x t is global solution furthermore an upper bound of the blow up time an upper estimate of the blow up rate and an upper estimate of the global solutions are given

energy and entropy preserving numerical approximations of thermodynamically consistent crystal growth models abstract we present a numerical scheme that preserves the total energy and the entropy production rate termed the energy and entropy production rate preserving scheme for a general class of thermodynamically consistent phase field models for dendritic crystal growth derived from the first and second law of thermodynamics the scheme is second order in time linear and energy and entropy production rate preserving for any time steps the scheme is first discretized in time aided by the energy quadratization eq method and then in space using compact finite difference methods the linear system resulting from the scheme is shown to be uniquely solvable at both the semi discrete and the fully discrete level mesh refinement tests are performed to show the second order time convergence rate in the scheme several numerical examples of dendritic crystal growth are provided to demonstrate the accuracy and efficiency of the scheme the effects of various model parameters on growth patterns of the crystal are further investigated in details with the numerical solver the approach to developing the energy and entropy production rate preserving numerical scheme proposed in this study is so general that it can be applied to a wide range of thermodynamically consistent models not limited to phase field models

computational approach for modeling intra and extracellular dynamics we introduce a phase field approach for diffusion inside and outside a closed cell with damping and with source terms at the stationary interface the method is compared to exact solutions where possible and the more traditional finite element method it is shown to be very accurate easy to implement and computationally inexpensive we apply our method to a recently introduced model for chemotaxis by rappel et al biophys j 83 1361 2002

morphological transitions and bistability in turing systems it is well known that in two dimensions turing systems produce spots stripes and labyrinthine patterns and in three dimensions lamellar and spherical structures or their combinations are observed in this paper we study transitions between these states in both two and three dimensions first we derive the regions of stability for different patterns using nonlinear bifurcation analysis then we apply large scale computer simulations to analyze the pattern selection in a bistable system by studying the effect of parameter selection on morphological clustering and the appearance of topological defects the method elaborated in this paper presents a probabilistic approach for studying pattern selection in a bistable reaction diffusion system

a diffuse interface model for alloys with multiple components and phases a nonisothermal phase field model for alloys with multiple phases and components is derived the model allows for arbitrary phase diagrams we relate the model to classical sharp interface models by formally matched asymptotic expansions in addition we discuss several examples and relate our model to the ones already existing

efficient computation of dendritic growth with r adaptive finite element methods this paper deals with the application of a moving grid method to the solution of a phase field model for dendritic growth in two and three dimensions a mesh is found as the solution of an optimization problem that automatically includes the boundary conditions and is solved using a multi grid approach the governing equations are discretized in space by linear finite elements and a split time level scheme is used to numerically integrate in time one novel aspect of the method is the choice of a regularized monitor function the moving grid method enables us to obtain accurate numerical solutions with much less degree of freedoms it is demonstrated numerically that the tip velocity obtained by our method is in good agreement with the previously published results

script pscript t symmetric models in more dimensions and solvable square well versions of their angular schrodinger equations from the partial differential calogero s three body and smorodinsky winternitz superintegrable hamiltonians in two variables we separate the respective angular schrodinger equations and study the possibilities of their minimal symmetric complexification the simultaneous loss of the hermiticity and solvability of the respective angular potentials v is compensated by their replacement by solvable purely imaginary and piece wise constant multiple wells v0 we demonstrate that the spectrum remains real and that it exhibits a rich four series structure in the double well case