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

an adaptive moving mesh method for forced curve shortening flow we propose a novel adaptive moving mesh method for the numerical solution of a forced curve shortening geometric evolution equation control of the mesh quality is obtained using a tangential mesh velocity derived from a mesh equidistribution principle where a positive adaptivity measure or monitor function is approximately equidistributed along the evolving curve central finite differences are used to discretize in space the governing evolution equation for the position vector and a second order implicit scheme is used for the temporal integration simulations are presented indicating the generation of meshes which resolve areas of high curvature and are of second order accuracy furthermore the new method delivers improved solution accuracy compared to the use of uniform arc length meshes

moving mesh partial differential equations to describe nematic order dynamics nematic liquid crystals are aggregates of calamitic molecules and most related experimental phenomena are described well by their mean molecular orientation i e by the director and by the scalar order parameter considering a perfect uniaxial symmetry however when the nematic distortion is very strong and it occurs over a length scale comparable with the nematic coherence length the molecular order may be significantly altered as in the case of the core of a defect or in the case of highly frustrated nematic systems such systems where spatial and or temporal changes of the nematic order are relevant require a full landau de gennes q tensor description in this work we will present the implementation of a q tensor numerical model based on a one dimensional finite element method with a r type moving mesh technique capable of describing the nematic order dynamics inside a p cell submitted to a strong electric pulse the use of the moving grid technique ensures no waste of computational effort in the area of low spatial order variability in fact the technique concentrates the grid points in regions of large q maintaining constant the total number of nodes in the domain

on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one dimensional partial differential equations pdes the particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by w huang y ren and r d russell 1994 siam j numer anal 31 709 we show how the accuracy of the computations is strongly dependent on the choice of monitor function and we present a monitor function that yields an optimal rate of convergence motivated by efficiency considerations for problems in two or more space dimensions we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical pde the accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time step control mechanism is integrated into the scheme

on simple moving grid methods for one dimensional evolutionary partial differential equations abstract two moving grid algorithms for the numerical integration of evolutionary one dimensional partial differential equations are considered both algorithms discretize the equation on trapezoidal space time elements and combine a prediction step with a remeshing technique in order to determine the orientation of the sides of the trapezoids joining nodes at consecutive time levels the behavior of the discretizations employed is forecast by means of the modified equation technique and then studied in a series of numerical experiments

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

tensor product adaptive grids based on coordinate transformations in this paper we discuss a two dimensional adaptive grid method that is based on a tensor product approach adaptive grids are a commonly used tool for increasing the accuracy and reducing computational costs when solving both partial differential equations pdes and ordinary differential equations a traditional and widely used form of adaptivity is the concept of equidistribution which is well defined and well understood in one space dimension the extension of the equidistribution principle to two or three space dimensions however is far from trivial and has been the subject of investigation of many researchers during the last decade besides the nonsingularity of the transformation that defines the nonuniform adaptive grid the smoothness of the grid or transformation plays an important role as well we will analyse these properties and illustrate their importance with numerical experiments for a set of time dependent pde models with steep moving pulses fronts and boundary layers

practical aspects of formulation and solution of moving mesh partial differential equations moving mesh partial differential equations mmpdes are used in the mmpde moving mesh method to generate adaptive moving meshes for the numerical solution of time dependent problems how mmpdes are formulated and solved is crucial to the efficiency and robustness of the method in this paper several practical aspects of formulating and solving mmpdes are studied they include spatial balance scaling invariance effective control of mesh concentration bounds on time steps multiple sub steps and two level mesh movement numerical results are also given c 2001 academic press

bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication we show that entanglement guarantees difficulty in the discrimination of orthogonal multipartite states locally the number of pure states that can be discriminated by local operations and classical communication is bounded by the total dimension over the average entanglement a similar general condition is also shown for pure and mixed states these results offer a rare operational interpretation for three abstractly defined distancelike measures of multipartite entanglement

adaptive pseudospectral solution of a diffuse interface model a diffuse interface type model using an energy based variational formulation with a free energy that is a function of the density and its gradients is presented all of the boundary terms are retained and related to external surface forces which can be of particular interest when considering the fluid fluid solid region the numerical solution of these types of problems can be troublesome if a thin transition layer is desired here chebyshev pseudospectral methods with mesh adaptation for the solution of diffuse interface type problems are studied a mesh adaptation algorithm based in the equidistribution principle following a continuation process is derived in order to achieve high precision for problems exhibiting thin transition layers a modified version of the arc length monitor function is proposed which yields a sufficiently smooth coordinate transformation at every step of the continuation process a fixed number of iterations is implemented so that the equidistribution equations are not solved completely at each step which saves a considerable amount of computational effort numerical results for the static phase field model exhibiting thin transition layers are presented

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

adaptive grid modelling for cancer cells in the early stage of invasion modelling the interaction of the urokinase plasminogen activator system with a model for cancer cells dynamics lead to a system of five coupled partial differential equations the solution of which over a one dimensional domain can be obtained using the finite element method as the discretization of the integration domain is crucial particularly in the presence of strong variations of the domain variables we tackle the problem solution implementing an adaptive grid numerical technique our results improve previous numerical simulations performed on uniformly discretized domains allowing to capture finer spatial details characterizing the interface between cancer and healthy cells which can be related to malignancy low values of the cancer cells motility induce a high spatial heterogeneity at the cancer healthy cells interface while the dynamical evolution over the whole domain shows branched spatio temporal patterns on the contrary higher motility values smooth the cancer cells spatial profile letting cancer cell concentration to evolve according to a less heterogeneous pattern

an adaptive moving grid method for one dimensional systems of partial differential equations we describe a fully adaptive moving grid method for solving initial boundary value problems for systems of one space dimensional partial differential equations whose solutions exhibit rapid variations in space and time the method based on finite differences is of the lagrangian type and has been derived through a co ordinate transformation which leads to equidistribution in space of the second derivative our technique is intermediate between static regridding methods where nodes remain fixed for intervals of time and continuously moving grid methods where the node movement and the pde integration are fully coupled in our approach the computation of the moving grids and the solution on these grids are carried out separately while the nodes are moved at each time step two error monitors have been implemented one to govern the time step selection and the other to eventually adapt the number of moving nodes the method allows the use of different moving grids for different components in the pde system numerical experiments are presented for a set of five sample problems from the literature including two problems from combustion

an error indicator monitor function for an r adaptive finite element method an r adaptive finite element method based on moving mesh partial differential equations pdes and an error indicator is presented the error indicator is obtained by applying a technique developed by bank and weiser to elliptic equations which result in this case from temporal discretization of the underlying physical pdes on moving meshes the construction of the monitor function based on the error indicator is discussed numerical results obtained with the current method and the commonly used method based on solution gradients are presented and analyzed for several examples c 2001 academic press

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

movcol4 a moving mesh code for fourth order time dependent partial differential equations in this paper we develop and analyze a moving mesh code for the adaptive simulation of fourth order pdes based on collocation the scheme is shown to enforce discrete conservation for problems written in a generalized conservation form to demonstrate the breadth of applicability we present examples from both cahn hilliard and thin film type equations exhibiting metastable behavior finite time solution blow up finite time extinction and moving interfaces

a computational method for the coupled solution of reaction diffusion equations on evolving domains and manifolds in this paper we devise a moving mesh finite element method for the approximate solution of coupled bulk surface reaction diffusion equations on an evolving two dimensional domain fundamental to the success of the method is the robust generation of bulk and surface meshes for this purpose we use a novel moving mesh partial differential equation mmpde approach the developed method is applied to model problems with known analytical solutions these experiments indicate second order spatial and temporal accuracy coupled bulk surface problems occur frequently in many areas in particular in the modelling of eukaryotic cell migration and chemotaxis we apply the method to a model of the two way interaction of a migrating cell in a chemotactic field where the bulk region corresponds to the extracellular region and the surface to the cell membrane

solution of evolutionary partial differential equations using adaptive finite differences with pseudospectral post processing a coordinate transformation approach is described that enables pseudospectral methods to be applied efficiently to unsteady differential problems with steep solutions the work is an extension of a method presented by mulholland huang and sloan for the adaptive pseudospectral solution of steady problems a coarse grid is generated by a moving mesh finite difference method that is based on equidistribution and this grid is used to construct a time dependent coordinate transformation a sequence of spatial transformations may be generated at discrete points in time or a single transformation may be generated as a continuous function of space and time the differential problem is transformed by the coordinate transformation and then solved using a method that combines pseudospectral discretisation in space with a suitable integrator in time numerical results are presented for unsteady problems in one space dimension

efficient moving mesh methods for q tensor models of nematic liquid crystals this paper describes a robust and efficient numerical scheme for solving the system of six coupled partial differential equations which arises when using q tensor theory to model the behavior of a nematic liquid crystal cell under the influence of an applied electric field the key novel feature is the use of a full moving mesh partial differential equation approach to generate an adaptive mesh which accurately resolves important solution features this includes the use of a new monitor function based on a local measure of biaxiality in addition adaptive time step control is used to ensure the accurate predicting of the switching time which is often critical in the design of liquid crystal cells we illustrate the behavior of the method on a one dimensional time dependent problem in a pi cell geometry which admits two topologically different equilibrium states modeling the order reconstruction which occurs on the application of an electric field our numerical results show that in addition to achie

a pseudospectral method of solution of fisher s equation in this paper we develop an accurate and efficient pseudospectral solution of fisher s equation a prototypical reaction diffusion equation the solutions of fisher s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient there is an ongoing effort to better adapt pseudospectral methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations the collocation method employed is based on chebyshev gauss lobatto quadrature points we compare results for a single domain as well as for a subdivision of the main domain into subintervals instabilities that occur in the numerical solution for a single domain analogous to those found by others are attributed to round off errors arising from numerical features of the discrete second derivative matrix operator however accurate stable solutions of fisher s equation are obtained with a multidomain pseudospectral method a detailed comparison of the present approach with the use of the sine interpolation is also carried out

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

review of some adaptive node movement techniques in finite element and finite difference solutions of partial differential equations abstract this article summarizes the results of a literature search for adaptive numerical methods of solving partial differential equations the methods discussed involve the adaptive movement of nodes so as to obtain a low level of solution truncation error while minimizing the number of nodes used in the calculation such methods are applicable to the solution of nonstationary flow problems that contain moving regions of rapid change in the flow variables surrounded by regions of relatively smooth variation flows with shock waves contact surfaces slip streams phase change interfaces and boundary layers can be modelled with great precision by these methods it will be shown that significant economies of execution can be attained if nodes are moved so that they remain concentrated in regions of rapid variation of the flow variables

hermite collocation solution of near singular problems using numerical coordinate transformations based on adaptivity a coordinate transformation approach is described that enables hermite collocation methods to be applied efficiently in one space dimension to steady and unsteady differential problems with steep solutions the work is an extension of earlier work by mulholland et al j comput phys 131 1997 280 a coarse grid is generated by an adaptive finite difference method and this grid is used to construct a steady or unsteady coordinate transformation that is based on monotonic cubic spline approximation an uneven grid is generated by means of the coordinate transformation and the differential problem is solved on this grid using hermite collocation numerical results are presented for steady and unsteady problems

a grid redistribution method for singular problems many physical phenomena develop singular or nearly singular behavior in localized regions e g boundary layers or blowup solutions using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity ren and wang developed a semi static adaptive grid method w ren x p wang an iterative grid redistribution method for singular problems in multiple dimensions j comput phys 159 2000 246 273 for the solution of these problems known as the iterative grid redistribution igr method in this study we develop a theoretical basis for semi static adaptive grid method for singular problems based on this theory we obtain the key result of this study a methodology for designing robust weight functionals which ensures grid resolution in the singular region as well as control of the maximal grid spacing in the outer region using this methodology we introduce a semi static adaptive grid method which does not involve an iterative procedure for grid redistribution as in the igr method we demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude

on a uniformly accurate finite difference approximation of a singulary peturbed reaction diffusion problem using grid equidistribution we examine the convergence properties of a finite difference approximation of a singularly perturbed reaction diffusion boundary value problem using a nonuniform grid the grid is based on the equidistribution of a positive monitor function that is a linear combination of a constant floor and a power of the second derivative of the solution analysis shows how the monitor function can be chosen to ensure that the accuracy of the numerical approximation is insensitive to the size of the singular perturbation parameter the use of equidistribution principles appears in many practical grid adaption schemes and our analysis provides insight into the convergence behaviour on such grids numerical results are given that confirm the uniform convergence rates

a two dimensional time domain near zone to far zone transformation a 2d version of a time domain transformation useful for extrapolating 3d near zone finite difference time domain fdtd results to the far zone is outlined while the 3d transformation produced a physically observable far zone time domain field this is not convenient for the 2d case however a representative 2d far zone time domain result can be obtained directly this result can then be transformed to the frequency domain using a fast fourier transform corrected with a simple multiplicative factor and used for example to calculate the complex wideband scattering width of a target if an actual time domain far zone result is required it can be obtained by inverse fourier transform of the final frequency domain result

computational solution of two dimensional unsteady pdes using moving mesh methods numerical experiments are described which illustrate some important features of the performance of moving mesh methods for solving two dimensional partial differential equations pdes here we are concerned with algorithms based on moving mesh methods proposed by w huang and r d russell siam j sci comput 20 998 1999 we show that the accuracy of the computations is strongly influenced by the choice of monitor function and we present a monitor function which yields a higher rate of convergence than those that are commonly used in an earlier paper g beckett j a mackenzie a ramage and d m sloan j comput phys 167 372 2001 we demonstrated a robust and efficient algorithm for problems in one space dimension in which the mesh equation is decoupled from the physical pde and the time step is controlled automatically the present work extends this algorithm to deal with problems in two space dimensions

a robust moving mesh finite volume method applied to 1d hyperbolic conservation laws from magnetohydrodynamics in this paper we describe a one dimensional adaptive moving mesh method and its application to hyperbolic conservation laws from magnetohydrodynamics mhd the method is robust because it employs automatic control of mesh adaptation when a new model is considered without manually set parameters adaptive meshes are a common tool for increasing the accuracy and reducing computational costs when solving time dependent partial differential equations pdes mesh points are moved towards locations where they are needed the most to obtain a time dependent adaptive mesh monitor functions are used to automatically monitor the importance of the various parts of the domain by assigning a weight value to each location based on the equidistribution principle all mesh points are distributed according to their assigned weights we use a sophisticated monitor function that tracks both small local phenomena as well as large shocks in the same solution the combination of the moving mesh method and a high resolution finite volume solver for hyperbolic pdes yields a serious gain in accuracy at relatively no extra costs the results of several numerical experiments including comparisons with h refinement are presented which cover many intriguing aspects typifying nonlinear magneto fluid dynamics with higher accuracy than often seen in similar publications