adaptive grid generation from harmonic maps on reimannian manifolds abstract in this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on riemannian manifolds the reliability of the method is assured by an existence and uniqueness theorem for one to one maps between multidimensional multiconnected domains we formulate an adaptive riemannian metric consistent with this theorem several examples demonstrating application of the developed procedure are provided

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

focusing blow up for quasilinear parabolic equations we study the behaviour of the non negative blowing up solutions to the quasilinear parabolic equation with a typical reaction diffusion right hand side and with a singularity in the space variable which takes the form where m 1 p 1 are arbitrary constants in the critical exponent case q p 1 m 0 we impose zero dirichlet boundary conditions at the singular point x 0 and at x 1 and take large initial data for a class of concave initial functions we prove focusing at the origin of the solutions as t approaches the blow up time t in the sense that x 0 belongs to the blow up set the proof is based on an application of the intersection comparison method with an explicit separable solution which has the same blow up time as u the method has a natural generalisation to the case of more general nonlinearities in the equation a description of different fine structures of blow up patterns in the semilinear case m 1 and in the quasilinear one m 1 is also presented a numerical study of the semilinear equation is also made using an adaptive collocation method this is shown to give very close agreement with the fine structure predicted and allows us to make some conjectures about the general behaviour

on error estimates of the penalty method for unsteady navier stokes equations the penalty method has been widely used for numerical computations of the unsteady navier stokes equations however the best error estimates available to the author s knowledge were not optimal an

numerical solution of fisher s equation using a moving mesh method the paper investigates the viability of using moving mesh methods to simulate travelling wave solutions of fisher s equation results are presented that illustrate the weaknesses in moving mesh methods based on equidistribution of some popular monitor functions it is shown that knowledge of the differential equation and the travelling wave solution may be used to construct a monitor function that yields accurate results with suitably chosen moving mesh methods a comparison is made between a moving mesh partial differential equation and a moving mesh differential algebraic equation for the evolution in time

a moving finite element method with error estimation and refinement for one dimensional time dependent partial differential equations we discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension the mesh is moved so as to equidistribute the spatial component of the discretization error in h 1 we present a method of estimating this error by using p hierarchic finite elements the error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement we discretize the partial differential equations in space using a galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error a system of ordinary differential equations for mesh velocities are used to control element motions we use existing software for stiff ordinary differential equations for the temporal integration of the solution the error estimate and the mesh motion computational results using a code based on our method are presented for several examples

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

phase field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing this paper describes a novel numerical algorithm for simulating interfacial dynamics of non newtonian fluids the interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously the property and evolution of the interfacial layer is governed by a phase field variable that obeys a cahn hilliard type of convection diffusion equation this circumvents the task of directly tracking the interface and produces the correct interfacial tension from the free energy stored in the mixing layer viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase field framework the greatest challenge of this approach is in resolving the sharp gradients at the interface this is achieved by using an efficient adaptive meshing scheme governed by the phase field variable the finite element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large scale two phase systems of complex fluids in two dimensional and axisymmetric three dimensional implementations the numerical toolkit is applied here to drop deformation in shear and elongational flows rise of drops and retraction of drops and torii some of these solutions serve as validation of the method and illustrate its key features while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces

an asymptotic and numerical description of self similar blow up in quasilinear parabolic equations we study the blow up behaviour of two reaction diffusion problems with a quasilinear degenerate diffusion and a superlinear reaction we show that in each case the blow up is self similar in contrast to the linear diffusion limit of each in which the diffusion is only approximately self similar we then investigate the limit of the self similar behaviour and describe the transition from a stable manifold blow up behaviour for quasilinear diffusion to a centre manifold one for the linear diffusion

on the unnormalized solution of the filtering problem with counting process observations a general procedure to solve filtering problems for counting process observations is discussed linear nonstochastic integro differential equations describe the evolution of unnormalized conditional distribution of the state process between observation jump times while at jump times a linear updating is required final normalization is the only nonlinear operation to be implemented quite general situations may be accommodated in the present setup the state can be virtually any markov semimartingale the observation process may affect the dynamics of the state and vice versa and there is complete freedom in correlating state and observation martingale terms

a study of moving mesh methods applied to a thin flame propagating in a detonator delay element we study the application of moving mesh methods to a one dimensional time dependent detonator delay element problem we consider moving mesh methods based on the equidistribution principle derived by huang et al 1 adaptive mesh methods have been widely used recently to solve time dependent partial differential equations having large solution gradients significant improvements in accuracy and efficiency are achieved by adapting the nodes mesh points so that they are concentrated about areas of large solution variations each system of equations for the moving mesh methods is solved in conjunction with the detonator problem in this paper the system of ordinary differential equations that results after discretising in space is solved using the double precision version of the stiff ordinary differential equation solver dassl the numerical results clearly demonstrate that the moving mesh methods are capable of tracking the deflagration wave as it travels down the detonator delay element more accurately and more efficiently than a fixed mesh method

on the robustness of linear continuous time dynamic systems in this note for a given linear continuous time dynamic system dot x ax bu we determine sufficient conditions for the existence of a constant feedback matrix k such that the closed loop system matrix a bk admits a diagonal positive solution of the lyapunov equation an algorithm for the determination of such k matrix is given and some examples are solved

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

a numerical investigation of blow up in reaction diffusion problems with traveling heat sources this paper studies the numerical solution of a reaction diffusion differential equation with traveling heat sources according to the fact that the locations of heat sources are known we add auxiliary mesh points exactly at heat sources and present a novel moving mesh algorithm for solving the problem several examples are provided to demonstrate the efficiency of the new moving mesh method especially in the case of two or three traveling heat sources moreover numerical results illustrate that the speed of the movement of the heat source is critical for blow up when there is only one traveling heat source for the case of two traveling heat sources blow up depends not only on the speed but also on the distance between the two traveling heat sources

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

linear and unconditionally energy stable schemes for the binary fluid surfactant phase field model abstract in this paper we consider the numerical solution of a binary fluid surfactant phase field model in which the free energy contains a nonlinear coupling entropy a ginzburg landau double well potential and a logarithmic flory huggins potential the resulting system consists of two nonlinearly coupled cahn hilliard type equations we develop a first and a second order time stepping schemes for this system using the invariant energy quadratization approach in particular the system is transformed into an equivalent one by introducing appropriate auxiliary variables and all nonlinear terms are then treated semi explicitly both schemes are linear and lead to symmetric positive definite systems in space at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable in the discrete sense various 2d and 3d numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

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

a geometric discretization and a simple implementation for variational mesh generation and adaptation we present a simple direct discretization for functionals used in the variational mesh generation and adaptation meshing functionals are discretized on simplicial meshes and the jacobian matrix of the continuous coordinate transformation is approximated by the jacobian matrices of affine mappings between elements the advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional moreover the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form which allows a simple implementation of variational mesh generation and adaptation on computer since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods the result in this work can be used to develop simple implementations of those methods numerical examples are given

stability of newton tvd runge kutta scheme for one dimensional euler equations with adaptive mesh abstract in this paper we propose a moving mesh method with a newton total variation diminishing tvd runge kutta scheme for the euler equations our scheme improves time discretization in the moving mesh algorithms by analyzing the semi discrete euler equations with the discrete moving mesh equations as constraints the stability of the newton tvd runge kutta scheme is proved thus we can conclude that the proposed algorithm can generate a weak solution to the euler equations finally numerical examples are presented to verify the theoretical results and demonstrate the accuracy of the proposed scheme

an adaptive grid with directional control abstract a variational formulation of winslow s variable diffusion method is combined with a directional control functional to derive an adaptive grid generator for adaptive node movement the addition of directional control allows one to align mesh lines with a prescribed vector field while adapting the spacing to resolve the data it is shown that harmonic function theory can be applied to define the conditions under which unique solutions of the resulting elliptic equations exist the grid generator is applied to the complex problem of magnetic field reconnection driven by fluid instability the results demonstrate that the cost of a computation with an adaptive grid with directional control is only one tenth of the cost of either a calculation on an eulerian grid or on an adaptive grid without directional control

a numerical study of three moving grid methods for one dimensional partial differential equations which are based on the method of lines in recent years several sophisticated packages based on the method of lines mol have been developed for the automatic numerical integration of time dependent problems in partial differential equations pdes notably for problems in one space dimension these packages greatly benefit from the very successful developments of automatic stiff ordinary differential equation solvers however from the pde point of view they integrate only in a semiautomatic way in the sense that they automatically adjust the time step sizes but use just a fixed space grid chosen a priori for the entire calculation for solutions possessing sharp spatial transitions that move e g travelling wave fronts or emerging boundary and interior layers a grid held fixed for the entire calculation is computationally inefficient since for a good solution this grid often must contain a very large number of nodes in such cases methods which attempt automatically to adjust the sizes of both the space and the time steps are likely to be more successful in efficiently resolving critical regions of high spatial and temporal activity methods and codes that operate this way belong to the realm of adaptive or moving grid methods following the mol approach this paper is devoted to an evaluation and comparison mainly based on extensive numerical tests of three moving grid methods for 1d problems viz the finite element method of miller and co workers the method published by petzold and a method based on ideas adopted from dorii and drury our examination of these three methods is aimed at assessing which is the most suitable from the point of view of retaining the acknowledged features of reliability robustness and efficiency of the conventional mol approach therefore considerable attention is paid to the temporal performance of the methods

decoupled energy stable schemes for phase field vesicle membrane model we consider the numerical approximations of the classical phase field vesicle membrane models proposed a decade ago in du et al 2004 6 we first reformulate the model derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish the energy dissipation law then we develop a stabilized decoupled time discretization scheme for the coupled nonlinear system the scheme is unconditionally energy stable and leads to linear and decoupled elliptic equations to be solved at each time step stability analysis and ample numerical simulations are presented thereafter

implementation of arbitrary inner product in the global galerkin method for incompressible navier stokes equations the global galerkin or weighted residuals method applied to the incompressible navier stokes equations is considered the basis functions are assumed to be divergence free and satisfy all the boundary conditions the method is formulated for an arbitrary inner product so that the pressure cannot be eliminated by galerkin projections on a divergence free basis a proposed straightforward procedure for the elimination of the pressure reduces the problem to an ode system without algebraic constraints to illustrate the applicability and the robustness of the numerical approach and to show that numerical solutions with unit and non unit weight functions yield similar results the driving lid cavity and natural convection benchmark problems are solved using the unit and chebyshev weight functions further implications of the proposed galerkin formulation are discussed

exponential splitting time integration for pseudospectral methods on moving meshes moving meshes are successfully used in many fields here we investigate how a recently proposed approach to combine the strang splitting method for time integration with pseudospectral spatial discretization by orthogonal polynomials can be extended to include moving meshes a double representation of a function by coefficients of polynomial expansion and by values at the mesh nodes associated with a suitable quadrature formula is an essential part of the numerical integration before numerical implementation the original pde is transformed into a suitable form the approach is illustrated on the linear heat transfer equation

simple adaptive grids for 1 d initial value problems abstract in solving 1 d initial value problems by implicit finite difference methods it is advantageous to use an adaptive grid to provide local regions of high resolution and allow larger time steps to be taken a simple method of constructing such grids is presented and the benefits are demonstrated by calculations of sod s shock tube problem g a sod j comput phys 27 1 1978 and of a supernova explosion

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

optimality in accelerated life tests we formulate a stochastic control problem arising from the optimal design of accelerated life tests the model is obtained in a natural way in the setup of point processes the cost functional depends on the total number of items under test at time t the observed number of items failed up to time t and the stress level applied to the items under test at time t the optimal policy minimizing the cost functional is characterized via the dynamic programming equation after a further specification of the model we prove that the solution is of bang bang type in some specific cases we also prove that the control problem is equivalent to an optimal stopping problem this amounts to simply running the test with the number of items and the stress level at their highest available value up to the exit time of the state process from a given region in this special case we exhibit an explicit solution and give a criterion in order to decide whether to let the experiment run or to stop it

high order and mass conservative methods for the conservative allen cahn equation the conservative allen cahn ac equation has been studied analytically and numerically our mathematical analysis and numerical experiment however show that previous numerical methods are not second order accurate in time and or do not conserve the initial mass the aim of this paper is to propose high order and mass conservative methods for solving the conservative ac equation in the methods we discretize the conservative ac equation by using a fourier spectral method in space and first second and third order implicit explicit runge kutta schemes in time we show that the methods inherit the mass conservation numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods

a time stepping scheme involving constant coefficient matrices for phase field simulations of two phase incompressible flows with large density ratios we present an efficient time stepping scheme for simulations of the coupled navier stokes cahn hilliard equations for the phase field approach the scheme has several attractive characteristics i it is suitable for large density ratios and numerical experiments with density ratios up to 1000 have been presented ii it involves only constant time independent coefficient matrices for all flow variables which can be pre computed during pre processing so it effectively overcomes the performance bottleneck induced by variable coefficient matrices associated with the variable density and variable viscosity iii it completely de couples the computations of the velocity pressure and the phase field function strategy for spectral element type spatial discretizations to overcome the difficulty associated with the large spatial order of the cahn hilliard equation is also discussed ample numerical simulations demonstrate that the current algorithm together with the navier stokes cahn hilliard phase field approach is an efficient and effective method for studying two phase flows involving large density ratios moving contact lines and interfacial topology changes

adaptive residual subsampling methods for radial basis function interpolation and collocation problems we construct a new adaptive algorithm for radial basis functions rbfs method applied to interpolation boundary value and initial boundary value problems with localized features nodes can be added and removed based on residuals evaluated at a finer point set we also adapt the shape parameters of rbfs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix the performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains

numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh a finite difference method using a nonuniform triangle mesh is described for the numerical solution of the nonlinear two dimensional poisson equation s 0 where is a function of or its derivatives sis a function of position and or its normal derivative is specified on the boundary the finite difference equations are solved by successive overrelaxation the triangle mesh which is constructed numerically by solving laplace s equation is easily adapted to nonrectangular boundaries and interfaces examples of numerical results are given for the magnetostatic problem with iron and other possible applications are mentioned

on sequential construction of solutions of stochastic differential equations with jump terms on etudie l existence et l unicite des solutions d une equation differentielle stochastique sur un espace de probabilite avec filtration continue a droite

multiphase image segmentation using a phase field model in this paper we propose a new fast and stable hybrid numerical method for multiphase image segmentation using a phase field model the proposed model is based on the allen cahn equation with a multiple well potential and a data fitting term the model is computationally superior to the previous multiphase image segmentation via modica mortola phase transition and a fitting term we split its numerical solution algorithm into linear and a nonlinear equations the linear equation is discretized using an implicit scheme and the resulting discrete system of equations is solved by a fast numerical method such as a multigrid method the nonlinear equation is solved analytically due to the availability of a closed form solution we also propose an initialization algorithm based on the target objects for the fast image segmentation finally various numerical experiments on real and synthetic images with noises are presented to demonstrate the efficiency and robustness of the proposed model and the numerical method

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

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

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

moving mesh methods with locally varying time steps the time steps associated with moving mesh methods are proportional to the smallest mesh size in space and as a result they are very small at each time level for some practical problems the physical phenomena develop dynamically singular or nearly singular solutions in fairly localized regions and therefore the smallest time step at each time level occurs only in these localized regions in this work we will develop a local time stepping algorithm for the moving mesh methods the principal idea will be demonstrated by investigating the nonlinear hyperbolic conservation laws numerical experiments are carried out to demonstrate the efficiency and robustness of the proposed methods

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

on the robustness of optimal regulators for nonlinear discrete time systems in this paper the robustness of nonlinear discrete time systems is analyzed the nominal plant is supposed to be controlled by means of a feedback control law which is optimal with respect to some given criterion the robustness of the closed loop system is studied for two different classes of perturbations in the control law which are called gain and additive nonlinear perturbations the results are entirely based on the existence of a stationary solution of the dynamic programming equation dpe which provides directly a lyapunov function associated to the closed loop system the convexity of that solution and the use of the taylor formula appear to be the key to establish the robustness properties of the nominal plant two examples are solved in order to show an interesting fact the existence of a compromise between the robustness of the system subjected to the two different classes of perturbations

on imposing dynamic contact angle boundary conditions for wall bounded liquid gas flows we present an efficient scheme within the phase field framework for imposing dynamic contact angle boundary conditions for wall bounded flows of two immiscible incompressible fluids with large density ratios first we develop an algorithm for imposing the dynamic contact angle boundary conditions to the cahn hilliard equation our algorithm consists of two components i we ignore the boundary conditions and transform the cahn hilliard equation into two nominally de coupled helmholtz type equations ii we treat the dynamic contact angle boundary conditions in such a manner that the two helmholtz type equations are truly de coupled then we combine this algorithm together with a scheme for variable density navier stokes equations we developed recently to form an efficient method for the coupled system of navier stokes and cahn hilliard equations for contact line problems involving large density ratios the overall method can deal with moving contact lines under dynamic and also static contact angle boundary conditions it is endowed with several attractive features that make the method very efficient in particular computations for all flow variables are completely decoupled the resultant linear algebraic systems after discretization for all flow variables involve only constant and time independent coefficient matrices which can be pre computed during pre processing even though the coupled navier stokes cahn hilliard system involves variable density and variable viscosity ample numerical simulations of wall bounded air water two phase flows have been presented to demonstrate the capability of the method for dealing with contact line problems under dynamic and static contact angle conditions involving large density ratios

robustness of nonlinear state feedback a survey a survey of robustness of nonlinear state feedback is given for series perturbations there are fairly complete results showing that under mild restrictions an optimal controller can tolerate an infinite increase in gain for gain reductions there are some results for systems linear in the control in particular there is a 50 reduction tolerance if the control penalty is quadratic usually the optimal controller cannot be computed exactly there are some results showing the effects of truncation on the robustness essentially robustness is maintained but in a reduced computable part of the state space

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

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

unique characterization of conditional distributions in nonlinear filtering on definit un probleme de martingale filtree pour un generateur a et on montre etant donne des hypotheses appropriees sur a que la distribution conditionnelle est la solution unique au probleme

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

an approximation method for controlled discrete jump processes under partial observations the paper is concerned with control and filtering of a discrete jump markov process when only the total number of jumps is observed a partially observable control problem is considered and equivalence between this problem and the separated one is proved a technique to approximate the value function under some suitable assumptions is presented

optimal rate convergence analysis of a second order scheme for a thin film model with slope selection abstract in this paper an energy stable second order mixed finite element scheme is proposed and analyzed for the thin film epitaxial growth model with slope selection we employ second order backward differentiation formula bdf scheme with a second order stabilized term which guarantees the long time energy stability to approximate the continuous model in terms of the convergence analysis the key difficulty to derive an optimal rate spatial estimate is associated with the appearance of the gradient operator in the nonlinear terms which may lead to a loss of optimal accuracy order to overcome this well known difficulty we make use of some auxiliary techniques over triangular elements and obtain an optimal convergence rate o h q 1 t 2 in comparison with o h q t 2 rate from a standard projection estimate furthermore we use an efficient preconditioned steepest descent psd solver for the numerical implementation a few numerical examples are presented to validate the stability and convergence

numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method abstract the molecular beam epitaxial model is derived from the variation of a free energy that consists of either a fourth order ginzburg landau double well potential or a nonlinear logarithmic potential in terms of the gradient of a height function one challenge in solving the mbe model numerically is how to develop proper temporal discretization for the nonlinear terms in order to preserve energy stability at the time discrete level in this paper we resolve this issue by developing a first and second order time stepping scheme based on the invariant energy quadratization ieq method the novelty is that all nonlinear terms are treated semi explicitly and the resulted semi discrete equations form a linear system at each time step moreover the linear operator is symmetric positive definite and thus can be solved efficiently we then prove that all proposed schemes are unconditionally energy stable the semi discrete schemes are further discretized in space using finite difference methods and implemented on gpus for high performance computing various 2d and 3d numerical examples are presented to demonstrate stability and accuracy of the proposed schemes

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

an efficient moving mesh spectral method for the phase field model of two phase flows we develop in this paper a moving mesh spectral method for the phase field model of two phase flows with non periodic boundary conditions the method is based on a variational moving mesh pde for the phase function coupled with efficient semi implicit treatments for advancing the mesh function the phase function and the velocity and pressure in a decoupled manner ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method

variational mesh adaptation isotropy and equidistribution we present a new approach for developing more robust and error oriented mesh adaptation methods specifically assuming that a regular in cell shape and uniform in cell size computational mesh is used as is commonly done in computation we develop a criterion for mesh adaptation based on an error function whose definition is motivated by the analysis of function variation and local error behavior for linear interpolation the criterion is then decomposed into two aspects the isotropy or conformity and uniformity or equidistribution requirements each of which can be easier to deal with the functionals that satisfy these conditions approximately are constructed using discrete and continuous inequalities a new functional is finally formulated by combining the functionals corresponding to the isotropy and uniformity requirements the features of the functional are analyzed and demonstrated by numerical results in particular unlike the existing mesh adaptation functionals the new functional has clear geometric meanings of minimization a mesh that has the desired properties of isotropy and equidistribution can be obtained by properly choosing the values of two parameters the analysis presented in this article also provides a better understanding of the increasingly popular method of harmonic mapping in two dimensions

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

an adaptive moving mesh method for thin film flow equations with surface tension we present an adaptive moving mesh method for the numerical solution of thin liquid film spreading flows with surface tension we follow the r adaptive moving mesh technique which utilises a mesh density function and moving mesh partial differential equations mmpdes to adapt and move the mesh coupled to the pde s describing the thin film flow problem numerical experiments are performed on two one dimensional thin film flow equations to test the accuracy and efficiency of the method this technique accurately resolves the multiple one dimensional structures observed in these test problems moreover it reduces the computational effort in comparison to the numerical solution using the finite difference scheme on a fixed uniform mesh we are the first to implement r adaptive schemes to higher order parabolic pdes this method accurately resolves the solution and reduces the computational effort we introduce a new mesh density function that resolves multiple solution structures this mesh density function is also adapted to include multiple solution components

efficient and accurate numerical schemes for a hydro dynamically coupled phase field diblock copolymer model in this paper we consider numerical approximations of a hydro dynamically coupled phase field diblock copolymer model in which the free energy contains a kinetic potential a gradient entropy a ginzburglandau double well potential and a long range nonlocal type potential we develop a set of second order time marching schemes for this system using the invariant energy quadratization approach for the double well potential the projection method for the navierstokes equation and a subtle implicit explicit treatment for the stress and convective term the resulting schemes are linear and lead to symmetric positive definite systems at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable various numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

capturing material interfaces capturing material interfaces using the maccormack difference scheme with flux corrected transport over both lagrangian and eulerian grids without complicated transport formulations are investigated several simple techniques of damping spurious oscillations resulting from capturing material interfaces using eulerian grids are applied locally formulations based on actual physical conditions are found to be superior to artificial dissipative mechanisms

linear second order and unconditionally energy stable schemes for the viscous cahn hilliard equation with hyperbolic relaxation using the invariant energy quadratization method abstract in this paper we consider numerical approximations for the viscous cahn hilliard equation with hyperbolic relaxation this type of equations processes energy dissipative structure the main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms allowing large time marching step while preserving the energy stability i e the energy dissipative structure at the time discrete level we resolve this issue by developing two second order time marching schemes using the recently developed invariant energy quadratization approach where all nonlinear terms are discretized semi explicitly in each time step one only needs to solve a symmetric positive definite spd linear system all the proposed schemes are rigorously proven to be unconditionally energy stable and the second order convergence in time has been verified by time step refinement tests numerically various 2d and 3d numerical simulations are presented to demonstrate the stability accuracy and efficiency of the proposed schemes

a robust moving mesh method for spectral collocation solutions of time dependent partial differential equations this work extends the machinery of the moving mesh partial differential equation mmpde method to the spectral collocation discretization of time dependent partial differential equations unlike previous approaches which bootstrap the moving grid from a lower order finite difference discretization this work uses a consistent spectral collocation discretization for both the grid movement problem and the underlying physical partial differential equation additionally this work develops an error monitor function based on filtering in the spectral domain which concentrates grid points in areas of locally poor resolution without relying on an assumption of locally steep gradients this makes the mmpde method more robust in the presence of rarefaction waves which feature rapid change in higher order derivatives

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

a phase field method for shape optimization of incompressible flows abstract in this paper we present a phase field method applied to the fluid based shape optimization the fluid flow is governed by the incompressible navier stokes equations a phase field variable is used to represent material distributions and the optimized shape of the fluid is obtained by minimizing the certain objective functional regularized the shape sensitivity analysis is presented in terms of phase field variable which is the main contribution of this paper it saves considerable amount of computational expense when the meshes are locally refined near the interfaces compared to the case of fixed meshes numerical results on some benchmark problems are reported and it is shown that the phase field approach for fluid shape optimization is efficient and robust

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

quasi variational inequalities and first order hamilton jacobi equations on considere des inegalites quasi variationnelles de hamilton jacobi dans r n on introduit pour ce probleme la notion de solution viscosite

optimal mass transport based adaptive mesh method for phase field models of two phase fluid flows in this paper we present an efficient adaptive mesh method for solving phase field model of a mixture of two phase incompressible fluid flows the adaptive mesh is generated by a coordinate transformation that is determined as the solution of the classical problem of the optimal mass transportation also known as mongekantorovich problem mkp the numerical solution of the mkp is computed by taking the gradient of the steady state solution of a parabolic mongeampre equation pmae several numerical experiments are conducted to demonstrate the performance of the pmae method for solving phase field models the numerical results illustrate that the adaptive mesh computed with pmae method captures well the moving interface of the phase field function

on the stability of mesh equidistribution strategies for time dependent partial differential equations abstract we study the stability of several mesh equidistribution schemes for time dependent partial differential equations in one space dimension the schemes move a finite difference or finite element mesh so that a given quantity is uniform over the domain we consider mesh moving methods that are based on solving a system of ordinary differential equations for the mesh velocities and show that some of these methods are unstable with respect to an equidistributing mesh when the partial differential system is dissipative using linear perturbation techniques we are able to develop simple criteria for determining the stability of a particular method and show how to construct stable differential systems for the mesh velocities several examples illustrating stable and unstable mesh motions are presented

an implementation of a moving finite element method abstract the moving finite element method an adaptive gridding procedure for systems of partial differential equations whose solutions contain steep gradients has been implemented in a very straightforward way accurate results can be obtained using fewer nodes than that required by fixed mesh solution methods performance of the method using some minor changes to a standard ordinary differential equation algorithm is illustrated with solutions to burger s equation

numerical approximations for a phase field moving contact line model with variable densities and viscosities we consider the numerical approximations of a two phase hydrodynamics coupled phase field model that incorporates the variable densities viscosities and moving contact line boundary conditions the model is a nonlinear coupled system that consists of incompressible navierstokes equations with the generalized navier boundary condition and the cahnhilliard equations with moving contact line boundary conditions by some subtle explicitimplicit treatments to nonlinear terms we develop two efficient unconditionally energy stable numerical schemes in particular a linear decoupled energy stable scheme for the system with static contact line condition and a nonlinear energy stable scheme for the system with dynamic contact line condition an efficient spectral galerkin spatial discretization is implemented to verify the accuracy and efficiency of proposed schemes various numerical results show that the proposed schemes are efficient and accurate

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

the geometric integration of scale invariant ordinary and partial differential equations this review paper examines a synthesis of adaptive mesh methods with the use of symmetry to solve ordinary and partial differential equations it looks at the effectiveness of numerical methods in preserving geometric structures of the underlying equations such as scaling invariance conservation laws and solution orderings studies are made of a series of examples including the porous medium equation and the nonlinear schrodinger equation

a study of monitor functions for two dimensional adaptive mesh generation in this paper we study the problem of two dimensional adaptive mesh generation using a variational approach and specifically the effect that the monitor function has on the resulting mesh behavior the basic theoretical tools employed are green s function for elliptic problems and the eigendecomposition of symmetric positive definite matrices based upon this study a general strategy is suggested for how to choose the monitor function and numerical results are presented for illustrative purposes the three dimensional case is also briefly discussed it is noted that the strategy used here can be applied to other elliptic mesh generation techniques as well

on the gain margin of nonlinear and optimal regulators the robustness of nonlinear regulators for nonlinear systems with respect to variations in gain is investigated it is shown that there exist regulators that produce asymptotically stable closed loop systems but do not tolerate any variation in gain without instability however if the linearized closed loop system is also asymptotically stable then there is always some gain margin for a wide class of optimal regulators it is shown that the gain margin is infinite with respect to increases in gain and that decreases down to 0 5 can be tolerated the robustness properties of linear quadratic control laws are thus generalized

a quantitative comparison between c0 and c1 elements for solving the cahn hilliard equation the cahn hilliard ch equation is a time dependent fourth order partial differential equation pde when solving the ch equation via the finite element method fem the domain is discretized by c 1 continuous basis functions or the equation is split into a pair of second order pdes and discretized via c 0 continuous basis functions in the current work a quantitative comparison between c 1 hermite and c 0 lagrange elements is carried out using a continuous galerkin fem formulation the different discretizations are evaluated using the method of manufactured solutions solved with newton s method and jacobian free newton krylov it is found that the use of linear lagrange elements provides the fastest computation time for a given number of elements while the use of cubic hermite elements provides the lowest error the results offer a set of benchmarks to consider when choosing basis functions to solve the ch equation in addition an example of microstructure evolution demonstrates the different types of elements for a traditional phase field model

on the robustness of nonlinear regulators and its application to nonlinear systems stabilization in this note we develop a new robustness property of optimal regulators of nonlinear dynamic systems the stability condition we establish presents a great advantage when compared to other stability conditions already available in the literature it does not depend explicitly on the optimal control problem solution allowing then a very simple and a priori analysis of the closed loop system stability based on that we propose a methodology for determining linear decentralized controllers which stabilize asymptotically a wide class of nonlinear dynamic systems as an application example we present and discuss in detail the control design for a two pendulum system 4