approximation of cahn hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with c1 elements a variational formulation and c1 finite element scheme with adaptive mesh refinement and coarsening are developed for phase separation processes described by the cahn hilliard diffuse interface model of transport in a mixture or alloy the adaptive scheme is guided by a laplacian jump indicator based on the corresponding term arising from the weak formulation of the fourth order non linear problem and is implemented in a parallel solution framework it is then applied to resolve complex evolving interfacial solution behavior for 2d and 3d simulations of the classic spinodal decomposition problem from a random initial mixture and to other phase transformation applications of interest simulation results and adaptive performance are discussed the scheme permits efficient robust multiscale resolution and interface characterization copyright 2008 john wiley sons ltd

optimal multinational capital budgeting under uncertainty this paper discusses the multinational capital budgeting problem when there are some candidate foreign projects which project s should the investor choose in the paper special cash flows and value sources of foreign projects are introduced regarding project parameters such as construction costs annual net operating cash flows terminal values of the projects as well as the foreign exchange rates as uncertain variables the paper proposes one new uncertain zero one integer model for optimal multinational project selection to solve the problem a hybrid intelligent algorithm integrating the 99 methods and genetic algorithm is provided as an illustration an application example is also presented

spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

two phase fluid simulation using a diffuse interface model with peng robinson equation of state in this paper two phase fluid systems are simulated using a diffusive interface model with the peng robinson equation of state eos a widely used realistic eos for hydrocarbon fluid in the petroleum industry we first utilize the gradient theory of thermodynamics and variational calculus to derive a generalized chemical equilibrium equation which is mathematically a second order elliptic partial differential equation pde in molar density with a strongly nonlinear source term to solve this pde we convert it to a time dependent parabolic pde with the main interest in its final steady state solution a lagrange multiplier is used to enforce mass conservation the parabolic pde is then solved by mixed finite element methods with a semi implicit time marching scheme convex splitting of the energy functional is proposed to construct this time marching scheme where the volume exclusion effect of an eos is treated implicitly while the pairwise attraction effect of eos is calculated explicitly this sch

an adaptive finite element method for simulating surface tension with the gradient theory of fluid interfaces the gradient theory for the surface tension of simple fluids and mixtures is rigorously analyzed based on mathematical theory the finite element approximation of surface tension is developed and analyzed and moreover an adaptive finite element method based on a physical based estimator is proposed and it can be coupled efficiently with newton s method as well the numerical tests are carried out both to verify the proposed theory and to demonstrate the efficiency of the proposed method

linear programming with fuzzy variables in practice there are many problems in which all decision parameters are fuzzy numbers and such problems are usually solved by either possibilistic programming or multiobjective programming methods unfortunately all these methods have shortcomings in this note using the concept of comparison of fuzzy numbers we introduce a very effective method for solving these problems then we propose a new method for solving linear programming problems with fuzzy variables

redundancy optimization problems with uncertainty of combining randomness and fuzziness by using random fuzzy lifetimes as basic parameters three types of system performance expected system lifetime system lifetime and system reliability are presented some random fuzzy simulations are employed to estimate the system performance furthermore a spectrum of random fuzzy models are constructed for redundancy optimization problems in order to solve these models a hybrid intelligent algorithm is used finally some numerical examples are provided for the sake of illustration

solving the regularized strongly anisotropic cahn hilliard equation by an adaptive nonlinear multigrid method abstract we present efficient second order accurate and adaptive finite difference methods to solve the regularized strongly anisotropic cahn hilliard equation in 2d and 3d when the surface energy anisotropy is sufficiently strong there are missing orientations in the equilibrium level curves of the diffuse interface solutions corresponding to those missing from the sharp interface wulff shape and the anisotropic cahn hilliard equation becomes ill posed to regularize the equation a higher order derivative term is added to the energy this leads to a sixth order nonlinear parabolic equation for the order parameter an implicit time discretization is used to remove the high order time step stability constraints dynamic block structured cartesian mesh refinement is used to highly resolve narrow interfacial layers a multilevel nonlinear multigrid method is used to solve the nonlinear equations at the implicit time level one of the keys to the success of the method is the treatment of the anisotropic term this term is discretized in conservation form in space and is discretized fully implicitly in time numerical simulations are presented that confirm the accuracy efficiency and stability of the scheme we study the dynamics of interfaces under strong anisotropy and compare near equilibrium diffuse interface solutions to the sharp interface wulff shapes in 2d and 3d we also simulate large scale coarsening of a corrugated surface in 3d evolving by anisotropic surface diffusion we show the emergence of long range order during coarsening and an interesting mechanism of ordered coarsening

an integrated decision support approach for project investors in risky and uncertain environments in project investment decisions it is often assumed that estimated values of project parameters are certain and they would not deviate by the time however project parameters normally change during a life cycle of the project therefore an existence of a deviation or gap between forecasted values and actual values is inevitable because of the uncertainty of the future forecasting the true and exact values of project parameters is almost impossible in this study an integrated decision support approach based on simulation and fuzzy set theory is proposed for project investors in risky and uncertain environments this approach determines the risk levels of the projects and helps investors to make investment decisions in the scope of the study a flowchart is presented to guide to decision maker in different situations of information uncertainty that belongs to project parameter values via this flowchart the values of project parameters can be chosen depending on how they are determined deterministic stochastic or fuzzy by project analyst besides calculating and analyzing the project risk in all possible situations would be easier illustrative examples are given to demonstrate the application of this approach

a second order accurate non linear difference scheme for the n component cahn hilliard system a b s t r a c t we consider a second order conservative nonlinear numerical scheme for the ncomponent cahn hilliard system modeling the phase separation of a n component mixture the scheme is based on a crank nicolson finite difference method and is solved by an efficient and accurate nonlinear multigrid method we numerically demonstrate the second order accuracy of the numerical scheme we observe that our numerical solutions are consistent with the exact solutions of linear stability analysis results we also describe numerical experiments such as the evolution of triple junctions and the spinodal decomposition in a quaternary mixture we investigate the effects of a concentration dependent mobility on phase separation

second order convex splitting schemes for gradient flows with ehrlich schwoebel type energy application to thin film epitaxy we construct unconditionally stable unconditionally uniquely solvable and second order accurate in time schemes for gradient flows with energy of the form int omega f nabla phi bf x frac epsilon 2 2 delta phi bf x 2 d bf x the construction of the schemes involves the appropriate combination and extension of two classical ideas i appropriate convex concave decomposition of the energy functional and ii the secant method as an application we derive schemes for epitaxial growth models with slope selection f bf y frac14 bf y 2 1 2 or without slope selection f bf y frac12 ln 1 bf y 2 two types of unconditionally stable uniquely solvable second order schemes are presented the first type inherits the variational structure of the original continuous in time gradient flow while the second type does not preserve the variational structure we present numerical simulations for the case with slope selection which verify well known physical scaling laws for the long time coarsening process

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

expected value of fuzzy variable and fuzzy expected value models this paper will present a novel concept of expected values of fuzzy variables which is essentially a type of choquet integral and coincides with that of random variables in order to calculate the expected value of general fuzzy variable a fuzzy simulation technique is also designed finally we construct a spectrum of fuzzy expected value models and integrate fuzzy simulation neural network and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models

some inequalities of random fuzzy variableswith application to moment convergence random fuzzy variables are functions from possibility spaces to the set of random variables in this paper in order further to discuss the mathematical properties of random fuzzy variables some inequalities for random fuzzy variables are first proved based on the concepts of chance measures and expected value operators finally as applications of inequalities obtained moment convergence theorems are proved

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

chance constrained programming models for capital budgeting with npv as fuzzy parameters in an uncertain economic environment experts knowledge about outlays and cash inflows of available projects consists of much vagueness instead of randomness investment outlays and annual net cash flows of a project are usually predicted by using experts knowledge fuzzy variables can overcome the difficulties in predicting these parameters in this paper capital budgeting problem with fuzzy investment outlays and fuzzy annual net cash flows is studied based on credibility measure net present value npv method is employed and two fuzzy chance constrained programming models for capital budgeting problem are provided a fuzzy simulation based genetic algorithm is provided for solving the proposed model problems two numerical examples are also presented to illustrate the modelling idea and the effectiveness of the proposed algorithm

systems of cahn hilliard equations the phase separation of alloys with two or more components is studied with emphasis on more than two components particular attention is given to differences between multicomponent and binary alloys specific topics of the paper include equilibrium theory aspects of the dynamics and numerical simulations in the equilibrium theory it is found that there is an enriched equilibrium structure that allows for multiple coexisting phases and the presence of triple points in the solution dynamic results include the characterization of the spinodal region and of the concentration variations that lead to spinodal decomposition unlike the binary theory not all composition variations lead to separation and the compositions are not restricted to the convex hull of the equilibrium concentrations linear analysis is used to predict that a pseudo binary will initially result from spinodal decomposition numerical simulations of the dynamics for a ternary alloy verify this initially but more than two phases often

diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix a model for phase separation in multicomponent systems is studied in particular the possibility of a concentration dependence of the mobility matrix is taken into account this leads to a system of fourth order degenerate parabolic partial differential equations we derive these equations from balance laws show some properties of the model and prove a global existence result for the degenerate system

transportation projects selection process using fuzzy sets theory government transportation agencies are faced with the problem of efficiently selecting a subset of transportation projects for implementation this selection process is based on multiple objectives which are often measured in incommensurable units usually the problem is treated by neglecting or biasing the qualitative characteristics of the various projects moreover the usual selection methods cannot deal effectively with the decision makers preferences or vagueness fuzzy sets theory is able to cope with inexact information and therefore is believed to be an appropriate tool for use in the projects selection process this work presents an efficient technique for the selection of transportation projects using fuzzy sets theory the selection procedure is a multiple objectives process and projects are rated both on a quantitative and qualitative basis using linguistic variables in order to describe appropriately a given transportation policy both fuzzy weighted average and noncompensatory fuzzy decision rules are used in the proposed approach in addition this work contains a case study of a selection process of interurban road projects in israel the results of the proposed method obtained by a fuzzy expert system are compared with the results obtained by an ordinary crisp process

a practical numerical scheme for the ternary cahn hilliard system with a logarithmic free energy abstract we consider a practically stable finite difference method for the ternary cahn hilliard system with a logarithmic free energy modeling the phase separation of a three component mixture the numerical scheme is based on a linear unconditionally gradient stable scheme by eyre and is solved by an efficient and accurate multigrid method the logarithmic function has a singularity at zero to remove the singularity we regularize the function near zero by using a quadratic polynomial approximation we perform a convergence test a linear stability analysis and a robustness test of the ternary cahn hilliard equation we observe that our numerical solutions are convergent consistent with the exact solutions of linear stability analysis and stable with practically large enough time steps using the proposed numerical scheme we also study the temporal evolution of morphology patterns during phase separation in one two and three dimensional spaces

computationally efficient approach for the minimization of volume constrained vector valued ginzburg landau energy functional the minimization of volume constrained vector valued ginzburg landau energy functional is considered in the present study it has many applications in computational science and engineering like the conservative phase separation in multiphase systems such as the spinodal decomposition phase coarsening in multiphase systems color image segmentation and optimal space partitioning a computationally efficient algorithm is presented to solve the space discretized form of the original optimization problem the algorithm is based on the constrained nonmonotone l 2 gradient flow of ginzburg landau functional followed by a regularization step which is resulted from the tikhonov regularization term added to the objective functional that lifts the solution from the l 2 function space into h 1 space the regularization step not only improves the convergence rate of the presented algorithm but also increases its stability bound the step size selection based on the barzilai borwein approach is adapted to improve the convergence rate of the introduced algorithm the success and performance of the presented approach is demonstrated throughout several numerical experiments to make it possible to reproduce the results presented in this work the matlab implementation of the presented algorithm is provided as the supplementary material developing an efficient algorithm to solve conservative gradient flow of multiphase ginzburg landau functional performing the constrained minimization of energy functional on the h 1 space using non monotone optimization path by applying the barzilai borwein step size selection approach the matlab implementation of the presented algorithms is given as supplementary material

sharp bounds for the signless laplacian spectral radius of digraphs abstract let g v e be a digraph with n vertices and m arcs without loops and multiarcs and vertex set v v 1 v 2 v n denote the outdegree and average 2 outdegree of the vertex v i by d i and m i respectively let a g be the adjacency matrix and d g diag d 1 d 2 d n be the diagonal matrix with outdegree of the vertices of the digraph g then we call q g d g a g the signless laplacian matrix of g let q g denote the signless laplacian spectral radius of the digraph g in this paper we present several improved bounds in terms of outdegree and average 2 outdegree for the signless laplacian spectral radius of digraphs then we give an example to compare the bounds

the efficiency of investment information we investigate how the description of a correlated information v improves the investment in the stock market x the objective is to maximize the growth rate of wealth in repeated investments we find a single letter characterization of the incremental growth rate spl delta r the maximum increase in growth rate when v is described to the investor at rate r the incremental growth rate specialized to the horse race market is related to source coding with side information of wyner and ahlswede kormer we provide two horse race examples jointly binary and jointly gaussian the initial efficiency spl delta 0 is the maximum possible increase in the growth rate per bit of description we show that the initial efficiency is related to the dependency between v and the market in particular for the horse race market the initial efficiency is the square of the hirschfeld gebelein renyi maximal correlation between v and x this provides a connection with the hypercontraction of the markov operator of ahlswede and gacs for the general market the initial efficiency is 1 when the side information v is equal to the stock market outcome x

a parametric finite element method for fourth order geometric evolution equations we present a finite element approximation of motion by minus the laplacian of curvature and related flows the proposed scheme covers both the closed curve case and the case of curves that are connected via triple junctions on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches it turns out that the new scheme has very good properties with respect to area conservation and the equidistribution of mesh points we state also an extension of our scheme to willmore flow of curves and discuss possible further generalizations

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

a nonstiff adaptive mesh refinement based method for the cahn hilliard equation we present a nonstiff fully adaptive mesh refinement based method for the cahn hilliard equation the method is based on a semi implicit splitting in which linear leading order terms are extracted and discretized implicitly combined with a robust adaptive spatial discretization the fully discretized equation is written as a system which is efficiently solved on composite adaptive grids using the linear multigrid method without any constraint on the time step size we demonstrate the efficacy of the method with numerical examples both the transient stage and the steady state solutions of spinodal decompositions are captured accurately with the proposed adaptive strategy employing this approach we also identify several stationary solutions of that decomposition on the 2d torus

multi component cahn hilliard system with different boundary conditions in complex domains abstract we propose an efficient phase field model for multi component cahn hilliard ch systems in complex domains the original multi component cahn hilliard system with a fixed phase is modified in order to make it suitable for complex domains in the cartesian grid along with contact angle or no mass flow boundary conditions on the complex boundaries the proposed method uses a practically unconditionally gradient stable nonlinear splitting numerical scheme further a nonlinear full approximation storage multigrid algorithm is used for solving semi implicit formulations of the multi component ch system incorporated with an adaptive mesh refinement technique the robustness of the proposed method is validated through various numerical simulations including multi phase separations via spinodal decomposition equilibrium contact angle problems and multi phase flows with a background velocity field in complex domains

improved upper and lower bounds for the spectral radius of digraphs let g be a digraph with n vertices and m arcs without loops and multiarcs the spectral radius r g of g is the largest eigenvalue of its adjacency matrix in this paper sharp upper and lower bounds on r g are given we show that some known bounds can be obtained from our bounds

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

sharp bounds on the spectral radius of nonnegative matrices and digraphs abstract the spectral radius a of a matrix a is the maximum modulus of its eigenvalues we present bounds on a that are often tighter and are applicable to a larger class of nonnegative matrices than previously reported the bounds are particularly suited to matrices which are sparse we complete the paper by applying these bounds to digraphs deriving the associated equality conditions which relate to the outdegree regularity of the digraph finally we show that the equality conditions may be achieved only for very specific values of the digraph s spectral radius

a robust approach to the decision rules of npv and irr for simple projects we propose a robust approach to the decision rules of net present value npv and internal rate of return irr by building an analogy between robust optimization approach proposed in the literature and decision rules of npv and irr by considering uncertainty in cash flows as proposed in the literature we assume that the parameters cash inflows and cash outflows in our case belong to a symmetric and bounded interval set and define a series of decision rules of npv and irr by considering robustness

minimax chance constrained programming models for fuzzy decision systems abstract the existing chance constrained programming for fuzzy decision systems is essentially a kind of maximax models optimistic models which maximize the maximum possible return this paper presents a spectrum of minimax models as opposed to maximax models based on chance constrained programming as well as chance constrained multi objective programming and chance constrained goal programming in which the minimax models will select the alternative that provides the best of the worst possible return finally a fuzzy simulation based genetic algorithm will be designed for solving minimax models and illustrated by some numerical examples

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

diffuse interface model for incompressible two phase flows with large density ratios we investigate the applicability of an incompressible diffuse interface model for two phase incompressible fluid flows with large viscosity and density contrasts diffuse interface models have been used previously primarily for density matched fluids and it remains unclear to what extent such models can be used for fluids of different density thereby potentially limiting the application of these models in this paper the convective cahn hilliard equation and the condition that the velocity field is divergence free are derived from the conservation law of mass of binary mixtures in a straightforward way for fluids with large density and viscosity ratios differences in the equations of motion with a previously derived quasi incompressible model are shown to result from the respective assumptions made regarding the relationship between the diffuse fluxes of two species the convergence properties of the model are investigated for cases with large density ratio quantitative comparisons are made with results from previous studies to validate the model and its numerical implementation tests show that the variation in volume during the computation is of the order of machine accuracy which is consistent with our use of a conservative discretization scheme finite volume methods for the cahn hilliard equation results of the method are compared with previous work for the change in topology of rising bubbles and rayleigh taylor instability additional results are presented for head on droplet collision and the onset of droplet entrainment in stratified flows

bounds on the spectral radii of digraphs in terms of walks abstract let g v e be a digraph with n vertices and m arcs without loops and multiarcs v v 1 v 2 v n denote by g the largest eigenvalue of its adjacency matrix w k i the number of k walks from vertex v i in this paper we prove that the inequalities min w p q i w q i v i v p g max w p q i w q i v i v hold for every integers p 1 q 0 if g is strongly connected then each equality holds iff w p q 1 w q 1 w p q 2 w q 2 w p q n w q n furthermore we have lim p w p q i w q i 1 p g for every integers p 1 q 0 and each vertex v i v g

a fourth order spatial accurate and practically stable compact scheme for the cahn hilliard equation abstract we present a fourth order spatial accurate and practically stable compact difference scheme for the cahn hilliard equation the compact scheme is derived by combining a compact nine point formula and linearly stabilized splitting scheme the resulting system of discrete equations is solved by a multigrid method numerical experiments are conducted to verify the practical stability and fourth order accuracy of the proposed scheme we also demonstrate that the compact scheme is more robust and efficient than the non compact fourth order scheme by applying to parallel computing and adaptive mesh refinement

coupling discontinuous galerkin and mixed finite element discretizations using mortar finite elements discontinuous galerkin dg and mixed finite element mfe methods are two popular methods that possess local mass conservation in this paper we investigate dg dg and dg mfe domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface the subdomain grids need not match and the mortar grid may be much coarser giving a two scale method convergence results in terms of the fine subdomain scale h and the coarse mortar scale h are established for both types of couplings in addition a nonoverlapping parallel domain decomposition algorithm is developed which reduces the coupled system to an interface mortar problem the properties of the interface operator are analyzed

an efficient algorithm for incompressible n phase flows abstract we present an efficient algorithm within the phase field framework for simulating the motion of a mixture of n n 2 immiscible incompressible fluids with possibly very different physical properties such as densities viscosities and pairwise surface tensions the algorithm employs a physical formulation for the n phase system that honors the conservations of mass and momentum and the second law of thermodynamics we present a method for uniquely determining the mixing energy density coefficients involved in the n phase model based on the pairwise surface tensions among the n fluids our numerical algorithm has several attractive properties that make it computationally very efficient i it has completely de coupled the computations for different flow variables and has also completely de coupled the computations for the n 1 phase field functions ii the algorithm only requires the solution of linear algebraic systems after discretization and no nonlinear algebraic solve is needed iii for each flow variable the linear algebraic system involves only constant and time independent coefficient matrices which can be pre computed during pre processing despite the variable density and variable viscosity of the n phase mixture iv within a time step the semi discretized system involves only individual de coupled helmholtz type including poisson equations despite the strongly coupled phase field system of fourth spatial order at the continuum level v the algorithm is suitable for large density contrasts and large viscosity contrasts among the n fluids extensive numerical experiments have been presented for several problems involving multiple fluid phases large density contrasts and large viscosity contrasts in particular we compare our simulations with the de gennes theory and demonstrate that our method produces physically accurate results for multiple fluid phases we also demonstrate the significant and sometimes dramatic effects of the gravity density ratios pairwise surface tensions and drop sizes on the n phase configurations and dynamics the numerical results show that the method developed herein is capable of dealing with n phase systems with large density ratios large viscosity ratios and pairwise surface tensions and that it can be a powerful tool for studying the interactions among multiple types of fluid interfaces

efficient numerical solution of discrete multi component cahn hilliard systems in this work we develop preconditioners for the iterative solution of the large scale algebraic systems arising in finite element discretizations of microstructures with an arbitrary number of components described by the diffusive interface model the suggested numerical techniques are applied to the study of ternary fluid flow processes

sharp bounds for the spectral radius of digraphs abstract let g v e be a digraph with n vertices and m arcs without loops and multiarcs the spectral radius g of g is the largest eigenvalue of its adjacency matrix in this paper the following sharp bounds on g have been obtained min t i t j v i v j e g max t i t j v i v j e where g is strongly connected and t i is the average 2 outdegree of vertex v i moreover each equality holds if and only if g is average 2 outdegree regular or average 2 outdegree semiregular

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

phase field computations for ternary fluid flows we present a new phase field model for three component immiscible liquid flows with surface tension in the phase field approach the classical sharp interface between the two immiscible fluids is replaced by a transition region across which the properties of fluids change continuously the proposed method incorporates a chemical potential which can eliminate the unphysical phase field profile and a continuous surface tension force formulation from which we can calculate the pressure field directly from the governing equations the capabilities of the method are demonstrated with several examples we compute the ternary phase separation via spinodal decomposition equilibrium phase field profiles pressure field distribution and a three interface contact angle resulting from a spreading liquid lens on an interface the numerical results show excellent agreement with analytical solutions

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

an augmented lagrangian method for the phase field approach for pressurized fractures abstract in the modeling of pressurized fractures using phase field approaches the irreversibility of crack growth is enforced through an inequality constraint on the temporal derivative of the phase field function in comparison to the classical case in elasticity the presence of the pressure requires the additional constraint and makes the problem much harder to analyze after temporal discretization this induces a minimization problem in each time step over a solution dependent admissible set to avoid solving the resulting variational inequality corresponding to the first order necessary conditions a penalization approach is used commonly to remove the inequality constraint it is well known that for large penalty parameters the algorithm suffers from numerical instabilities in the solution process consequently to avoid such a drawback we propose an augmented lagrangian algorithm for the discrete in time and continuous in space phase field problems the final set of equations is solved in a decoupled fashion the proposed method is substantiated with several benchmark and prototype tests in two and three dimensions

a locally conservative finite element method based on piecewise constant enrichment of the continuous galerkin method this paper presents a locally conservative finite element method based on enriching the approximation space of the continuous galerkin method with elementwise constant functions the proposed method has a smaller number of degrees of freedom than the discontinuous galerkin method numerical examples on coupled flow and transport in porous media are provided to illustrate the advantages of this method we also present a theoretical analysis of the method and establish optimal convergence of numerical solutions

multiphase allen cahn and cahn hilliard models and their discretizations with the effect of pairwise surface tensions in this paper the mathematical properties and numerical discretizations of multiphase models that simulate the phase separation of an n component mixture are studied for the general choice of phase variables the unisolvent property of the coefficient matrix involved in the n phase models based on the pairwise surface tensions is established moreover the symmetric positive definite property of the coefficient matrix on an n 1 n 1 dimensional hyperplane which is of fundamental importance to the well posedness of the models can be proved equivalent to some physical condition for pairwise surface tensions the n phase allen cahn and n phase cahn hilliard equations can then be derived from the free energy functional a natural property is that the resulting dynamics of concentrations are independent of phase variables chosen finite element discretizations for n phase models can be obtained as a natural extension of the existing discretizations for the two phase model the discrete energy law of the numerical schemes can be proved and numerically observed under some restrictions pertaining to time step size numerical experiments including the spinodal decomposition and the evolution of triple junctions are described in order to investigate the effect of pairwise surface tensions

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

a finite element method for the numerical solution of the coupled cahn hilliard and navier stokes system for moving contact line problems in this paper a semi implicit finite element method is presented for the coupled cahn hilliard and navier stokes equations with the generalized navier boundary condition for the moving contact line problems in our method the system is solved in a decoupled way for the cahn hilliard equations a convex splitting scheme is used along with a p1 p1 finite element discretization the scheme is unconditionally stable a linearized semi implicit p2 p0 mixed finite element method is employed to solve the navier stokes equations with our method the generalized navier boundary condition is extended to handle the moving contact line problems with complex boundary in a very natural way the efficiency and capacity of the present method are well demonstrated with several numerical examples

unconditionally stable methods for simulating multi component two phase interface models with peng robinson equation of state and various boundary conditions in this paper we consider multi component dynamic two phase interface models which are formulated by the cahn hilliard system with peng robinson equation of state and various boundary conditions these models can be derived from the minimum problems of helmholtz free energy or grand potential in the realistic thermodynamic systems the resulted cahn hilliard systems with various boundary conditions are fully coupled and strongly nonlinear a linear transformation is introduced to decouple the relations between different components and as a result the models are simplified from this we further propose a semi implicit unconditionally stable time discretization scheme which allows us to solve the cahn hilliard system by a decoupled way and thus our method can significantly reduce the computational cost and memory requirements the mixed finite element methods are employed for the spatial discretization and the approximate errors are also analyzed for both space and time numerical examples are tested to demonstrate the efficiency of our proposed methods

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

error estimates for a mixed finite element discretization of some degenerate parabolic equations we consider a numerical scheme for a class of degenerate parabolic equations including both slow and fast diffusion cases a particular example in this sense is the richards equation modeling the flow in porous media the numerical scheme is based on the mixed finite element method mfem in space and is of one step implicit in time the lowest order raviart thomas elements are used here we extend the results in radu et al siam j numer anal 42 1452 1478 2004 schneid et al numer math 98 353 370 2004 to a more general framework by allowing for both types of degeneracies we derive error estimates in terms of the discretization parameters and show the convergence of the scheme the features of the mfem especially of the lowest order raviart thomas elements are now fully exploited in the proof of the convergence the paper is concluded by numerical examples

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

decoupled energy stable schemes for phase field models of two phase complex fluids we consider in this paper numerical approximations of phase field models for two phase complex fluids we first reformulate the phase field models derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish their energy laws then we construct two classes stabilized and convex splitting of decoupled time discretization schemes for the coupled nonlinear systems these schemes are unconditionally energy stable and lead to decoupled elliptic equations to solve at each time step furthermore these elliptic equations are linear for the stabilized version stability analysis and ample numerical simulations are presented

mixed finite element based fully conservative methods for simulating wormhole propagation abstract wormhole propagation during reactive dissolution of carbonates plays a very important role in the product enhancement of oil and gas reservoir because of high velocity and nonuniform porosity the darcy forchheimer model is applicable for this problem instead of conventional darcy framework we develop a mixed finite element scheme for numerical simulation of this problem in which mixed finite element methods are used not only for the darcy forchheimer flow equations but also for the solute transport equation by introducing an auxiliary flux variable to guarantee full mass conservation in theoretical analysis aspects based on the cut off operator of solute concentration we construct an analytical function to control and handle the change of porosity with time we treat the auxiliary flux variable as a function of velocity and establish its properties we employ the coupled analysis approach to deal with the fully coupling relation of multivariables from this the stability analysis and a priori error estimates for velocity pressure concentration and porosity are established in different norms numerical results are also given to verify theoretical analysis and effectiveness of the proposed scheme

on anisotropic order parameter models for multi phase system and their sharp interface limits abstract for a general class of diffuse anisotropic multi phase order parameter or phase field models we use formally matched asymptotic expansions to determine the asymptotic limit when a small parameter related to the thickness of the interface tends to zero in the case of anisotropic allen cahn systems we obtain in the limit that the interface moves by anisotropic mean curvature flow at triple junctions a force balance holds which in the anisotropic case includes shear forces herring torque terms acting normal to the interface we further identify the singular limit of anisotropic cahn hilliard systems

characterization of optimal solutions of uncertainty investment problem in this paper we characterize the optimal profit best return of an investment problem with trapezoidal fuzzy numbers we give a solution algorithm to determine the optimal profit for investing of f money in the n investment polices alternative approach is considered for people who need to be more precise in their requirements illustrative example to determine the optimal profits of such problem is presented

fuzzy quadratic minimum spanning tree problem in this paper a fuzzy quadratic minimum spanning tree problem is formulated as expected value model chance constrained programming and dependent chance programming according to different decision criteria then the crisp equivalents are derived when the fuzzy costs are characterized by trapezoidal fuzzy numbers furthermore a simulation based genetic algorithm using prufer number representation is designed for solving the proposed fuzzy programming models as well as their crisp equivalents finally a numerical example is provided for illustrating the effectiveness of the genetic algorithm

preconditioning for vector valued cahn hilliard equations the solution of vector valued cahn hilliard systems is of interest in many applications we discuss strategies for the handling of smooth and nonsmooth potentials as well as for different types of constant mobilities concerning the nonsmooth systems the necessary bound constraints are incorporated via the moreau yosida regularization technique we develop effective preconditioners for the efficient solution of the linear systems in saddle point form numerical results illustrate the efficiency of our approach in particular we numerically show mesh and phase independence of the developed preconditioner in the smooth case the results in the nonsmooth case are also satisfying and the preconditioned version always outperforms the unpreconditioned one

unconditionally energy stable time stepping scheme for cahn morral equation an unconditionally energy stable time stepping scheme is introduced to solve cahn morral like equations in the present study it is constructed based on the combination of david eyre s time stepping scheme and schur complement approach although the presented method is general and independent of the choice of homogeneous free energy density function term logarithmic and polynomial energy functions are specifically considered in this paper the method is applied to study the spinodal decomposition in multi component systems and optimal space tiling problems a penalization strategy is developed in the case of later problem to avoid trivial solutions extensive numerical experiments demonstrate the success and performance of the presented method according to the numerical results the method is convergent and energy stable independent of the choice of time stepsize its matlab implementation is included in the appendix for the numerical evaluation of algorithm and reproduction of the presented results extension of eyre s convex concave splitting scheme to multiphase systems efficient solution of spinodal decomposition in multi component systems efficient solution of least perimeter periodic space partitioning problem developing a penalization strategy to avoid trivial solutions presentation of matlab implementation of the introduced algorithm

handling uncertainty in route choice models from probabilistic to possibilistic approaches abstract in order to design effective advanced traffic information systems atis suitable mathematical models have to be defined to simulate the effects of information on users route choice behaviour and then to incorporate it into traffic assignment models to estimate how traffic demand loads the roads network to face this problem it is necessary to deal with uncertainty that plays a crucial role in the users decision making processes to this purpose this paper first analyses how uncertainty affects users route choice process and how traffic assignment models may take it into account in literature route choice behaviour modelling is widely solved within the random utility theory framework but we show in this paper that such an approach only considers one type of uncertainty more precisely the consideration of randomness of traffic by drivers is for example hardly ever represented in classical models in spite of its importance in the management of information by drivers starting from the presented analysis a new route choice model is also proposed to represent explicitly the uncertainty lying in users route choice behaviour it is based on recent developments in possibility theory which is an alternate way to probability theory in order to represent or measure uncertainty

an efficient fully discrete local discontinuous galerkin method for the cahn hilliard hele shaw system in this paper we develop an efficient and energy stable fully discrete local discontinuous galerkin ldg method for the cahn hilliard hele shaw chhs system the semi discrete energy stability of the ldg method is proved firstly due to the strict time step restriction dt o dx 4 of explicit time discretization methods for stability we introduce a semi implicit time integration scheme which is based on a convex splitting of the discrete cahn hilliard energy the unconditional energy stability has been proved for this fully discrete ldg scheme the fully discrete equations at the implicit time level are nonlinear thus the nonlinear full approximation scheme fas multigrid method has been applied to solve this system of algebraic equations which has been shown the nearly optimal complexity numerically numerical results are also given to illustrate the accuracy and capability of the ldg method coupled with the multigrid solver

fuzzy capital budgeting fuzzy equivalents of all the classical capital budgeting investment choice methods are generalised or proposed these equivalents can be used to evaluate and compare projects in which the cash ows duration time and required rate of return cost of capital are given imprecisely in the form of a fuzzy number c 2000 elsevier science b v all rights reserved

chance constrained programming with fuzzy parameters this paper extends chance constrained programming from stochastic to fuzzy environments analogous to stochastic programming some crisp equivalents of chance constraints in fuzzy environments are presented we also propose a technique of fuzzy simulation for the chance constraints which are usually hard to be converted to their crisp equivalents finally a fuzzy simulation based genetic algorithm is designed for solving this kind of problems and some numerical examples are discussed

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

symmetric and nonsymmetric discontinuous galerkin methods for reactive transport in porous media for solving reactive transport problems in porous media we analyze three primal discontinuous galerkin dg methods with penalty namely symmetric interior penalty galerkin sipg nonsymmetric interior penalty galerkin nipg and incomplete interior penalty galerkin iipg a cut off operator is introduced in dg to treat general kinetic chemistry error estimates in l2 h1 are established which are optimal in h and nearly optimal in p we develop a parabolic lift technique for sipg which leads to h optimal and nearly p optimal error estimates in the l2 l2 and negative norms numerical results validate these estimates we also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces

analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits we propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science the primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method in particular by focusing on the dependence of the error bounds on the parameter e known as the measure of the interface thickness optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size h and the time step size k in particular it is shown that all error bounds depend on 1 a only in some lower polynomial order for small e the cruxes of the analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of chen and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect finally as a nontrivial byproduct the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients the sharp interface limits include the classical stefan problem the generalized stefan problems with surface tension and surface kinetics the motion by mean curvature flow and the hele shaw model

fuzzy capital rationing model in this paper we study the fuzzification of weingartner s pure capital rationing model and its analysis we develop a primal dual pair based on t norm t conorm relation for the constraints and objective function for a fully fuzzified pure capital rationing problem except project selection variables we define the a interval under which the weak duality is proved we perform sensitivity analysis for a change in a budget level or in a cash flow level of a non basic as well as a basic variable we analyze the problem based on duality and complementary slackness results we illustrate the proposed model by computational analysis and interpret the results

a diffuse interface model for axisymmetric immiscible two phase flow a diffuse interface model is considered for solving axisymmetric immiscible two phase flow with surface tension the navier stokes ns equations are modified by the addition of a continuum forcing the interface between the two fluids is considered as the half level set of a mass concentration c which is governed by the cahn hilliard ch equation a fourth order degenerate nonlinear parabolic diffusion equation in this work we develop a nonlinear multigrid method to solve the ch equation with degenerate mobility and couple this to a projection method for the incompressible ns equations the diffuse interface method can deal with topological transitions such as breakup and coalescence smoothly without ad hoc cut and connect or other artificial procedures we present results for rayleigh s capillary instability up to forming satellite drops the results agree well with the linear stability theory

efficient numerical methods for simulating surface tension of multi component mixtures with the gradient theory of fluid interfaces abstract surface tension significantly impacts subsurface flow and transport and it is the main cause of capillary effect a major immiscible two phase flow mechanism for systems with a strong wettability preference in this paper we consider the numerical simulation of the surface tension of multi component mixtures with the gradient theory of fluid interfaces major numerical challenges include that the system of the euler lagrange equations is solved on the infinite interval and the coefficient matrix is not positive definite we construct a linear transformation to reduce the euler lagrange equations and naturally introduce a path function which is proven to be a monotonic function of the spatial coordinate variable by using the linear transformation and the path function we overcome the above difficulties and develop the efficient methods for calculating the interface and its interior compositions moreover the computation of the surface tension is also simplified the proposed methods do not need to solve the differential equation system and they are easy to be implemented in practical applications numerical examples are tested to verify the efficiency of the proposed methods

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

efficient numerical algorithm for multiphase field simulations phase field models have emerged as a successful class of models in a wide variety of applications in computational materials science multiphase field theories as a subclass of phase field theories have been especially useful for studying nucleation and growth in polycrystalline materials in theory an infinite number of phase field variables are required to represent grain orientations in a rotationally invariant free energy however limitations on available computational time and memory have restricted the number of phase field variables used in the simulations we present an approach by which the time and memory requirements are drastically reduced relative to standard algorithms the proposed algorithm allows us the use of an unlimited number of phase field variables to perform simulations without the associated burden on computational time or memory we present the algorithm in the context of coalescence free grain growth

a discontinuous galerkin method for the cahn hilliard equation a discontinuous galerkin finite element method has been developed to treat the high order spatial derivatives appearing in the cahn hilliard equation the cahn hilliard equation is a fourth order nonlinear parabolic partial differential equation originally proposed to model phase segregation of binary alloys the developed discontinuous galerkin approach avoids the need for mixed finite element methods coupled equations or interpolation functions with a high degree of continuity that have been employed in the literature to treat the fourth order spatial derivatives the variational formulation of the discontinuous galerkin method its implementation and numerical examples are presented in this communication it is also shown under what conditions the method is stable and an error estimate in an energy type norm is presented the method is evaluated by comparison with a standard finite element treatment in which the cahn hilliard equation is decomposed into two coupled partial differential equations

local discontinuous galerkin methods for the cahn hilliard type equations in this paper we develop local discontinuous galerkin ldg methods for the fourth order nonlinear cahn hilliard equation and system the energy stability of the ldg methods is proved for the general nonlinear case numerical examples for the cahn hilliard equation and the cahn hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capability of the methods