an unconditionally energy stable scheme based on an implicit auxiliary energy variable for incompressible two phase flows with different densities involving only precomputable coefficient matrices abstract we present an energy stable scheme for numerically approximating the governing equations for incompressible two phase flows with different densities and dynamic viscosities for the two fluids the proposed scheme employs a scalar valued auxiliary energy variable in its formulation and it satisfies a discrete energy stability property more importantly the scheme is computationally efficient within each time step it computes two copies of the flow variables velocity pressure phase field function by solving individually a linear algebraic system involving a constant and time independent coefficient matrix for each of these field variables the coefficient matrices involved in these linear systems only need to be computed once and can be pre computed additionally within each time step the scheme requires the solution of a nonlinear algebraic equation about a scalar valued number using the newton s method the cost for this nonlinear solver is very low accounting for only a few percent of the total computation time per time step because this nonlinear equation is about a scalar number not a field function extensive numerical experiments have been presented for several two phase flow problems involving large density ratios and large viscosity ratios comparisons with theory show that the proposed method produces physically accurate results simulations with large time step sizes demonstrate the stability of computations and verify the robustness of the proposed method an implication of this work is that energy stable schemes for two phase problems can also become computationally efficient and competitive eliminating the need for expensive re computations of coefficient matrices even at large density ratios and viscosity ratios

numerical methods for a multicomponent two phase interface model with geometric mean influence parameters in this paper we consider an interface model for multicomponent two phase fluids with geometric mean influence parameters which is popularly used to model and predict surface tension in practical applications for this model there are two major challenges in theoretical analysis and numerical simulation the first one is that the influence parameter matrix is not positive definite the second one is the complicated structure of the energy function which requires us to find out a physically consistent treatment to overcome these two challenging problems we reduce the formulation of the energy function by employing a linear transformation and a weighted molar density and furthermore we propose a local minimum grand potential energy condition to establish the relation between the weighted molar density and mixture compositions from this we prove the existence of the solution under proper conditions and prove the maximum principle of the weighted molar density for numerical simulation we propose a

a space time fractional phase field model with tunable sharpness and decay behavior and its efficient numerical simulation abstract we present a space time fractional allen cahn phase field model that describes the transport of the fluid mixture of two immiscible fluid phases the space and time fractional order parameters control the sharpness and the decay behavior of the interface via a seamless transition of the parameters although they are shown to provide more accurate description of anomalous diffusion processes and sharper interfaces than traditional integer order phase field models do fractional models yield numerical methods with dense stiffness matrices consequently the resulting numerical schemes have significantly increased computational work and memory requirement we develop a lossless fast numerical method for the accurate and efficient numerical simulation of the space time fractional phase field model numerical experiments shows the utility of the fractional phase field model and the corresponding fast numerical method

phase field modeling and simulation of three phase flow on solid surfaces phase field models are widely used to describe the two phase system the evolution of the phase field variables is usually driven by the gradient flow of a total free energy functional the generalization of the approach to an n phase n 3 system requires some extra consistency conditions on the free energy functional in order for the model to give physically relevant results a projection approach is proposed for the derivation of a consistent free energy functional for the three phase cahn hilliard equations the system is then coupled with the navier stokes equations to describe the three phase flow on solid surfaces with moving contact line an energy stable scheme is developed for the three phase flow system the discrete energy law of the numerical scheme is proved which ensures the stability of the scheme we also show some numerical results for the dynamics of triple junctions and four phase contact lines

mass conservative and energy stable finite difference methods for the quasi incompressible navier stokes cahn hilliard system primitive variable and projection type schemes abstract in this paper we describe two fully mass conservative energy stable finite difference methods on a staggered grid for the quasi incompressible navier stokes cahn hilliard q nsch system governing a binary incompressible fluid flow with variable density and viscosity both methods namely the primitive method finite difference method in the primitive variable formulation and the projection method finite difference method in a projection type formulation are so designed that the mass of the binary fluid is preserved and the energy of the system equations is always non increasing in time at the fully discrete level we also present an efficient practical nonlinear multigrid method comprised of a standard fas method for the cahn hilliard equation and a method based on the vanka type smoothing strategy for the navier stokes equation for solving these equations we test the scheme in the context of capillary waves rising droplets and rayleigh taylor instability quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes moreover in all cases mass of the single component and the binary fluid was conserved up to 1 0 8 and energy decreases in time

an incremental pressure correction finite element method for the time dependent oldroyd flows abstract in this paper we present an incremental pressure correction finite element method for the time dependent oldroyd flows this method is a fully discrete projection method as we all know most projection methods have been studied without space discretization then the ensuing analysis may not extend to this case we also give the stability analysis and the optimal error analysis the analysis is based on a time discrete error and a spatial discrete error in order to show the effectiveness of the method we also present some numerical results the numerical results confirm our analysis and show clearly the stability and optimal convergence of the incremental pressure correction finite element method for the time dependent oldroyd flows

numerical approximations to a new phase field model for two phase flows of complex fluids abstract we derive a new phase field theory for immiscible mixtures of nematic liquid crystals and viscous fluids using the variational principle coupled with the generalized onsager principle a novel phase transition mechanism is implemented to couple the nematic liquid crystal phase with the viscous fluid phase to arrive at the dissipative hydrodynamic model for incompressible fluid mixtures through a delicate explicit implicit numerical discretization we develop a decoupled linear scheme for a simplified version of the phase field model as well as a coupled nonlinear scheme for the full model both schemes are shown as unconditionally energy stable with consistent discrete dissipative energy laws several numerical examples are presented to show the effectiveness of the new model and the new numerical schemes developed for it

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

efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model abstract in this paper we study efficient numerical schemes of the classical phase field elastic bending energy model that has been widely used to describe the shape deformation of biological lipid vesicles in which the free energy of the system consists of an elastic bending energy a surface area constraint and a volume constraint one major challenge in solving such model numerically is how to design appropriate temporal discretizations in order to preserve energy stability with large time step sizes at the semi discrete level we develop a first order and a second order time stepping scheme for this highly nonlinear and stiff parabolic pde system based on the invariant energy quadratization approach in particular the resulted semi discretizations lead to linear systems in space with symmetric positive definite operators at each time step thus can be efficiently solved in addition the proposed schemes are rigorously proved to be unconditionally energy stable various numerical experiments in 2d and 3d are presented to demonstrate the stability and accuracy of the proposed schemes

a phase field model and its numerical approximation for two phase incompressible flows with different densities and viscosities modeling and numerical approximation of two phase incompressible flows with different densities and viscosities are considered a physically consistent phase field model that admits an energy law is proposed and several energy stable efficient and accurate time discretization schemes for the coupled nonlinear phase field model are constructed and analyzed ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios

decoupled energy stable schemes for phase field models of two phase incompressible flows in this paper we construct two classes based on stabilization and convex splitting of decoupled unconditionally energy stable schemes for cahn hilliard phase field models of two phase incompressible flows at each time step these schemes require solving only a sequence of elliptic equations including a pressure poisson equation furthermore all of these elliptic equations are linear for the schemes based on stabilization making them the first to the best of the authors knowledge totally decoupled linear unconditionally energy stable schemes for phase field models of two phase incompressible flows thus the schemes constructed in this paper are very efficient and easy to implement

isogeometric analysis of the navier stokes cahn hilliard equations with application to incompressible two phase flows abstract in this work we provide a unified and comparative description of the most prominent phase field based two phase flow models and present our numerical results of the application of galerkin based isogeometric analysis iga to incompressible navier stokes cahn hilliard nsch equations in velocity pressure phase field chemical potential formulation for the approximation of the velocity and pressure fields lbb compatible non uniform rational b spline spaces are used which can be regarded as smooth generalizations of taylor hood pairs of finite element spaces the one step scheme is used for the discretization in time the static and rising bubble in addition to the nonlinear rayleigh taylor instability flow problems are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme

uniquely solvable and energy stable decoupled numerical schemes for the cahn hilliard stokes darcy system for two phase flows in karstic geometry we propose and analyze two novel decoupled numerical schemes for solving the cahn hilliard stokes darcy chsd model for two phase flows in karstic geometry in the first numerical scheme we explore a fractional step method operator splitting to decouple the phase field cahn hilliard equation from the velocity field stokes darcy fluid equations to further decouple the stokes darcy system we introduce a first order pressure stabilization term in the darcy solver in the second numerical scheme so that the stokes system is decoupled from the darcy system and hence the chsd system can be solved in a fully decoupled manner we show that both decoupled numerical schemes are uniquely solvable energy stable and mass conservative ample numerical results are presented to demonstrate the accuracy and efficiency of our schemes

a coupled ericksen allen cahn model for liquid crystal droplets abstract we present a model and discretization that couples the ericksen model of liquid crystals with variable degree of orientation to the allen cahn equations with a mass constraint the coupled system models liquid crystal droplets with anisotropic surface tension effects due to the liquid crystal molecular alignment the total energy consists of the ericksen energy phase field allen cahn energy and a weak anchoring energy that couples the liquid crystal to the diffuse interface we describe our discretization of the total energy along with a method to compute minimizers via a discrete gradient flow algorithm which has a strictly monotone energy decreasing property numerical experiments are given in three dimensions that illustrate a wide variety of droplet shapes that result from their interaction with defects

a gpu parallelized spectral method for elliptic equations in rectangular domains we design and implement a polynomial based spectral method on graphic processing units gpus the key to success lies in the seamless integration of the matrix diagonalization technique and the new generation cuda tools the method is applicable to elliptic equations in rectangular domains with general boundary condition we show remarkable speedups of up to 15 times in the 2 d case and more than 35 times in the 3 d case

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

fast provably unconditionally energy stable and second order accurate algorithms for the anisotropic cahn hilliard model abstract in this paper we consider numerical approximations for solving the anisotropic cahn hilliard model we combine the scalar auxiliary variable sav approach with the stabilization technique to arrive at a novel stabilized sav approach where three linear stabilization terms which are shown to be crucial to remove the oscillations caused by the anisotropic coefficient are added to enhance the stability while keeping the required accuracy the schemes are very easy to implement and fast in the sense that all nonlinear terms are treated in a semi explicit way and one only needs to solve three decoupled linear equations with constant coefficients at each time step we further prove the unconditional energy stabilities rigorously and present numerous 2d and 3d numerical simulations to demonstrate the stability and accuracy

a finite difference method for a conservative allencahn equation on non flat surfaces we present an efficient numerical scheme for the conservative allencahn cac equation on various surfaces embedded in a narrow band domain in the three dimensional space we apply a quasi neumann boundary condition on the narrow band domain boundary using the closest point method this boundary treatment allows us to use the standard cartesian laplacian operator instead of the laplacebeltrami operator we apply a hybrid operator splitting method for solving the cac equation first we use an explicit euler method to solve the diffusion term second we solve the nonlinear term by using a closed form solution third we apply a spacetime dependent lagrange multiplier to conserve the total quantity the overall scheme is explicit in time and does not need iterative steps therefore it is fast a series of numerical experiments demonstrate the accuracy and efficiency of the proposed hybrid scheme

a novel linear second order unconditionally energy stable scheme for a hydrodynamic q tensor model of liquid crystals abstract the hydrodynamic q tensor model has been used for studying flows of liquid crystals and liquid crystal polymers it can be derived from a variational approach together with the generalized onsager principle in which the total energy decreases in time in this paper we develop a novel linear second order semi discrete scheme in time to solve the governing system of equations in the model the scheme is developed following the novel energy quadratization strategy so that it is linear and unconditionally energy stable at the semi discrete level this scheme is further discretized in space using a second order finite difference method and implemented on a gpu for high performance computing the convergence rate in time is established using a mesh refinement test several numerical examples are presented to demonstrate the usefulness of the model and the effectiveness of the numerical scheme in simulating defect dynamics in flows of liquid crystals

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

efficient non iterative and second order accurate numerical algorithms for the anisotropic allen cahn equation with precise nonlocal mass conservation abstract we propose two efficient non iterative and second order accurate algorithms to solving the anisotropic allen cahn equation with the nonlocal mass conservation the first scheme is the stabilized sav approach which combines the sav approach with the stabilization technique in which three linear stabilization terms are added to enhance the stability and keep the required accuracy while using large time steps the scheme not only can conserve the mass precisely but also is very easy to implement and non iterative where one only needs to solve three decoupled fourth order biharmonic equations with constant coefficients at each time step we further prove the scheme is unconditionally energy stable rigorously the second scheme is based on the linear stabilization approach where all nonlinear terms are treated in an explicit way therefore the scheme is quite efficient and stable that allows for large time steps in computations for both schemes we present a number of 2d and 3d numerical simulations to show stability and accuracy

gpav based unconditionally energy stable schemes for the cahn hilliard equation stability and error analysis abstract we present several first order and second order numerical schemes for the cahn hilliard equation with unconditional energy stability in terms of a discrete energy these schemes stem from the generalized positive auxiliary variable gpav idea and require only the solution of linear algebraic systems with a constant coefficient matrix more importantly the computational complexity operation count per time step of these schemes is approximately a half of those of the gpav and the scalar auxiliary variable sav methods in previous works we investigate the stability properties of the proposed schemes to establish stability bounds for the field function and the auxiliary variable and also provide their error analyses numerical experiments are presented to verify the theoretical analyses and also demonstrate the stability of the schemes at large time step sizes

convergence analysis and error estimates for a second order accurate finite element method for the cahn hilliard navier stokes system in this paper we present a novel second order in time mixed finite element scheme for the cahn hilliard navier stokes equations with matched densities the scheme combines a standard second order crank nicolson method for the navier stokes equations and a modification to the crank nicolson method for the cahn hilliard equation in particular a second order adams bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the cahn hilliard equation we show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the pde system specifically the discrete phase variable is shown to be bounded in ell infty left 0 t l infty right and the discrete chemical potential bounded in ell infty left 0 t l 2 right for any time and space step sizes in two and three dimensions and for any finite final time t we subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions

error estimates for time discretizations of cahn hilliard and allen cahn phase field models for two phase incompressible flows we carry out rigorous error analysis for some energy stable time discretization schemes developed in shen and yang siam j sci comput 32 3 1159 1179 2010 for a cahn hilliard phase field model and in shen and yang chin ann math ser b 31 5 743 758 2010 for an allen cahn phase field model

error analysis of a fractional time stepping technique for incompressible flows with variable density in this paper we analyze the convergence properties of a new fractional time stepping technique for the solution of the variable density incompressible navier stokes equations the main feature of this method is that contrary to other existing algorithms the pressure is determined by just solving one poisson equation per time step first order error estimates are proved and stability of a formally second order variant of the method is established

first order system least squares and the energetic variational approach for two phase flow this paper develops a first order system least squares fosls formulation for equations of two phase flow the main goal is to show that this discretization along with numerical techniques such as nested iteration algebraic multigrid and adaptive local refinement can be used to solve these types of complex fluid flow problems in addition from an energetic variational approach it can be shown that an important quantity to preserve in a given simulation is the energy law we discuss the energy law and inherent structure for two phase flow using the allen cahn interface model and indicate how it is related to other complex fluid models such as magnetohydrodynamics finally we show that using the fosls framework one can still satisfy the appropriate energy law globally while using well known numerical techniques

decoupled non iterative and unconditionally energy stable large time stepping method for the three phase cahn hilliard phase field model abstract in this paper we consider numerical approximations for a three phase phase field model where three fourth order cahn hilliard equations are nonlinearly coupled together through a lagrange multiplier term and a sixth order polynomial bulk potential by combining the recently developed sav approach with the linear stabilization technique we arrive at a novel stabilized sav scheme at each time step the scheme requires solving only four linear biharmonic equations with constant coefficients making it the first to the best of the author s knowledge totally decoupled second order accurate linear and unconditionally energy stable scheme for the model we further prove the unconditional energy stability rigorously and demonstrate the stability and the accuracy of the scheme numerically through the comparisons with the non stabilized sav scheme for simulating numerous benchmark numerical examples in 2d and 3d

the ericksen model of liquid crystals with colloidal and electric effects we present a robust discretization of the ericksen model of liquid crystals with variable degree of orientation coupled with colloidal effects and electric fields the total energy consists of the ericksen energy a weak anchoring or penalized dirichlet energy to model colloids and an electrical energy for a given electric field we describe our special discretization of the total energy along with a method to compute minimizers via a discrete quasi gradient flow algorithm which has a strictly monotone energy decreasing property numerical experiments are given in two and three dimensions to illustrate that the method is able to capture non trivial defect patterns such as the saturn ring defect we conclude with a rigorous proof of the convergence of our discrete energy to the continuous energy

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

a least squares finite element method for the numerical solution of the navier stokes cahn hilliard system modeling the motion of the contact line in this article we discuss the numerical solution of the navier stokes cahn hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall the method we employ combines a finite element space approximation with a time discretization by operator splitting to solve the cahn hilliard part of the problem we use a least squares conjugate gradient method we also show that the scheme has the total energy decaying in time property under certain conditions our numerical experiments indicate that the method discussed here is accurate stable and efficient

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

efficient energy stable numerical schemes for a phase field moving contact line model in this paper we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line the model is a coupled system that consists of incompressible navier stokes equations with a generalized navier boundary condition and cahn hilliard equation in conserved form in both schemes the projection method is used to deal with the navier stokes equations and stabilization approach is used for the non convex ginzburg landau bulk potential by some subtle explicit implicit treatments we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions an efficient spectral galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes numerical results show that the proposed schemes are very efficient and accurate we proposed two linear energy stable numerical schemes for a phase field moving contact line model discrete energy laws that are consistent to the continuous ones are proved the efficiency and fast convergence of the proposed schemes with spectral spatial approximations are numerically verified

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

linearly first and second order unconditionally energy stable schemes for the phase field crystal model in this paper we develop a series of linear unconditionally energy stable numerical schemes for solving the classical phase field crystal model the temporal discretizations are based on the first order euler method the second order backward differentiation formulas bdf2 and the second order crank nicolson method respectively the schemes lead to linear elliptic equations to be solved at each time step and the induced linear systems are symmetric positive definite we prove that all three schemes are unconditionally energy stable rigorously various classical numerical experiments in 2d and 3d are performed to validate the accuracy and efficiency of the proposed schemes

a theorem on the first heteroclinic tangency in two dimensional maps orientation preserving cases using the properties of the jordan curve the following theorem on the heteroclinic tangency in orientation preserving two dimensional maps is proved lett r 2 r 2 be a one parameter family ofc 1 diffeomorphisms andj detdt be such that 0 j 1 or 1 j letw be the unstable manifold of a hyperbolicn cycle andw the stable manifold of a hyperbolicm cycle suppose that for c w andw have no common points and that for c w andw s m have a transversal heteroclinic point then at c w andw are in the first asymptotic heteroclinic tangency except for the following three cases 1 n m both cycles are without reflection 2 m 2n then andm cycles are with and without reflection respectively 3 n 2m then andm cycles are without and with reflection respectively

a family of second order energy stable schemes for cahn hilliard type equations abstract we focus on the numerical approximation of the cahn hilliard type equations and present a family of second order unconditionally energy stable schemes by reformulating the equation into an equivalent system employing a scalar auxiliary variable we approximate the system at the time step n n denoting the time step index and is a real valued parameter and devise a family of corresponding approximations that are second order accurate and unconditionally energy stable this family of approximations contains the often used crank nicolson scheme and the second order backward differentiation formula as particular cases we further develop two efficient solution algorithms for the resultant discrete system of equations to overcome the difficulty caused by the unknown scalar auxiliary variable within each time step our method requires only the solution of either four de coupled individual helmholtz type equations or two separate individual systems with each system consisting of two coupled helmholtz type equations all the resultant linear algebraic systems involve only constant and time independent coefficient matrices that can be pre computed a number of numerical examples are presented to demonstrate the performance of the family of schemes developed herein we note that this family of second order approximations can be readily applied to devise energy stable schemes for other types of gradient flows when combined with the auxiliary variable approaches

improved design of subcritical and supercritical cascades using complex characteristics and boundary layer correction the method of complex characteristics and hodograph transformation for the design of shockless airfoils was extended to design supercritical cascades with high solidities and large inlet angles this capability was achieved by introducing a conformal mapping of the hodograph domain onto an ellipse and expanding the solution in terms of tchebycheff polynomials a computer code was developed based on this idea a number of airfoils designed with the code are presented various supercritical and subcritical compressor turbine and propeller sections are shown the lag entrainment method for the calculation of a turbulent boundary layer was incorporated to the inviscid design code the results of this calculation are shown for the airfoils described the elliptic conformal transformation developed to map the hodograph domain onto an ellipse can be used to generate a conformal grid in the physical domain of a cascade of airfoils with open trailing edges with a single transformation a grid generated with this transformation is shown for the korn airfoil previously announced in star as n83 24474

on convergent schemes for diffuse interface models for two phase flow of incompressible fluids with general mass densities we are concerned with convergence results for fully discrete finite element schemes suggested in grun and klingbeil j comput phys in press doi 10 1016 j jcp 2013 10 028 they were developed for the diffuse interface model in h abels h garcke and g grun math models methods appl sci 22 2012 1150013 which describes two phase flow of immiscible incompressible viscous fluids we formulate general conditions on discretization spaces and projection operators which allow us to prove compactness of discrete solutions with respect to both time and space and which hence permit us to establish convergence of the scheme to a generalized solution we identify a simple quantitative and physical criterion to decide whether this generalized solution is in fact a weak solution in this case our analysis provides another pathway to establish existence of weak solutions to the aforementioned model in two and in three space dimensions our argument is particularly based on higher regularity results

fully decoupled linear and unconditionally energy stable time discretization scheme for solving the magneto hydrodynamic equations abstract in this paper we consider numerical approximations for solving the magneto hydrodynamic equations which couples the navier stokes equations and maxwell equations together a challenging issue to solve this model numerically is the time discretization i e how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level we solve this issue in this paper by developing a linear fully decoupled first order time marching scheme by combining the projection method for navier stokes equations and some subtle implicit explicit treatments for nonlinear coupling terms we further prove that the scheme is unconditional energy stable and derive the optimal error estimates of the semi discretization rigorously various numerical simulations are implemented to demonstrate the stability and the accuracy

an adaptive pressure correction method without spurious velocities for diffuse interface models of incompressible flows in this article we propose to study two issues associated with the use of the incremental projection method for solving the incompressible navier stokes equation the first one is the combination of this time splitting algorithm with an adaptive local refinement method the second one is the reduction of spurious velocities due to the right hand side of the momentum balance we propose a new variant of the incremental projection method for solving the navier stokes equations with variable density and illustrate its behavior with the example of two phase flows simulations using a cahn hilliard navier stokes model