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

new exact solutions of heat conduction in metals we consider the way in which a solution to a class of nonlinear partial differential equationss u ut k u ux x approaches the similarity form the problem we solve is chosen for two main reasons first the equation above is of widespread use in modeling physical situations and second it provides a tractable but significant example of a free boundary problem

energy law preserving c0 finite element schemes for phase field models in two phase flow computations we use the idea in 33 to develop the energy law preserving method and compute the diffusive interface phase field models of allen cahn and cahn hilliard type respectively governing the motion of two phase incompressible flows we discretize these two models using a c 0 finite element in space and a modified midpoint scheme in time to increase the stability in the pressure variable we treat the divergence free condition by a penalty formulation under which the discrete energy law can still be derived for these diffusive interface models through an example we demonstrate that the energy law preserving method is beneficial for computing these multi phase flow models we also demonstrate that when applying the energy law preserving method to the model of cahn hilliard type un physical interfacial oscillations may occur we examine the source of such oscillations and a remedy is presented to eliminate the oscillations a few two phase incompressible flow examples are computed to show the good performance of our method

a stable and conservative finite difference scheme for the cahn hilliard equation we propose a stable and conservative finite difference scheme to solve numerically the cahn hilliard equation which describes a phase separation phenomenon numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill posed problem we design a new difference scheme based on a general strategy proposed recently by furihata and mori the new scheme inherits characteristic properties the conservation of mass and the decrease of the total energy from the equation the decrease of the total energy implies boundedness of discretized sobolev norm of the solution this in turn implies by discretized sobolev s lemma boundedness of max norm of the solution and hence the stability of the solution an error estimate for the solution is obtained and the order is o delta x 2 delta t 2 numerical examples demonstrate the effectiveness of the proposed scheme

a roadmap for discretely energy stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity abstract we present a framework for devising discretely energy stable schemes for general dissipative systems based on a generalized auxiliary variable the auxiliary variable a scalar number can be defined in terms of the energy functional by a general class of functions not limited to the square root function adopted in previous approaches the current method has another remarkable property the computed values for the generalized auxiliary variable are guaranteed to be positive on the discrete level regardless of the time step sizes or the external forces this property of guaranteed positivity is not available in previous approaches a unified procedure for treating the dissipative governing equations and the generalized auxiliary variable on the discrete level has been presented the discrete energy stability of the proposed numerical scheme and the positivity of the computed auxiliary variable have been proved for general dissipative systems the current method termed gpav generalized positive auxiliary variable requires only the solution of linear algebraic equations within a time step with appropriate choice of the operator in the algorithm the resultant linear algebraic systems upon discretization involve only constant and time independent coefficient matrices which only need to be computed once and can be pre computed several specific dissipative systems are studied in relative detail using the gpav framework ample numerical experiments are presented to demonstrate the performance of the method and the robustness of the scheme at large time step sizes

analytical and numerical solution of a generalized stefan problem exhibiting two moving boundaries with application to ocean delta formation abstract a model associated with the formation of sedimentary ocean deltas is presented this model is a generalized one dimensional stefan problem bounded by two moving boundaries the shoreline and the alluvial bedrock transition the sediment transport is a non linear diffusive process the diffusivity modeled as a power law of the fluvial slope dimensional analysis shows that the first order behavior of the moving boundaries is determined by the dimensionless parameter 0 r a b 1 the ratio of the fluvial slope to bedrock slope at the alluvial bedrock transition a similarity form of the governing equations is derived and a solution that tracks the boundaries obtained via the use of a numerical ode solver in the cases where the exponent in the diffusivity model is zero linear diffusion or infinite closed from solutions are found for the full range of the diffusivity exponents 0 the similarity solution shows that when r a b 0 4 there is no distinction in the predicted speeds of the moving boundaries further within the range of physically meaningful values of the diffusivity exponent i e 0 2 reasonable agreement in predictions extents up to r a b 0 7 in addition to the similarity solution a fixed grid enthalpy like solution is also proposed predictions obtained with this solution closely match those obtained with the similarity solution

brunn minkowski inequality for multiplicities this is a convex subset of the real vector space p zr in fact it is a convex polytope see br where this polytope is discussed from an algebraic point of view it is known that the total mass of m is a polynomial in m for su ciently large m denote by k its degree and m k m m weak d where d is the lebesgue measure supported on and the density is a piecewise polynomial function we will not use this piecewise polynomiality in this paper recall that a real function f de ned on a convex subset u of a vector space v is called concave if

new exact solutions for a free boundary system a method for constructing explicit exact solutions to nonlinear evolution equations is further developed the method is based on consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation the method is then applied to a free boundary problem based on the process of precipitant assisted protein crystal growth

tropical critical points of the superpotential of a flag variety abstract in this paper we investigate the idea of a tropical critical point of the superpotential for the full flag variety of type a recall that associated to an irreducible representation of g s l n c are various polytopes whose integral points parameterize a basis for the representation e g the gelfand zetlin polytope such polytopes can be constructed via the theory of geometric crystals by tropicalising a certain function and in fact the function involved coincides with the superpotential from the landau ginzburg model for g b coming from mirror symmetry in mirror symmetry a special role is played by the critical points of the superpotential and motivated by this we give a definition of the tropical critical points and use it to find a canonical point in each polytope we then characterise the highest weights for which this tropical critical point is integral and therefore corresponds to a basis vector of the corresponding representation finally we give an interpretation of the tropical critical point by constructing a special vector in the representation using borel weil theory and conjecturing a correspondence between this vector and the tropical critical point

existence of an exact solution for a one phase stefan problem with nonlinear thermal coefficients from tirskii s method abstract the mathematical analysis of a one phase lame clapeyron stefan problem with nonlinear thermal coefficients following g a tirskii two exact solutions of stefan s nonlinear problem sov phys dokl 4 1959 288 292 is obtained two related cases are considered one of them has a temperature condition on the fixed face x 0 and the other one has a flux condition of the type q 0 t q 0 0 we obtain in both cases sufficient conditions for data in order to have the existence of an explicit solution of a similarity type which is given by using a double fixed point

the scalar auxiliary variable sav approach for gradient flows abstract we propose a new approach which we term as scalar auxiliary variable sav approach to construct efficient and accurate time discretization schemes for a large class of gradient flows the sav approach is built upon the recently introduced ieq approach it enjoys all advantages of the ieq approach but overcomes most of its shortcomings in particular the sav approach leads to numerical schemes that are unconditionally energy stable and extremely efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step the scheme is not restricted to specific forms of the nonlinear part of the free energy so it applies to a large class of gradient flows numerical results are presented to show that the accuracy and effectiveness of the sav approach over the existing methods

the intersection of opposed big cells in real flag varieties for any reductive algebraic group over c split over r a graph is defined in type a and d it is shown that the number of connected components of the graph coincides with the number of connected components of the intersection of two opposed big cells in the corresponding real flag variety this gives a combinatorial description for the number of connected components of such a variety

a second order splitting method for the cahn hilliard equation a semi discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary fourth order in space cahn hilliard equation optimal order error bounds are derived in various norms for an implementation which uses mass lumping the continuous problem has an energy based lyapunov functional it is proved that this property holds for the discrete problem

finite element approximation of nematic liquid crystal flows using a saddle point structure in this work we propose finite element schemes for the numerical approximation of nematic liquid crystal flows based on a saddle point formulation of the director vector sub problem it introduces a lagrange multiplier that allows to enforce the sphere condition in this setting we can consider the limit problem without penalty and the penalized problem using a ginzburg landau penalty function in a unified way further the resulting schemes have a stable behavior with respect to the value of the penalty parameter a key difference with respect to the existing schemes two different methods have been considered for the time integration first we have considered an implicit algorithm that is unconditionally stable and energy preserving the linearization of the problem at every time step value can be performed using a quasi newton method that allows to decouple fluid velocity and director vector computations for every tangent problem then we have designed a linear semi implicit algorithm i e it does not involve nonlinear iterations and proved that it is unconditionally stable verifying a discrete energy inequality finally some numerical simulations are provided

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

new non lie ansatze and exact solutions of nonlinear reaction diffusion convection equations a constructive method for obtaining new exact solutions of nonlinear evolution equations is further developed the method is based on the consideration of a fixed nonlinear partial differential equation together with an additional generating condition in the form of a linear high order ordinary differential equation using this method new non lie ansatze and exact solutions are obtained for two classes of diffusion equations with power and exponential nonlinearities which describe real processes in physics chemistry and biology the analysis of the found solutions and the relation of the proposed method to some approaches which have been suggested in several recently published papers are presented

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

an energy stable scheme for incompressible navier stokes equations with periodically updated coefficient matrix abstract we present an energy stable scheme for simulating the incompressible navier stokes equations based on the generalized positive auxiliary variable gpav framework in the gpav reformulated system the original nonlinear term is replaced by a linear term plus a correction term where the correction term is put under control by an auxiliary variable the proposed scheme incorporates a pressure correction type strategy into the gpav procedure and it satisfies a discrete energy stability property the scheme entails the computation of two copies of the velocity and pressure within a time step by solving an individual de coupled linear equation for each of these field variables upon discretization the pressure linear system involves a constant coefficient matrix that can be pre computed while the velocity linear system involves a coefficient matrix that is updated periodically once every k 0 time steps in the current work where k 0 is a user specified integer the auxiliary variable being a scalar valued number is computed by a well defined explicit formula which guarantees the positivity of its computed values it is observed that the current method can produce accurate simulation results at large or fairly large time step sizes for the incompressible navier stokes equations the impact of the periodic coefficient matrix update on the overall cost of the method is observed to be small in typical numerical simulations several flow problems have been simulated to demonstrate the accuracy and performance of the method developed herein

an energy law preserving c0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics in this paper we use finite element methods to simulate the hydrodynamical systems governing the motions of nematic liquid crystals in a bounded domain w we reformulate the original model in the weak form which is consistent with the continuous dissipative energy law for the flow and director fields in w 1 2 s w s 0 is an arbitrarily small number this enables us to use convenient conformal c 0 finite elements in solving the problem moreover a discrete energy law is derived for a modified midpoint time discretization scheme a fixed iterative method is used to solve the resulted nonlinear system so that a matrix free time evolution may be achieved and velocity and director variables may be solved separately a number of hydrodynamical liquid crystal examples are computed to demonstrate the effects of the parameters and the performance of the method

a mirror symmetric construction of qht g p q abstract let g be a simple simply connected complex algebraic group we give a lie theoretic construction of a conjectural mirror family associated to a general flag variety g p and show that it recovers the peterson variety presentation for the t equivariant quantum cohomology rings q h t g p q with quantum parameters inverted for sl n b we relate our construction to the mirror family defined by givental and its t equivariant analogue due to joe and kim

the non classical solution of the inhomogeneous non linear diffusion equation a complete classification of the symmetry reductions of the inhomogeneous nonlinear diffusion inld equation using the non classical method leads to one family of symmetry generators the resulting ordinary differential equation is of second order and non linear new solutions are obtained for some physical processes corresponding to particular values of n and p m

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

an eigen based high order expansion basis for structured spectral elements we present an eigen based high order expansion basis for the spectral element approach with structured elements the new basis exhibits a numerical efficiency significantly superior in terms of the conditioning of coefficient matrices and the number of iterations to convergence for the conjugate gradient solver to the commonly used jacobi polynomial based expansion basis this basis results in extremely sparse mass matrices and it is very amenable to the diagonal preconditioning ample numerical experiments demonstrate that with the new basis and a simple diagonal preconditioner the number of conjugate gradient iterations to convergence has essentially no dependence or only a very weak dependence on the element order the expansion bases are constructed by a tensor product of a set of special one dimensional 1d basis functions the 1d interior modes are constructed such that the interior mass and stiffness matrices are simultaneously diagonal and have identical condition numbers the 1d vertex modes are constructed to be orthogonal to all the interior modes the performance of the new basis has been investigated and compared with other expansion bases

preliminary group classification of equations vtt f x vx vxx g x vx a classification is given of equations vtt f v vx vxx g x vx admitting an extension by one of the principal lie algebra of the equation under consideration the paper is one of few applications of a new algebraic approach to the problem of group classification the method of preliminary group classification the result of the work is a wide class of equations summarized in table ii

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

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

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

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

wall bounded multiphase flows of n immiscible incompressible fluids we present an effective method for simulating wall bounded multiphase flows consisting of n n2 immiscible incompressible fluids with different densities viscosities and pairwise surface tensions the n phase physical formulation is based on a modified thermodynamically consistent phase field model that is more general than in a previous work and it is developed by considering the reduction consistency if some of the fluid components were absent from the system we propose an n phase contact angle boundary condition that is reduction consistent between n phases and m phases 2mn1 we also present a numerical algorithm for solving the n phase governing equations together with the contact angle boundary conditions developed herein extensive numerical experiments are presented for several flow problems involving multiple fluid components and solid wall boundaries to investigate the wettability effects with multiple types of contact angles in particular we compare simulation results with the de gennes theory for the contact angle effects on the liquid drop spreading on wall surfaces and demonstrate that our method produces physically accurate results

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

provably unconditionally stable second order time accurate mixed variational methods for phase field models we introduce provably unconditionally stable mixed variational methods for phase field models our formulation is based on a mixed finite element method for space discretization and a new second order accurate time integration algorithm the fully discrete formulation inherits the main characteristics of conserved phase dynamics namely mass conservation and nonlinear stability with respect to the free energy we illustrate the theory with the cahn hilliard equation but our method may be applied to other phase field models we also propose an adaptive time stepping version of the new time integration method we present some numerical examples that show the accuracy stability and robustness of the new method

a free energy stable nodal discontinuous galerkin approximation with summation by parts property for the cahn hilliard equation abstract we present a nodal discontinuous galerkin dg scheme for the cahn hilliard equation that satisfies the summation by parts simultaneous approximation term sbp sat property the latter permits us to show that the discrete free energy is bounded and as a result the scheme is provably stable the scheme and the stability proof are presented for general curvilinear three dimensional hexahedral meshes we use the bassi rebay 1 br1 scheme to compute interface fluxes and a first order implicit explicit imex scheme to integrate in time we provide a semi discrete stability study and a fully discrete proof subject to a positivity condition on the solution lastly we test the theoretical findings using numerical cases that include two and three dimensional problems

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

the global dynamics of discrete semilinear parabolic equations a class of scalar semilinear parabolic equations possessing absorbing sets a lyapunov functional and a global attractor are considered the gradient structure of the problem implies that provided all steady states are isolated solutions approach a steady state as t to infty the dynamical properties of various finite difference and finite element schemes for the equations are analysed the existence of absorbing sets bounded independently of the mesh size is proved for the numerical methods discrete lyapunov functions are constructed to show that under appropriate conditions on the mesh parameters numerical orbits approach steady state solutions as discrete time increases however it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented furthermore it is also shown that the explicit euler scheme introduces spurious solutions with period 2 in the timestep as a result the absorbing set is destroyed and there is initial data leading to blow up of the scheme however small the mesh parameters are taken to obtain stabilization to a steady state for this scheme it is necessary to restrict the timestep in terms of the initial data and the space step implicit schemes are constructed for which absorbing sets and lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step both one step and multistep bdf methods are studied

exact solutions of a class of nonlinear boundary value problems with moving boundaries presents a method for the construction of the exact solutions of nonlinear bvps in melting and evaporation processes the case of smooth functions of thermal conductivity is studied and illustrations of some specific nonlinearities given

exact solutions of nonlinear boundary value problems of the stefan type the 1 1 dimensional nonlinear boundary value problem modelling the process of melting and evaporation of metals is studied by means of the classical lie symmetry method all possible lie operators of the nonlinear heat equation which allow us to reduce the problem to the boundary value problem for the system of ordinary differential equations are found the forms of heat conductivity coefficients are established when the given problem can be analytically solved in an explicit form

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

on linear schemes for a cahn hilliard diffuse interface model numerical schemes to approximate the cahn hilliard equation have been widely studied in recent times due to its connection with many physically motivated problems in this work we propose two type of linear schemes based on different ways to approximate the double well potential term the first idea developed in the paper allows us to design a linear numerical scheme which is optimal from the numerical dissipation point of view meanwhile the second one allows us to design unconditionally energy stable linear schemes for a modified energy we present first and second order in time linear schemes to approximate the ch problem detailing their advantages over other linear schemes that have been previously introduced in the literature furthermore we compare all the schemes through several computational experiments

lie symmetries and reductions of multi dimensional boundary value problems of the stefan type a new definition of lie invariance for nonlinear multi dimensional boundary value problems bvps is proposed by the generalization of known definitions to much wider classes of bvps the class of 1 3 dimensional nonlinear bvps of the stefan type modeling the process of melting and evaporation of metals is studied in detail using the definition proposed the group classification problem for this class of bvps is solved and some reductions with physical meaning to bvps of lower dimensionality are made examples of how to construct exact solutions of the 1 3 dimensional nonlinear bvp with the correctly specified coefficients are presented

error analysis of full discrete invariant energy quadratization schemes for the cahn hilliard type equation abstract in this paper we present the error analysis for a fully discrete scheme of the cahn hilliard type equation along with numerical verifications the numerical schedule is developed by first transforming the cahn hilliard type equation into an equivalent form using the invariant energy quadratization ieq technique then the equivalent form is discretized by using the linear implicit crank nicolson method for the time variable and the fourier pseudo spectral method for the spatial variables the resulted full discrete scheme is linear and unconditionally energy stable which makes it easy to implement by constructing an appropriate interpolation equation the uniform boundedness of the numerical solution is obtained theoretically then we prove that the numerical solutions converge with the order o t 2 h m where t is the temporal step and h is the spatial step with m the regularity of the exact solution several numerical examples are presented to confirm the theoretical results and demonstrate the effectiveness of the presented linear scheme the numerical strategies and analytical tools in this paper could be readily applied to study other phase field models or gradient flow problems

linear first and second order unconditionally energy stable numerical schemes for the phase field model of homopolymer blends abstract in this paper we develop a series of efficient numerical schemes to solve the phase field model for homopolymer blends the governing system is derived from the energetic variational approach of a total free energy that consists of a nonlinear logarithmic flory huggins potential and a gradient entropy with a concentration dependent de gennes type coefficient the main challenging issue to solve this kind of models numerically is about the time marching problem 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 the first and second order temporal approximation schemes based on the invariant energy quadratization method where all nonlinear terms are treated semi explicitly consequently the resulting numerical schemes lead to a symmetric positive definite linear system to be solved at each time step the unconditional energy stabilities are further proved various numerical simulations of 2d and 3d are presented to demonstrate the stability and the accuracy of the proposed schemes

fully discrete finite element approximations of the navier stokes cahn hilliard diffuse interface model for two phase fluid flows this paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the navier stokes equations and the cahn hilliard equation which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids in the model the two sets of equations are coupled through an extra phase induced stress term in the navier stokes equations and a fluid induced transport term in the cahn hilliard equation fully discrete mixed finite element methods are proposed for approximating the coupled system it is shown that the proposed numerical methods satisfy a mass conservation law and a discrete energy law which is analogous to the basic energy law for the phase field model the convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law as a by product the convergence result also provides a constructive proof of the existence of weak solutions to the navier stokes cahn hilliard phase field model numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

a gpav based unconditionally energy stable scheme for incompressible flows with outflow open boundaries abstract we present an unconditionally energy stable scheme for approximating the incompressible navier stokes equations on domains with outflow open boundaries the scheme combines the generalized positive auxiliary variable gpav approach and a rotational velocity correction type strategy and the adoption of the auxiliary variable simplifies the numerical treatment for the open boundary conditions the discrete energy stability of the proposed scheme has been proven irrespective of the time step sizes within each time step the scheme entails the computation of two velocity fields and two pressure fields by solving an individual de coupled helmholtz including poisson type equation with a constant pre computable coefficient matrix for each of these field variables the auxiliary variable being a scalar number is given by a well defined explicit formula within a time step which ensures the positivity of its computed values extensive numerical experiments with several flows involving outflow open boundaries in regimes where the backflow instability becomes severe have been presented to test the performance of the proposed method and to demonstrate its stability at large time step sizes

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

new classes of similarity solutions of the inhomogeneous nonlinear diffusion equations the lie similarity method has been used to extend the similarity solutions of the one dimensional inhomogeneous nonlinear diffusion equations we determine the lie point symmetry vector fields and calculate the similarity ansatz then we discuss the resulting nonlinear ordinary differential equations exact solutions are found and their relations to some real physical situations 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

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29