a second order operator splitting method for allen cahn type equations with nonlinear source terms abstract allen cahn ac type equations with nonlinear source terms have been applied to a wide range of problems for example the vector valued ac equation for phase separation and the phase field equation for dendritic crystal growth in contrast to the well developed first and second order methods for the ac equation not many second order methods are suggested for the ac type equations with nonlinear source terms due to the difficulties in dealing with the nonlinear source term numerically in this paper we propose a simple and stable second order operator splitting method a core idea of the method is to decompose the original equation into three subequations with the free energy evolution term the heat evolution term and a nonlinear source term respectively it is important to combine these three subequations in proper order to achieve the second order accuracy and stability we propose a method with a half time free energy evolution solver a half time heat evolution solver a full time midpoint solver for the nonlinear source term and a half time heat evolution solver followed by a final half time free energy evolution solver we numerically demonstrate the second order accuracy of the new numerical method through the simulations of the phase separation and the dendritic crystal growth

first and second order operator splitting methods for the phase field crystal equation in this paper we present operator splitting methods for solving the phase field crystal equation which is a model for the microstructural evolution of two phase systems on atomic length and diffusive time scales a core idea of the methods is to decompose the original equation into linear and nonlinear subequations in which the linear subequation has a closed form solution in the fourier space we apply a nonlinear newton type iterative method to solve the nonlinear subequation at the implicit time level and thus a considerably large time step can be used by combining these subequations we achieve the first and second order accuracy in time we present numerical experiments to show the accuracy and efficiency of the proposed methods

global weak solution to the inhomogeneous navier stokes vlasov equations abstract the inhomogeneous navier stokes vlasov equations for fluid particle flows are considered in the three dimensional space the coupling in the fluid particle system arises from the drag force in the fluid equations and the acceleration in the vlasov equation an initial boundary value problem is studied in a bounded domain with large initial data the existence of global weak solution is established through an approximation scheme a fixed point argument energy estimates and a weak convergence method

high accuracy solutions to energy gradient flows from material science models a computational framework is presented for materials science models that come from energy gradient flows the models of interest lead to the evolution of structure involving two or more phases the framework includes higher order derivative models and vector problems solutions are considered in periodic cells and standard fourier spectral discretization in space is used implicit time stepping is used with adaptivity based on local error estimates the implicit system at every time step is solved iteratively with newton s method the resulting linear systems are solved in inner iterations with the conjugate gradient method using a novel preconditioner that is a constant coefficient version of the system taking values for the coefficients at the pure phase states solutions with high spatial and temporal accuracy are obtained the dependence of the condition number of the preconditioned system on the size of the time step and the order parameter in the model that represents the scaled width of transition layers between phases is investigated numerically and with formal asymptotics in a simple setting the asymptotic results require a conjecture on the rank of a modified square distance matrix results from a fast graphical processing unit implementation for a three dimensional model are shown a comparison to time stepping with operator splitting into convex and concave parts that guarantees energy decrease in the numerical scheme is done

convergence analysis and numerical implementation of a second order numerical scheme for the three dimensional phase field crystal equation abstract in this paper we analyze and implement a second order in time numerical scheme for the three dimensional phase field crystal pfc equation the numerical scheme was proposed in hu et al 2009 with the unique solvability and unconditional energy stability established however its convergence analysis remains open we present a detailed convergence analysis in this article in which the maximum norm estimate of the numerical solution over grid points plays an essential role moreover we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain and various three dimensional numerical results are presented including the numerical convergence test complexity test of the multigrid solver and the polycrystal growth simulation

an energy stable fourth order finite difference scheme for the cahn hilliard equation abstract in this paper we propose and analyze an energy stable numerical scheme for the cahn hilliard equation with second order accuracy in time and the fourth order finite difference approximation in space in particular the truncation error for the long stencil fourth order finite difference approximation over a uniform numerical grid with a periodic boundary condition is analyzed via the help of discrete fourier analysis instead of the standard taylor expansion this in turn results in a reduced regularity requirement for the test function in the temporal approximation we apply a second order bdf stencil combined with a second order extrapolation formula applied to the concave diffusion term as well as a second order artificial douglas dupont regularization term for the sake of energy stability as a result the unique solvability energy stability are established for the proposed numerical scheme and an optimal rate convergence analysis is derived in the l 0 t l 2 l 2 0 t h h 2 norm a few numerical experiments are presented which confirm the robustness and accuracy of the proposed scheme

stable and efficient finite difference nonlinear multigrid schemes for the phase field crystal equation in this paper we present and compare two unconditionally energy stable finite difference schemes for the phase field crystal equation the first is a one step scheme based on a convex splitting of a discrete energy by wise et al s m wise c wang j s lowengrub an energy stable and convergent finite difference scheme for the phase field crystal equation siam j numer anal in press in this scheme which is first order in time and second order in space the discrete energy is non increasing for any time step the second scheme we consider is a new fully second order two step algorithm in the new scheme the discrete energy is bounded by its initial value for any time step in both methods the equations at the implicit time level are nonlinear but represent the gradients of strictly convex functions and are thus uniquely solvable regardless of time step size we solve the nonlinear equations using an efficient nonlinear multigrid method numerical simulations are presented and confirm the stability efficiency and accuracy of the schemes

convergence of a fast explicit operator splitting method for the epitaxial growth model with slope selection a fast explicit operator splitting method for the epitaxial growth model with slope selection has been presented in cheng et al j comput phys 303 2015 pp 45 65 the original problem is split into linear and nonlinear subproblems for the linear part the pseudospectral method is adopted for the nonlinear part a 33 point difference scheme is constructed here we give a compact center difference scheme involving fewer points for the nonlinear subproblem in addition we analyze the convergence rate of the algorithm the global error order mathcal o tau 2 h 4 in discrete l 2 norm is proved theoretically and verified numerically some numerical experiments show the robustness of the algorithm for small coefficients of the fourth order term for the one dimensional case in addition coarsening dynamics are simulated in large domains and the 1 3 power laws are observed for the two dimensional case

relationship of eigenvalues for usaor iterative method applied to a class of p cyclic matrices a new relationship of eigenvalues between the jacobi iterative matrix and the usaor iterative matrix for a block p cyclic case is derived many known results are its special cases

analysis of a symmetric extrapolated iterative method for solving linear systems abstract we investigate the symmetric extrapolated gauss seidel ii method for solving large linear systems the theoretical results demonstrate that the method plus the semi iterative method is competitive with the sor method and in particular when p lu 1 4 it shows an order of magnitude improvement over the sor method our analysis leads us to form the opinion that in some cases egs2 si would be better than ssor si

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

an energy stable and convergent finite difference scheme for the phase field crystal equation we present an unconditionally energy stable finite difference scheme for the phase field crystal equation the method is based on a convex splitting of a discrete energy and is semi implicit the equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable regardless of time step size we present local in time error estimates that ensure the convergence of the scheme while this paper is primarily concerned with the phase field crystal equation most of the theoretical results hold for the related swift hohenberg equation as well

a new conservative swift hohenberg equation and its mass conservative method abstract the swift hohenberg sh energy functional has been widely used to study pattern formation the l 2 and h 1 gradient flows for the sh energy functional are the sh and phase field crystal pfc equations respectively the sh equation is of lower order in space than the pfc equation but does not conserve the total mass furthermore when the sh energy functional does not have the cubic nonlinearity the sh equation unlike the pfc equation only shows striped patterns even for various parameter values in this study we introduce a new mass conservative sh equation by using a lagrange multiplier in order to solve the conservative sh equation that is an integro partial differential equation we propose operator splitting methods that are shown analytically to inherit the mass conservation numerical examples including standard tests in the pfc equation are presented to show the applicability of the proposed framework

an unconditionally energy stable second order finite element method for solving the allen cahn equation abstract in this paper we design analyze and numerically validate an unconditionally energy stable second order numerical method for solving the allen cahn equation which represents a model for anti phase domain coarsening in a binary alloy the proposed scheme inherits the property of the decrease of the total energy from the allen cahn equation an error estimate for the fully discretized scheme is also established numerical examples are presented to confirm the accuracy efficiency and stability of the proposed method in particular the method is shown to be unconditionally energy stable and second order accurate in both time and space discretizations

fast and accurate coarsening simulation with an unconditionally stable time step we present cahn hilliard and allen cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy controlled simulation our stability analysis is based on eyre s theorem and unconditional von neumann stability analysis both of which we present numerical tests confirm the accuracy of the von neumann approach which is straightforward and should be widely applicable in phase field modeling for the cahn hilliard case we show that accuracy can be controlled with an unbounded time step dt that grows with time t as dt t a we develop a classification scheme for the step exponenta and demonstrate that a class of simple linear algorithms gives a51 3 for this class the speedup relative to a fixed time step grows with n the linear size of the system as n lnn with conservative choices for the parameters controlling accuracy and finite size effects we find that an 8192 2 lattice can be integrated 300 times faster than with the euler method

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

simulation of the phase field cahn hilliard and tumor growth models via a numerical scheme element free galerkin method abstract the main aim of this research work is to find the numerical solution based on a meshless technique for both the time dependent cahn hilliard and tumor growth partial differential equations the temporal variable is discretized using a second order method based on semi implicit backward differential formula and the stabilized term is added to the considered time discretization also an adaptive time algorithm is used to reduce the number of iterations of the proposed time discretization besides to approximate the spatial variables the element free galerkin method is considered for both mathematical models the result of full discrete schemes in studied models is solved via biconjugate gradient stabilized algorithm some numerical simulations are reported to show the capability of the numerical scheme presented here

convex splitting rungekutta methods for phase field models in this paper we present the convex splitting rungekutta csrk methods which provide a simple unified framework to solve phase field models such as the allencahn cahnhilliard and phase field crystal equations the core idea of the csrk methods is the combination of convex splitting methods and multi stage implicitexplicit rungekutta methods our csrk methods are high order accurate in time and we investigate the energy stability numerically we present numerical experiments to show the accuracy and efficiency of the proposed methods up to the third order accuracy

a simple and efficient finite difference method for the phase field crystal equation on curved surfaces abstract we present a simple and efficient finite difference method for the phase field crystal pfc equation on curved surfaces embedded in r 3 we employ a narrow band neighborhood of a curved surface that is defined as a zero level set of a signed distance function the pfc equation on the surface is extended to the three dimensional narrow band domain by using the closest point method and applying a pseudo neumann boundary condition we can use the standard seven point discrete laplacian operator instead of the discrete laplace beltrami operator on the surface the pfc equation on the narrow band domain is discretized using an unconditionally stable scheme and the resulting implicit discrete system of equations is solved by using the jacobi iterative method computational results are presented to demonstrate the efficiency and usefulness of the proposed method

first and second order numerical methods based on a new convex splitting for phase field crystal equation the phase field crystal equation derived from the swifthohenberg energy functional is a sixth order nonlinear equation we propose numerical methods based on a new convex splitting for the phase field crystal equation the first order convex splitting method based on the proposed splitting is unconditionally gradient stable which means that the discrete energy is non increasing for any time step the second order scheme is unconditionally weakly energy stable which means that the discrete energy is bounded by its initial value for any time step we prove mass conservation unique solvability energy stability and the order of truncation error for the proposed methods numerical experiments are presented to show the accuracy and stability of the proposed splitting methods compared to the existing other splitting methods numerical tests indicate that the proposed convex splitting is a good choice for numerical methods of the phase field crystal equation

an efficient and stable compact fourth order finite difference scheme for the phase field crystal equation abstract in this paper we present a high order accurate compact scheme for the phase field crystal model in two and three dimensional spaces the proposed scheme is derived by combining a fourth order compact finite difference formula in space and a backward differentiation for the time derivative term which is second order accurate in time furthermore a nonlinearly stabilized splitting scheme is used and thus a larger time step can be allowed since the equations at the implicit time level are nonlinear we introduce a newton type iterative method and employ a fast and efficient nonlinear multigrid solver to solve the resulting discrete system in particular we implement the compact scheme in the adaptive mesh refinement framework an adaptive time step method for the phase field crystal model is also proposed various numerical experiments are presented and confirm the accuracy stability and efficiency of our proposed method

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 mass conservative adaptive fas multigrid solver for cell centered finite difference methods on block structured locally cartesian grids abstract we present a mass conservative full approximation storage fas multigrid solver for cell centered finite difference methods on block structured locally cartesian grids the algorithm is essentially a standard adaptive fas afas scheme but with a simple modification that comes in the form of a mass conservative correction to the coarse level force this correction is facilitated by the creation of a zombie variable analogous to a ghost variable but defined on the coarse grid and lying under the fine grid refinement patch we show that a number of different types of fine level ghost cell interpolation strategies could be used in our framework including low order linear interpolation in our approach the smoother prolongation and restriction operations need never be aware of the mass conservation conditions at the coarse fine interface to maintain global mass conservation we need only modify the usual fas algorithm by correcting the coarse level force function at points adjacent to the coarse fine interface we demonstrate through simulations that the solver converges geometrically at a rate that is h independent and we show the generality of the solver applying it to several nonlinear time dependent and multi dimensional problems in several tests we show that second order asymptotic h 0 convergence is observed for the discretizations provided that 1 at least linear interpolation of the ghost variables is employed and 2 the mass conservation corrections are applied to the coarse level force term

unconditionally stable methods for gradient flow using convex splitting runge kutta scheme abstract we propose a convex splitting runge kutta csrk scheme which provides a simple unified framework to solve a gradient flow in an unconditionally gradient stable manner the key feature of the scheme is a combination of a convex splitting method and a specially designed multi stage two additive runge kutta method our methods are high order accurate in time and assure the gradient energy stability for any time step size we provide detailed proof of the unconditional energy stability and present issues on the practical implementations we demonstrate the accuracy and stability of the proposed methods using numerical experiments of the cahn hilliard equation

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

linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints abstract we present a set of linear second order unconditionally energy stable schemes for the allen cahn model with nonlocal constraints for crystal growth that conserves the mass of each phase solvability conditions are established for the linear systems resulting from the schemes convergence rates are verified numerically dynamics obtained using the allen cahn model with nonlocal constraints are compared with the one obtained using the classic allen cahn model as well as the cahn hilliard model respectively demonstrating slower dynamics than that of the allen cahn model but faster dynamics than that of the cahn hilliard model thus the allen cahn model with nonlocal constraints can serve as an alternative to the cahn hilliard model in simulating crystal growth while conserving the mass of each phase two benchmark examples are presented to contrast the predictions made with the four models highlighting the accuracy and effectiveness of the allen cahn model with nonlocal constraints

a semi analytical fourier spectral method for the swift hohenberg equation abstract the swift hohenberg sh equation has been widely used as a model for the study of pattern formation the sh equation is a fourth order nonlinear partial differential equation and cannot generally be solved analytically therefore computer simulations play an essential role in understanding of nonequilibrium processing the aim of this paper is to present first and second order semi analytical fourier spectral methods as an accurate and efficient approach for solving the sh equation the methods are based on the operator splitting method and are to split the sh equation into linear and nonlinear subequations the linear and nonlinear subequations have closed form solutions in the fourier and physical spaces respectively the methods are simple to implement and computationally cheap to achieve high order time accuracy numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods

on the convergence of some generalized preconditioned iterative methods this paper generalizes the preconditioning techniques introduced by evans 2 for the numerical solution of the linear system au b and defines two extrapolated versions of the gauss siedel method as well as an extrapolated version of the successive overrelaxation method finally it establishes some convergence theorems for the considered iterative schemes when a has particular properties such as being consistently ordered positive definite having weak diagonal dominance etc

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

existence of global weak solutions to a hybrid vlasov mhd model for magnetized plasmas we prove the global in time existence of large data nite energy weak solutions to an incompressible hybrid vlasov magnetohydrodynamic model in three space dimensions the model couples three essential ingredients of magnetized plasmas a transport equation for the probability density function which models energetic rare ed particles of one species the incompressible navier stokes system for the bulk uid and a parabolic evolution equation involving magnetic di usivity for the magnetic eld the physical derivation of our model is given it is also shown that the weak solution whose existence is established has nonincreasing total energy and that it satis es a number of physically relevant properties including conservation of the total momentum conservation of the total mass and nonnegativity of the probability density function for the energetic particles the proof is based on a one level approximation scheme which is carefully devised to avoid increase of the total energy for the sequence of approximating solutions in conjunction with a weak compactness argument for the sequence of approximating solutions the key technical challenges in the analysis of the mathematical model are the nondissipative nature of the vlasov type particle equation and passage to the weak limits in the multilinear coupling terms

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 semi analytical fourier spectral method for the allen cahn equation in recent years fourier spectral methods have been widely used as a powerful tool for solving phase field equations to improve its effectiveness many researchers have employed stabilized semi implicit fourier spectral sifs methods which allow a much larger time step than a usual explicit scheme our mathematical analysis and numerical experiments however suggest that an effective time step is smaller than a time step specified in the sifs schemes in consequence the sifs scheme is inaccurate for a considerably large time step and may lead to incorrect morphologies in phase separation processes in order to remove the time step constraint and guarantee the accuracy in time for a sufficiently large time step we present a first and a second order semi analytical fourier spectral safs methods for solving the allen cahn equation the core idea of the methods is to decompose the original equation into linear and nonlinear subequations which have closed form solutions in the fourier and physical spaces respectively both the first and the second order methods are unconditionally stable and numerical experiments demonstrate that our proposed methods are more accurate than the stabilized semi implicit fourier spectral method

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

an energy stable time integrator for phase field models we introduce a provably energy stable time integration method for general classes of phase field models with polynomial potentials we demonstrate how taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy stability implicitly which are second order accurate in time the spatial discretization relies on a mixed finite element formulation and isogeometric analysis we also propose an adaptive time stepping discretization that relies on a first order backward approximation to give an error estimator this error estimator is accurate robust and does not require the computation of extra solutions to estimate the error this methodology can be applied to any second order accurate time integration scheme we present numerical examples in two and three spatial dimensions which confirm the stability and robustness of the method the implementation of the numerical schemes is done in petiga a high performance isogeometric analysis framework

maximally fast coarsening algorithms we present maximally fast numerical algorithms for conserved coarsening systems that are stable and accurate with a growing natural time step deltat at2 3s we compare the scaling structure obtained from our maximally fast conserved systems directly against the standard fixed time step euler algorithm and find that the error scales as square root of a so arbitrary accuracy can be achieved for nonconserved systems only effectively finite time steps are accessible for similar unconditionally stable algorithms

n th order operator splitting schemes and nonreversible systems this paper is concerned with partitioned nth order accurate split operator schemes built using m distinct solution operators for semidiscrete equations of the form frac d dt u j f j 1 u k cdots f j m u k which arise among others from constant coefficient parabolic partial differential equations we prove that for every n geq 3 and m geqslant 2 each solution operator must be applied for at least one backward fractional time step during each complete time step this result has important consequences for applications to the complex ginzburg landau equation with periodic boundary conditions and other partial differential equations with both reversible and nonreversible components furthermore for the special case of n 3 and m 2 we analytically determine all possible schemes

energy stable compact scheme for cahn hilliard equation with periodic boundary condition abstract we present a compact scheme to solve the cahn hilliard equation with a periodic boundary condition which is fourth order accurate in space we introduce schemes for two and three dimensions which are derived from the one dimensional compact stencil the energy stability is completely proven for the proposed scheme based on the application of the compact method and well known convex splitting methods detailed proofs of the mass conservation and unique solvability are also established numerical experiments are presented to demonstrate the accuracy and stability of the proposed methods

on the convergence of the saor method and the error bounds for its acceleration given a linear system with hermitian positive definite coefficient matrix a a splitting of varga s type 1 is considered and the corresponding generalised saor scheme is presented we obtain the convergence theorem for the saor method and give the error bound for the a norm of the error vector for the saor semi iterative method hence extending the saor theory

an efficient algorithm for solving the phase field crystal model we present and discuss the development of an unconditionally stable algorithm used to solve the evolution equations of the phase field crystal pfc model this algorithm allows for an arbitrarily large algorithmic time step as the basis for our analysis of the accuracy of this algorithm we determine an effective time step in fourier space we then compare our calculations with a set of representative numerical results and demonstrate that this algorithm is an effective approach for the study of the pfc models yielding a time step effectively 180 times larger than the euler algorithm for a representative set of material parameters as the pfc model is just a simple example of a wide class of density functional theories we expect this method will have wide applicability to modeling systems of considerable interest to the materials modeling communities

global weak solutions of vlasov maxwell systems we study here the vlasov maxwell system in its classical and relativistic form we prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data the main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work