markowitz s mean variance portfolio selection with regime switching a continuous time model a continuous time version of the markowitz mean variance portfolio selection model is proposed and analyzed for a market consisting of one bank account and multiple stocks the market parameters including the bank interest rate and the appreciation and volatility rates of the stocks depend on the market mode that switches among a finite number of states the random regime switching is assumed to be independent of the underlying brownian motion this essentially renders the underlying market incomplete a markov chain modulated diffusion formulation is employed to model the problem using techniques of stochastic linear quadratic control mean variance efficient portfolios and efficient frontiers are derived explicitly in closed forms based on solutions of two systems of linear ordinary differential equations related issues such as a minimum variance portfolio and a mutual fund theorem are also addressed all the results are markedly different from those for the case when there is no regime switching an interesting observation is however that if the interest rate is deterministic then the results exhibit rather unexpected similarity to their no regime switching counterparts even if the stock appreciation and volatility rates are markov modulated

a weak galerkin finite element method with polynomial reduction the weak galerkin wg is a novel numerical method based on variational principles for weak functions and their weak partial derivatives defined as distributions in the implementation the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom the accuracy and the computational complexity of the corresponding wg scheme is significantly impacted by the selection of such polynomials this paper presents an optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation for illustrative purpose the authors use the second order elliptic equation to demonstrate the basic ideas of polynomial reduction consequently a new weak galerkin finite element method is proposed and analyzed error estimates of optimal order are established for the corresponding wg approximations in both a discrete h 1 norm and the standard l 2 norm in addition the paper presents some numerical results to demonstrate the power of the wg method in dealing with finite element partitions with arbitrary polygons in 2d or polyhedra in 3d the numerical examples include various finite element partitions such as triangular mesh quadrilateral mesh honeycomb mesh in 2d and mesh with deformed cubes in 3d

high order maximum principle preserving semi lagrangian finite difference weno schemes for the vlasov equation in this paper we propose the parametrized maximum principle preserving mpp flux limiter originally developed in 37 to the semi lagrangian finite difference weighted essentially non oscillatory scheme for solving the vlasov equation the mpp flux limiter is proved to maintain up to fourth order accuracy for the semi lagrangian finite difference scheme without any time step restriction numerical studies on the vlasov poisson system demonstrate the performance of the proposed method and its ability in preserving the positivity of the probability distribution function while maintaining the high order accuracy

on the performance of exponential integrators for problems in magnetohydrodynamics exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems while the results of these investigations showed that exponential integrators can provide significant computational savings the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages resistive magnetohydrodynamic mhd modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas in many problems numerical solution of mhd equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest in this paper we evaluate the performance of exponential integrators on large mhd problems and compare them to a state of the art implicit time integrator both the variable and constant time step exponential methods of epirk type are used to simulate magnetic reconnection and the kevin helmholtz instability in plasma performance of these methods which are part of the epic software package is compared to the variable time step variable order bdf scheme included in the cvode part of sundials library we study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as reynolds lundquist and prandtl numbers we find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for mhd problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as mhd

primal dual weak galerkin finite element methods for elliptic cauchy problems abstract the authors propose and analyze a well posed numerical scheme for a type of ill posed elliptic cauchy problem by using a constrained minimization approach combined with the weak galerkin finite element method the resulting euler lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable and is thus an example of the primal dual weak galerkin finite element method this new primal dual weak galerkin algorithm is consistent in the sense that the system is symmetric well posed and is satisfied by the exact solution a certain stability and error estimates were derived in discrete sobolev norms including one in a weak l 2 topology some numerical results are reported to illustrate and validate the theory developed in the paper

a nystrom based finite element method on polygonal elements abstract we consider families of finite elements on polygonal meshes that are defined implicitly on each mesh cell as solutions of local poisson problems with polynomial data functions in the local space on each mesh cell are evaluated via nystrom discretizations of associated integral equations allowing for curvilinear polygons and non polynomial boundary data several experiments demonstrate the approximation quality of interpolated functions in these spaces

coldice a parallel vlasov poisson solver using moving adaptive simplicial tessellation abstract resolving numerically vlasov poisson equations for initially cold systems can be reduced to following the evolution of a three dimensional sheet evolving in six dimensional phase space we describe a public parallel numerical algorithm consisting in representing the phase space sheet with a conforming self adaptive simplicial tessellation of which the vertices follow the lagrangian equations of motion the algorithm is implemented both in six and four dimensional phase space refinement of the tessellation mesh is performed using the bisection method and a local representation of the phase space sheet at second order relying on additional tracers created when needed at runtime in order to preserve in the best way the hamiltonian nature of the system refinement is anisotropic and constrained by measurements of local poincare invariants resolution of poisson equation is performed using the fast fourier method on a regular rectangular grid similarly to particle in cells codes to compute the density projected onto this grid the intersection of the tessellation and the grid is calculated using the method of franklin and kankanhalli 65 66 67 generalised to linear order as preliminary tests of the code we study in four dimensional phase space the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuation composed of two sinusoidal waves we also perform a warm dark matter simulation in six dimensional phase space that we use to check the parallel scaling of the code

a class of quadrature based moment closure methods with application to the vlasov poisson fokker planck system in the high field limit quadrature based moment closure methods are a class of approximations that replace high dimensional kinetic descriptions with lower dimensional fluid models in this work we investigate some of the properties of a sub class of these methods based on bi delta bi gaussian and bi b spline representations we develop a high order discontinuous galerkin dg scheme to solve the resulting fluid systems finally via this high order dg scheme and strang operator splitting to handle the collision term we simulate the fluid closure models in the context of the vlasov poisson fokker planck system in the high field limit we demonstrate numerically that the proposed scheme is asymptotic preserving in the high field limit

a locking free weak galerkin finite element method for elasticity problems in the primal formulation this paper presents an arbitrary order locking free numerical scheme for linear elasticity on general polygonal polyhedral partitions by using weak galerkin wg finite element methods like other wg methods the key idea for the linear elasticity is to introduce discrete weak strain and stress tensors which are defined and computed by solving inexpensive local problems on each element such local problems are derived from weak formulations of the corresponding differential operators through integration by parts locking free error estimates of optimal order are derived in a discrete h 1 norm and the usual l 2 norm for the approximate displacement when the exact solution is smooth numerical results are presented to demonstrate the efficiency accuracy and the locking free property of the weak galerkin finite element method

a weak galerkin finite element method for a coupled stokes darcy problem on general meshes abstract in this paper we introduce and analyze a weak galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow flows are governed by the stokes equations in primal velocity pressure formulation and darcy equation in the second order primary formulation respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law by using the weak galerkin approach we consider the two dimensional problem with the usual polynomials of degree k 1 for the velocity and hydraulic head while polynomials of degree k 1 for the pressure the velocity and hydraulic head is enhanced by polynomials of degree k 1 on the edge of a finite element partition this new method has a lot of attractive computational features more general finite element partitions of arbitrary polygons with certain shape regularity fewer degrees of freedom and parameter free stability and error estimates of optimal order are obtained moreover numerical experiences are presented to illustrate the good performance confirm the optimal order of convergence and verify the efficiency of the proposed weak galerkin method in this paper we also deal with finite element partitions consisting of general meshes such as triangular mesh quadrilateral mesh hexagonal dominant mesh and voronoi mesh for the numerical weak galerkin approximation

maximum principle satisfying second order discontinuous galerkin schemes for convection diffusion equations on triangular meshes we propose second order accurate discontinuous galerkin dg schemes which satisfy a strict maximum principle for general nonlinear convection diffusion equations on unstructured triangular meshes motivated by genuinely high order maximum principle satisfying dg schemes for hyperbolic conservation laws perthame 1996 and zhang 2010 14 26 we prove that under suitable time step restriction for forward euler time stepping for general nonlinear convection diffusion equations the same scaling limiter coupled with second order dg methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy similar to the purely convection cases the limiters are mass conservative and easy to implement strong stability preserving ssp high order time discretizations will keep the maximum principle following the idea in zhang 2012 30 we extend the schemes to two dimensional convection diffusion equations on triangular meshes there are no geometric constraints on the mesh such as angle acuteness numerical results including incompressible navier stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods

a locking free numerical approximation for quasilinear poroelasticity problems abstract this paper is devoted to a new formulation which couples the weak galerkin method and the finite element method for approximating solutions of the equations of quasi static poroelasticity which model flow through elastic porous media it is assumed that the permeability of the elastic matrix depends nonlinearly on the dilatation of the porous medium the steady state version of the system is recast in terms of displacement pressure and volumetric stress and the well posedness of both the continuous and discrete three field formulations is proved error estimates for the proposed numerical method are obtained these show that the method is locking free numerical experiments presented further demonstrate the accuracy and the locking free characteristic of the proposed numerical method

on the benney lin and kawahara equations abstract we establish global well posedness for the initial value problem ivp associated to the so called benney lin equation this model is a korteweg de vries equation perturbed by dissipative and dispersive terms which appears in fluid dynamics we also study the limiting behaviour of solutions to this ivp when the parameters of the perturbed terms approach 0

weak galerkin finite element method for biots consolidation problem in this paper a fully discrete weak galerkin wg finite element method is proposed to solve biots consolidation problem where weakly defined gradient and divergence operators over discontinuous functions are introduced plpl l1 finite element combination is used for the displacement and pressure approximations in the interior of the elements and plpl1 combination for the corresponding trace approximations on the interfaces of the finite element partition the existence and uniqueness of the discrete linear system at each time step is derived and error estimates for the approximation of displacement and pressure are obtained numerical experiments confirm the theoretical results and show that the proposed wg method is capable of overcoming pressure oscillations

arbitrarily high order convected scheme solution of the vlasov poisson system abstract the convected scheme cs is a forward trajectory semi lagrangian method for solution of transport equations which has been most often applied to the kinetic description of plasmas and rarefied neutral gases in its simplest form the cs propagates the solution forward in time by advecting the so called moving cells along their characteristic trajectories and by remapping them on the mesh at the end of the time step the cs is conservative positivity preserving simple to implement and it is not subject to time step restriction to maintain stability recently guclu and hitchon 2012 1 a new methodology was introduced for reducing numerical diffusion based on a modified equation analysis the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping while the spatial accuracy was increased from 2nd to 4th order the new scheme retained the important properties of the original method and was shown to be extremely simple and efficient for constant advection problems here the cs is applied to the solution of the vlasov poisson system which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal coulomb interactions the vlasov equation is split into two constant advection equations one in configuration space and one in velocity space and high order time accuracy is achieved by proper composition of the operators the splitting procedure enables us to use the constant advection solver which we extend to arbitrarily high order of accuracy in time and space a new improved procedure is given which makes the calculation of the corrections straightforward focusing on periodic domains we describe a spectrally accurate scheme based on the fast fourier transform the proposed implementation is strictly conservative and positivity preserving the ability to correctly reproduce the system dynamics as well as resolving small scale features in the solution is shown in classical 1d 1v test cases both in the linear and the non linear regimes

a high order time splitting method based on integral deferred correction for semi lagrangian vlasov simulations semi lagrangian schemes with various splitting methods and with different reconstruction interpolation strategies have been applied to kinetic simulations for example the order of spatial accuracy of the algorithms proposed in qiu and christlieb 2010 29 is very high as high as ninth order however the temporal error is dominated by the operator splitting error which is second order for strang splitting it is therefore important to overcome such low order splitting error in order to have numerical algorithms that achieve higher orders of accuracy in both space and time in this paper we propose to use the integral deferred correction idc method to reduce the splitting error specifically the temporal order accuracy is increased by r with each correction loop in the idc framework where r 1 2r 1 2 for coupling the first order splitting and the strang splitting respectively the proposed algorithm is applied to the vlasov poisson system the guiding center model and two dimensional incompressible flow simulations in the vorticity stream function formulation we show numerically that the idc procedure can automatically increase the order of accuracy in time we also investigate numerical stability of the proposed algorithm via performing fourier analysis to a linear model problem

a splitting approach for the magnetic schrodinger equation the schrodinger equation in the presence of an external electromagnetic field is an important problem in computational quantum mechanics it also provides a nice example of a differential equation whose flow can be split with benefit into three parts after presenting a splitting approach for three operators with two of them being unbounded we exemplarily prove first order convergence of lie splitting in this framework the result is then applied to the magnetic schrodinger equation which is split into its potential kinetic and advective parts the latter requires special treatment in order not to lose the conservation properties of the scheme we discuss several options numerical examples in one two and three space dimensions show that the method of characteristics coupled with a nonequispaced fast fourier transform nfft provides a fast and reliable technique for achieving mass conservation at the discrete level

an exponential type integrator for the kdv equation we introduce an exponential type time integrator for the kdv equation and prove its first order convergence in h 1 for initial data in h 3 furthermore we outline the generalization of the presented technique to a second order method

a new p1 weak galerkin method for the biharmonic equation abstract a new weak galerkin wg finite element method is designed featuring in using the first order polynomial to approximate solution of biharmonic equation the proposed p 1 wg method achieves o h convergence in energy norm and o h 2 in l 2 norm in solving the biharmonic equation this is not possible for the traditional finite element method as the minimum polynomial degree is 2 in order to approximate the biharmonic equation numerical tests on various polygonal meshes verify the theory

ergodic control of switching diffusions we study the ergodic control problem of switching diffusions representing a typical hybrid system that arises in numerous applications such as fault tolerant control systems flexible manufacturing systems etc under fairly general conditions we establish the existence of a stable nonrandomized markov policy which almost surely minimizes the pathwise long run average cost we then study the corresponding hamilton jacobi bellman hjb equation and establish the existence of a unique solution in a certain class using this we characterize the optimal policy as a minimizing selector of the hamiltonian associated with the hjb equations as an example we apply the results to a failure prone manufacturing system and obtain closed form solutions for the optimal policy

on the error propagation of semi lagrange and fourier methods for advection problems in this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired the numerical methods considered are based on the fast fourier transform polynomial interpolation semi lagrangian methods using a lagrange or spline interpolation and a discontinuous galerkin semi lagrangian approach which is conservative and has to store more than a single value per cell we demonstrate by carrying out numerical experiments that the worst case error estimates given in the literature provide a good explanation for the error propagation of the interpolation based semi lagrangian methods for the discontinuous galerkin semi lagrangian method however we find that the characteristic property of semi lagrangian error estimates namely the fact that the error increases proportionally to the number of time steps is not observed we provide an explanation for this behavior and conduct numerical simulations that corroborate the different qualitative features of the error in the two respective types of semi lagrangian methods the method based on the fast fourier transform is exact but due to round off errors susceptible to a linear increase of the error in the number of time steps we show how to modify the cooley tukey algorithm in order to obtain an error growth that is proportional to the square root of the number of time steps finally we show for a simple model that our conclusions hold true if the advection solver is used as part of a splitting scheme

generalized integrating factor methods for stiff pdes the integrating factor if method for numerical integration of stiff nonlinear pdes has the disadvantage of producing large error coefficients when the linear term has large norm we propose a generalization of the if method and in particular construct multistep type methods with several orders of magnitude improved accuracy we also consider exponential time differencing etd methods and point out connections with a particular application of the commutator free lie group methods we present a new fourth order etdrk method with improved accuracy the methods considered are compared in several numerical examples

a finite element code for the simulation of one dimensional vlasov plasmas i theory abstract the galerkin method is used to obtain a finite element solution to the vlasov poisson equations over the two dimensional x v phase plane using bilinear element shape functions a set of linear equations for updated values of the distribution function is obtained in which time dependent and time independent coefficients are separated to reduce the computation involved the noise levels are low and the energy conservation good the finite element approach offers certain advantages over alternative methods in applications to problems in plasma physics in particular the flexibility in using elements of different size allows problems in which regions of the plasma are characterized by often widely differing scale lengths to be treated efficiently

arbitrary order time accurate semi lagrangian spectral approximations of the vlasov poisson system abstract the vlasov poisson system modeling the evolution of non collisional plasmas in the electrostatic limit is approximated by a semi lagrangian technique spectral methods of periodic type are implemented through a collocation approach groups of particles are represented by the fourier lagrangian basis and evolve for a single timestep along a high order accurate representation of the local characteristic lines the time advancing technique is based on truncated taylor series that can be in principle of any order of accuracy a variant is obtained by coupling the phase space discretization with high order accurate backward differentiation formulas bdf at each timestep particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs thus these techniques combine the excellent features of spectral approximations with high order time integration the resulting method has excellent conservation properties indeed it can be proven that the total number of particles proportional to the total mass and charge is conserved up to the machine precision series of numerical experiments are performed in order to assess the real performance in particular comparisons with standard benchmarks are examined

highly efficient numerical algorithm based on random trees for accelerating parallel vlasov poisson simulations an efficient numerical method based on a probabilistic representation for the vlasov poisson system of equations in the fourier space has been derived this has been done theoretically for arbitrary dimensional problems and particularized to unidimensional problems for numerical purposes such a representation has been validated theoretically in the linear regime comparing the solution obtained with the classical results of the linear landau damping the numerical strategy followed requires generating suitable random trees combined with a pade approximant for approximating accurately a given divergent series such series are obtained by summing the partial contributions to the solution coming from trees with arbitrary number of branches these contributions coming in general from multi dimensional definite integrals are efficiently computed by a quasi monte carlo method it is shown how the accuracy of the method can be effectively increased by considering more terms of the series the new representation was used successfully to develop a probabilistic domain decomposition method suited for massively parallel computers which improves the scalability found in classical methods finally a few numerical examples based on classical phenomena such as the non linear landau damping and the two streaming instability are given illustrating the remarkable performance of the algorithm when compared the results with those obtained using a classical method

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

spectral method for analysis of switching diffusions the analysis problem of switching diffusions is considered this paper presents a new approach based on the spectral method formalism for solving generalized fokker planck equations the proposed method allows to transform partial differential equations into the linear algebraic equations and to arrive at a solution in an explicit form the aspects of applications are discussed a numerical example is given to illustrate the efficiency of the proposed method

regular decomposition and a framework of order reduced methods for fourth order problems this paper is devoted to the construction of order reduced method of fourth order problems a constructive framework is presented such that a problem on the high regularity space can be deduced to an equivalent system on three low regularity spaces which are connected by a regular decomposition corresponding to a decomposition of the regularity of the high order space the generated numerical schemes based on the deduced problems can be of lower complicacy and the framework is fit for various fourth order problems three fourth order problems are then discussed under the framework including one in two dimension and two in three dimension they are each corresponding to a regular decomposition and thus are discretised based on the discretised analogues of the regular decomposition optimal error estimates are given

a high order semi lagrangian discontinuous galerkin method for vlasov poisson simulations without operator splitting abstract in this paper we develop a high order semi lagrangian sl discontinuous galerkin dg method for nonlinear vlasov poisson vp simulations without operator splitting in particular we combine two recently developed novel techniques one is the high order non splitting sldg transport method cai et al 2017 4 and the other is the high order characteristics tracing technique proposed in qiu and russo 2017 29 the proposed method with up to third order accuracy in both space and time is locally mass conservative free of splitting error positivity preserving stable and robust for large time stepping size the sldg vp solver is applied to classic benchmark test problems such as landau damping and two stream instabilities for vp simulations efficiency and effectiveness of the proposed scheme is extensively tested tremendous cpu savings are shown by comparisons between the proposed sl dg scheme and the classical runge kutta dg method

a hybridized formulation for the weak galerkin mixed finite element method abstract this paper presents a hybridized formulation for the weak galerkin mixed finite element method wg mfem which was introduced and analyzed in wang and ye 2014 for second order elliptic equations the wg mfem method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape the key to wg mfem is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element the hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising from the lagrange multiplier in hybridization optimal order error estimates are derived for the hybridized wg mfem approximations some numerical results are reported to confirm the theory and a superconvergence for the lagrange multiplier

highly accurate lagrangian flux calculation via algebraic quadratures on spline approximated donating regions abstract lagrangian flux through a fixed curve segment within a time interval t n t n k can be formulated as an integral at the initial time t n over a compact point set called donating region in which each particle will pass through the curve during the time interval and contribute to the flux based an explicit constructive and analytical solution of the donating region the author proposes algorithms of lagrangian flux calculation lfc as algebraic quadratures over the spline approximated donating regions this generic formulation yields high accuracies up to the 8th order both in time and space as another benefit lfc leads to a conservative semi lagrangian method whose time step size is free of the eulerian stability constraint c r 1 the high accuracy and robustness of the proposed lfc algorithms are demonstrated by various numerical tests with courant numbers ranging from 1 to 10 000 and with accuracies from the 2nd order to the 8th order lfc might also be useful in designing hybrid eulerian methods for complex geometries

discontinuous galerkin methods for the vlasov maxwell equations discontinuous galerkin methods are developed for solving the vlasov maxwell system methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy such properties in general are hard to achieve within other numerical method frameworks for simulating the vlasov maxwell system the proposed scheme employs discontinuous galerkin discretizations for both the vlasov and the maxwell equations resulting in a consistent description of the distribution function and electromagnetic fields it is proven up to some boundary effects that charge is conserved and the total energy can be preserved with suitable choices of the numerical flux for the maxwell equations and the underlying approximation spaces error estimates are established for several flux choices the scheme is tested on the streaming weibel instability the order of accuracy and conservation properties of the proposed method are verified

weak galerkin finite element method for viscoelastic wave equations abstract in this article we consider a weak galerkin finite element method wg fem for solving one type of viscoelastic wave equation a discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme we introduce both the semi and fully discrete wg fems and establish the corresponding stability estimates optimal order error estimate in h 1 norm is derived numerical experiments are preformed to verify the theoretical results

fourth order time stepping for kadomtsev petviashvili and davey stewartson equations purely dispersive partial differential equations such as the korteweg de vries equation the nonlinear schrodinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow up to numerically study such phenomena fourth order time stepping in combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region we compare the performance of several fourth order methods for the kadomtsev petviashvili and the davey stewartson equations two integrable equations in 2 1 dimensions these methods are exponential time differencing integrating factors time splitting implicit runge kutta and driscoll s composite runge kutta method the accuracy in the numerical conservation of integrals of motion is discussed

new discretization schemes for time harmonic maxwell equations by weak galerkin finite element methods abstract this paper introduces new discretization schemes for time harmonic maxwell equations in a connected domain by using the weak galerkin wg finite element method the corresponding wg algorithms are analyzed for their stability and convergence error estimates of optimal order in various discrete sobolev norms are established for the resulting finite element approximations

conservative semi lagrangian schemes for vlasov equations conservative methods for the numerical solution of the vlasov equation are developed in the context of the one dimensional splitting in the case of constant advection these methods and the traditional semi lagrangian ones are proven to be equivalent but the conservative methods offer the possibility to add adequate filters in order to ensure the positivity in the non constant advection case they present an alternative to the traditional semi lagrangian schemes which can suffer from bad mass conservation in this time splitting setting

an efficient numerical scheme for the biharmonic equation by weak galerkin finite element methods on polygonal or polyhedral meshes this paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak galerkin wg finite element methods the wg finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual h 2 semi norm weak partial derivatives and their approximations called discrete weak partial derivatives are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra the discrete weak partial derivatives serve as building blocks for the wg finite element method the resulting matrix from the wg method is symmetric positive definite and parameter free an error estimate of optimal order is derived in an h 2 equivalent norm for the wg finite element solutions error estimates in the usual l 2 norm are established yielding optimal order of convergence for all the wg finite element algorithms except the one corresponding to the lowest order i e piecewise quadratic elements some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme

a performance comparison of semi lagrangian discontinuous galerkin and spline based vlasov solvers in four dimensions abstract the purpose of the present paper is to compare two semi lagrangian methods in the context of the four dimensional vlasov poisson equation more specifically our goal is to compare the performance of the more recently developed semi lagrangian discontinuous galerkin scheme to cubic spline interpolation which is widely used in eulerian vlasov simulation to that end we perform simulations for nonlinear landau damping and a two stream instability and provide benchmarks for the selalib and sldg codes both on a workstation and using mpi on a cluster in addition we will present results for the graphic processing unit gpu implementation contained in sldg we find that the semi lagrangian discontinuous galerkin scheme shows a moderate improvement in run time for nonlinear landau damping and a substantial improvement for the two stream instability it should be emphasized that these results are markedly different from results obtained in the asymptotic regime which favor spline interpolation thus we conclude that the traditional approach of evaluating numerical methods is misleading even for short time simulations in addition the absence of any global communication in the semi lagrangian discontinuous galerkin method gives it a decisive advantage for scaling to more than 256 cores

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

adaptive multiresolution semi lagrangian discontinuous galerkin methods for the vlasov equations we develop adaptive numerical schemes for the vlasov equation by combining discontinuous galerkin discretisation multiresolution analysis and semi lagrangian time integration we implement a tree based structure in order to achieve adaptivity both multi wavelets and discontinuous galerkin rely on a local polynomial basis the schemes are tested and validated using vlasov poisson equations for plasma physics and astrophysics

a split step fourier discontinuous galerkin scheme for the kadomtsev petviashvili equation abstract in this paper we propose a method to solve the kadomtsev petviashvili equation based on splitting the linear part of the equation from the nonlinear part the linear part is treated using ffts while the nonlinear part is approximated using a semi lagrangian discontinuous galerkin approach of arbitrary order we demonstrate the efficiency and accuracy of the numerical method by providing a range of numerical simulations in particular we find that our approach can outperform the numerical methods considered in the literature by up to a factor of five although we focus on the kadomtsev petviashvili equation in this paper the proposed numerical scheme can be extended to a range of related models as well

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

well posedness for the fifth order shallow water equations abstract the cauchy problems for some kind of fifth order shallow water equations t u x 5 u x 3 u x u f u x u x 2 u 0 x t r r are considered by the fourier restriction norm method where nonlinear terms f u x u x 2 u are x u k k 2 3 u x 2 u or x u x 2 u respectively the local well posedness is established for data in h s r with s 7 4 for the kawahara equation f x u 2 and is established for data in h s r with s 1 4 for the modified kawahara equation f x u 3 respectively moreover the local result is established for data in h s r with s 0 if f u x 2 u and is established for data in h s r with s 1 4 if f x u x 2 u respectively

superconvergence of numerical gradient for weak galerkin finite element methods on nonuniform cartesian partitions in three dimensions abstract a superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak galerkin finite element scheme on the uniform and non uniform cubic partitions due to the loss of the symmetric property from two dimensions to three dimensions this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions li et al 0000 from rectangular partitions to cubic partitions the error estimate for the numerical gradient in the l 2 norm arrives at a superconvergence order of o h r 1 5 r 2 when the lowest order weak galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed a series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions

a new weak galerkin finite element scheme for general second order elliptic problems abstract a new weak galerkin wg finite element scheme is presented for general second order elliptic problems in this paper in this new scheme a skew symmetric form has been used for handling the convection term the advantage of the new scheme is that the system of linear equations from the scheme is positive definite and one might easily get the well posedness of the system in this scheme the wg elements are designed to have the form of p k t p k 1 e that is we choose the polynomials of degree k 1 on each element and the polynomials of degree k 1 on the edge face of each element as a result fewer degrees of freedom are generated in the new wg finite element scheme it is also worth pointing out that the wg finite element scheme is established on finite element partitions consisting of arbitrary shape of polygons polyhedra which are shape regular optimal order error estimates are presented for the corresponding numerical approximation in various norms some numerical results are reported to confirm the theory

a hierarchical o n force calculation algorithm a novel code for the approximate computation of long range forces between n mutually interacting bodies is presented the code is based on a hierarchical tree of cubic cells and features mutual cell cell interactions which are calculated via a cartesian taylor expansion in a symmetric way such that total momentum is conserved the code benefits from an improved and simple multipole acceptance criterion that reduces the force error and the computational effort for n 104 the computational costs are found empirically to rise sublinearly with n for applications in stellar dynamics this is the first competitive code with complexity o n it is faster than the standard tree code by a factor of 10 or more

a semi lagrangian discontinuous galerkin method for scalar advection by incompressible flows a new conservative semi lagrangian formulation is proposed for the discretization of the scalar advection equation in flux form the approach combines the accuracy and conservation properties of the discontinuous galerkin dg method with the computational efficiency and robustness of semi lagrangian sl techniques unconditional stability in the von neumann sense is proved for the proposed discretization in the one dimensional case a monotonization technique is then introduced based on the flux corrected transport approach this yields a multi dimensional monotonic scheme for the piecewise constant component of the computed solution that is characterized by a smaller amount of numerical diffusion than standard dg methods the accuracy and stability of the method are further demonstrated by two dimensional tracer advection tests in the case of incompressible flows the comparison with results obtained by standard sl and dg methods highlights several advantages of the new technique

a particle in cell method for the simulation of plasmas based on an unconditionally stable field solver abstract we propose a new particle in cell pic method for the simulation of plasmas based on a recently developed unconditionally stable solver for the wave equation this method is not subject to a cfl restriction limiting the ratio of the time step size to the spatial step size typical of explicit methods while maintaining computational cost and code complexity comparable to such explicit schemes we describe the implementation in one and two dimensions for both electrostatic and electromagnetic cases and present the results of several standard test problems showing good agreement with theory with time step sizes much larger than allowed by typical cfl restrictions

a positivity preserving high order semi lagrangian discontinuous galerkin scheme for the vlasov poisson equations the vlasov poisson equations describe the evolution of a collisionless plasma represented through a probability density function pdf that self interacts via an electrostatic force one of the main difficulties in numerically solving this system is the severe time step restriction that arises from parts of the pdf associated with moderate to large velocities the dominant approach in the plasma physics community for removing these time step restrictions is the so called particle in cell pic method which discretizes the distribution function into a set of macro particles while the electric field is represented on a mesh several alternatives to this approach exist including fully lagrangian fully eulerian and so called semi lagrangian methods the focus of this work is the semi lagrangian approach which begins with a grid based eulerian representation of both the pdf and the electric field then evolves the pdf via lagrangian dynamics and finally projects this evolved field back onto the original eulerian mesh in particular we develop in this work a method that discretizes the 1 1 vlasov poisson system via a high order discontinuous galerkin dg method in phase space and an operator split semi lagrangian method in time second order accuracy in time is relatively easy to achieve via strang operator splitting with additional work using higher order splitting and a higher order method of characteristics we also demonstrate how to push this scheme to fourth order accuracy in time we show how to resolve all of the lagrangian dynamics in such a way that mass is exactly conserved positivity is maintained and high order accuracy is achieved the poisson equation is solved to high order via the smallest stencil local discontinuous galerkin ldg approach we test the proposed scheme on several standard test cases

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

operator splitting for two dimensional incompressible fluid equations we analyze splitting algorithms for a class of two dimensional fluid equations which includes the incompressible navier stokes equations and the surface quasi geostrophic equation our main result is that the godunov and strang splitting methods converge with the expected rates provided the initial data are sufficiently regular

a lowest order weak galerkin method for linear elasticity abstract the lowest order weak galerkin wg method is considered for linear elasticity based on the displacement formulation the new method approximates the displacement using piecewise constant vector functions both inside and on the boundary of each mesh element and its bilinear form does not require a stabilization term for the existence and uniqueness of the solution a priori error estimates of optimal order in the discrete h 1 and l 2 norms for the displacement are proved when the solution is smooth the error estimates are independent of the lame constant thus the performance of the new method does not deteriorate as the elastic material becomes incompressible further a simple post processing technique to obtain a numerical approximation of the stress is presented a careful error analysis reveals that the l 2 norm error in the stress is also optimal and independent of several numerical experiments confirm the locking free property and optimal convergence rates of the new method

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

a weak galerkin finite element method for high dimensional time fractional diffusion equation abstract this paper is concerned with the application of a weak galerkin finite element method wg fem to the time fractional diffusion equation the wg fem with l1 formula and the fast evaluation scheme based on wg fem are designed the optimal convergence rates for both semi discrete and fully discrete wg fem schemes are obtained and the stability analysis for the semi discrete wg fem is derived numerical experiments are implemented to verify the theoretical results

discontinuous galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations this paper develops a discontinuous galerkin dg finite element differential calculus theory for approximating weak derivatives of sobolev functions and piecewise sobolev functions by introducing numerical one sided derivatives as building blocks various first and second order numerical operators such as the gradient divergence hessian and laplacian operator are defined and their corresponding calculus rules are established among the calculus rules are product and chain rules integration by parts formulas and the divergence theorem approximation properties and the relationship between the proposed dg finite element numerical derivatives and some well known finite difference numerical derivative formulas on cartesian grids are also established besides independent interest in numerical differentiation the primary motivation and goal of developing the dg finite element differential calculus is to solve partial differential equations it is shown that several existing finite element finite difference and discontinuous galerkin methods can be rewritten compactly using the proposed dg finite element differential calculus framework moreover new discontinuous galerkin methods for linear and nonlinear pdes are also obtained from the framework

convergence analysis of a discontinuous galerkin strang splitting approximation for the vlasov poisson equations a rigorous convergence analysis of the strang splitting algorithm with a discontinuous galerkin approximation in space for the vlasov poisson equations is provided it is shown that under suitable assumptions the error is of order mathcal o tau 2 h q h q tau where tau is the size of a time step h is the cell size and q the order of the discontinuous galerkin approximation in order to investigate the recurrence phenomena for approximations of higher order as well as to compare the algorithm with numerical results already available in the literature a number of numerical simulations are performed

bound preserving discontinuous galerkin methods for conservative phase space advection in curvilinear coordinates we extend the positivity preserving method of zhang and shu 49 to simulate the advection of neutral particles in phase space using curvilinear coordinates the ability to utilize these coordinates is important for non equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries the method achieves high order accuracy using discontinuous galerkin dg discretization of phase space and strong stability preserving runge kutta ssp rk time integration special care is taken to ensure that the method preserves strict bounds for the phase space distribution function f i e f 0 1 the combination of suitable cfl conditions and the use of the high order limiter proposed in 49 is sufficient to ensure positivity of the distribution function however to ensure that the distribution function satisfies the upper bound the discretization must in addition preserve the divergence free property of the phase space flow proofs that highlight the necessary conditions are presented for general curvilinear coordinates and the details of these conditions are worked out for some commonly used coordinate systems i e spherical polar spatial coordinates in spherical symmetry and cylindrical spatial coordinates in axial symmetry both with spherical momentum coordinates results from numerical experiments including one example in spherical symmetry adopting the schwarzschild metric demonstrate that the method achieves high order accuracy and that the distribution function satisfies the maximum principle

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

operator splitting for the kdv equation we provide a new analytical approach to operator splitting for equations of the type u t au b u where a is a linear operator and b is quadratic a particular example is the korteweg de vries kdv equation u t uu x u xxx 0 we show that the godunov and strang splitting methods converge with the expected rates if the initial data are sufficiently regular

a splitting approach for the kadomtsev petviashvili equation we consider a splitting approach for the kadomtsev petviashvili equation with periodic boundary conditions and show that the necessary interpolation procedure can be efficiently implemented the error made by this numerical scheme is compared to exponential integrators which have been shown in klein and roidot 2011 2 to perform best for stiff solutions of the kadomtsev petviashvili equation since many classic high order splitting methods do not perform well we propose a stable extrapolation method in order to construct an efficient numerical scheme of order four in addition the conservation properties and the possibility of order reduction for certain initial values for the numerical schemes under consideration are investigated

a mixed finite element method for the biharmonic equation we consider the mixed finite element approximation of the biharmonic problem on a smooth domain with either clamped or simply supported boundary conditions error estimates are proved for the ciarlet raviart formulation of the clamped plate problem and for a similar method for the simply supported plate problem