immersed boundary method for variable viscosity and variable density problems using fast constant coefficient linear solvers ii theory we analyze the stability and convergence of first order accurate and second order accurate timestepping schemes for the navier stokes equations with variable viscosity these schemes are characterized by a mixed implicit explicit treatment of the viscous term in which a numerical parameter lambda determines the degree of splitting between the implicit and explicit contributions the reason for this splitting is that it avoids the need to solve computationally expensive linear systems that may change at each timestep provided the parameter lambda is within a permissible range we prove that the first order accurate and second order accurate schemes are convergent we show further that the efficiency of the second order accurate scheme depends on how lambda is chosen within the permissible range and we discuss choices that work well in practice we use parameters motivated by this analysis to simulate internal gravity waves which arise in stratified fluids with variable density we examine h

weight adjusted discontinuous galerkin methods wave propagation in heterogeneous media time domain discontinuous galerkin dg methods for wave propagation require accounting for the inversion of dense elemental mass matrices where each mass matrix is computed with respect to a parameter weighted l 2 inner product in applications where the wavespeed varies spatially at a subelement scale these matrices are distinct over each element necessitating additional storage in this work we propose a weight adjusted dg wadg method that reduces storage costs by replacing the weighted l 2 inner product with a weight adjusted inner product this equivalent inner product results in an energy stable method but does not increase storage costs for locally varying weights a priori error estimates are derived and numerical examples are given illustrating the application of this method to the acoustic wave equation with heterogeneous wavespeed

time exponential splitting technique for the klein gordon equation with hagstrom warburton high order absorbing boundary conditions klein gordon equations on an unbounded domain are considered in one dimensional and two dimensional cases numerical computation is reduced to a finite domain by using the hagstrom warburton h w high order absorbing boundary conditions abcs time integration is made by means of exponential splitting schemes that are efficient and easy to implement in this way it is possible to achieve a negligible error due to the time integration and to study the behavior of the absorption error numerical experiments displaying the accuracy of the numerical solution for the two dimensional case are provided the influence of the dispersion coefficient on the error is also studied

orthogonal polynomials and some q beta integrals of ramanujan abstract two integrals of ramanujan are used to define a q analogue of the euler beta integral on the real line and of the cauchy beta integral on the complex unit circle such integrals are connected to orthogonal biorthogonal and laurent polynomials explicit examples of laurent orthogonal polynomials are given on the real line and on the circle

weak ill posedness of spatial discretizations of absorbing boundary conditions for schrodinger type equations when we wish to solve numerically a differential problem defined on an infinite domain it is necessary to consider a finite subdomain and to use artificial boundary conditions in such a way that the solutions in the finite subdomain approximate the original solution these boundary conditions are called absorbing when small reflections to the interior domain are allowed in this paper we develop a general class of absorbing boundary conditions for schrodinger type equations by using rational approximations to the transparent boundary conditions with this approach previous absorbing boundary conditions in the literature are included in this class we use the method of lines for the discretization of the initial boundary value problems obtained this way we show that the ordinary differential systems that arise after the spatial discretization are weakly ill posed explaining a previous conjecture of fevens and jiang the time discretization is carried out with a stable runge kutta methods where the high order ones may be used to compensate for the possible troubles present in the problems semidiscretized in space

basic hypergeometric polynomials with zeros on the unit circle the objective in this manuscript is to present two specific sequences of polynomials with elements given in terms of basic hypergeometric polynomials or q hypergeometric polynomials it is shown that the zeros of these polynomials are simple and that they lie exactly on the unit circle

robust approaches to handling complex geometries with galerkin difference methods abstract the galerkin difference gd basis is a set of continuous piecewise polynomials defined using a finite difference like grid of degrees of freedom the one dimensional gd basis functions are naturally extended to multiple dimensions using a tensor product construction on quadrilateral elements the gd basis can be used to define the solution space for a discontinuous galerkin finite element discretization of partial differential equations in this work we propose two approaches to handling complex geometries within this setting 1 using nonconforming curvilinear gd elements and 2 coupling affine gd elements with curvilinear simplicial elements in both cases the semidiscrete discontinuous galerkin method is provably energy stable even when variational crimes are committed additionally for both element types a weight adjusted mass matrix is used which ensures that only the reference mass matrix must be inverted we also present sufficient conditions on the treatment of metric terms for the curvilinear nonconforming gd elements to ensure that the scheme is both constant preserving and conservative numerical experiments confirm the stability results and demonstrate the accuracy of the coupled schemes

optimized strong stability preserving imex runge kutta methods we construct and analyze robust strong stability preserving implicit explicit runge kutta imex rk methods for models of flow with diffusion as they appear in astrophysics and in many other fields where equations with similar structure arise it turns out that besides the optimization of the region of absolute monotonicity some other properties of the methods are crucial for the success of such simulations in particular the models in our focus dictate to also take into account the step size limits associated with dissipativity positivity of the stiff parabolic terms which represent transport by diffusion the uniform convergence with respect to different stiffness properties of those same terms etc furthermore in the literature some other properties like the inclusion of a part of the imaginary axis in the stability region have been argued to be relevant in this paper we construct several new imex rk methods which differ from each other by taking various or even all of these constraints simultaneously into account it is demonstrated for some simple examples as well as for the problem of double diffusive convection that the newly constructed schemes provide a significant computational advantage over other methods from the literature due to their accumulation of different stability properties the optimized imex rk methods obtained in this paper are robust schemes that may also be useful for general models which involve the solution of advection diffusion equations or other transport equations with similar stability requirements

stability analysis of the numerical method of characteristics applied to a class of energy preserving hyperbolic systems part ii nonreflecting boundary conditions abstract we show that imposition of non periodic in place of periodic boundary conditions bc can alter stability of modes in the method of characteristics moc employing certain ordinary differential equation ode numerical solvers thus using non periodic bc may render some of the moc schemes stable for most practical computations even though they are unstable for periodic bc this fact contradicts a statement found in some textbooks and known as part of the babenko gelfand criterion that an instability detected by the von neumann analysis for a given numerical scheme implies an instability of that scheme with arbitrary i e non periodic bc we explain the mechanism behind this contradiction which lies in a certain property of the scheme s eigenmodes that is assumed by the babenko gelfand criterion but does not hold for eigenmodes of some of the moc based schemes we also show that and explain why for the moc employing some other ode solvers stability of the modes may indeed not be improved by non periodic bc as the babenko gelfand criterion implies

on order reduction for runge kutta methods applies to differential algebraic systems and to stiff systems of odes in this short note the order reduction results of petzold siam j numer anal 23 1986 pp 837 852 for implicit runge kutta methods applied to index 1 differential algebraic systems are extended to include a larger class of methods the relationship between the order reduction results for differential algebraic systems and recent results for stiff systems of ordinary differential equations is explained

high order operator splitting methods based on an integral deferred correction framework integral deferred correction idc methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties in this paper we construct high order operator splitting schemes using the idc procedure to solve initial value problems ivps we present analysis to show that the idc methods can correct for both the splitting and numerical errors lifting the order of accuracy by r with each correction where r is the order of accuracy of the method used to solve the correction equation we further apply this framework to solve partial differential equations pdes numerical examples in two dimensions of linear and nonlinear initial boundary value problems are presented to demonstrate the performance of the proposed idc approach

galerkin differences for acoustic and elastic wave equations in two space dimensions abstract the galerkin difference method a finite element method built using standard galerkin projection techniques but employing nonstandard basis functions was originally developed for one space dimension in 1 here the method is extended to two space dimensions using a tensor product construction theoretical and computational evidence shows the method behaves as expected for the acoustic wave equation for the elastic wave equation the approximations are found to be at least as accurate as predicted but with a free surface the scheme may exhibit an unexpected superconvergence extension to curvilinear mapped grids is also considered for acoustics in all cases the use of a tensor product construction allows for efficient solution of the linear system involving the mass matrix which implies optimal linear time solutions with respect to the number of degrees of freedom

bernstein bezier weight adjusted discontinuous galerkin methods for wave propagation in heterogeneous media abstract this paper presents an efficient discontinuous galerkin method to simulate wave propagation in heterogeneous media with sub cell variations this method is based on a weight adjusted discontinuous galerkin method wadg which achieves high order accuracy for arbitrary heterogeneous media 1 however the computational cost of wadg grows rapidly with the order of approximation in this work we propose a bernstein bezier weight adjusted discontinuous galerkin method bbwadg to address this cost by approximating sub cell heterogeneities by a fixed degree polynomial the main steps of wadg can be expressed as polynomial multiplication and l 2 projection which we carry out using fast bernstein algorithms the proposed approach reduces the overall computational complexity from o n 2 d to o n d 1 in d dimensions numerical experiments illustrate the accuracy of the proposed approach and computational experiments for a gpu implementation of bbwadg verify that this theoretical complexity is achieved in practice

a skew symmetric discontinuous galerkin spectral element discretization and its relation to sbp sat finite difference methods this paper shows that the discontinuous galerkin collocation spectral element method with gauss lobatto points dgsem gl satisfies the discrete summation by parts sbp property and can thus be classified as an sbp sat simultaneous approximation term scheme with a diagonal norm operator in the same way sbp sat finite difference schemes can be interpreted as discontinuous galerkin type methods with a corresponding weak formulation based on an inner product formulation common in the finite element community this relation allows the use of matrix vector notation common in the sbp sat finite difference community to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for dgsem gl and diagonal norm sbp sat based on this result a skew symmetric energy stable discretely conservative dgsem gl formulation applicable to general diagonal norm sbp sat schemes for the nonlinear burgers equation is constructed

runge kutta methods without order reduction for linear initial boundary value problems it is well known the loss of accuracy when a runge kutta method is used together with the method of lines for the full discretization of an initial boundary value problem we show that this phenomenon called order reduction is caused by wrong boundary values in intermediate stages with a right choice the order reduction can be avoided and the optimal order of convergence in time is achieved we prove this fact for time discretizations of abstract initial boundary value problems based on implicit runge kutta methods moreover we apply these results to the full discretization of parabolic problems by means of galerkin finite element techniques we present some numerical examples in order to confirm that the optimal order is actually achieved

a low storage curvilinear discontinuous galerkin method for wave problems the low storage curvilinear discontinuous galerkin lsc dg method reduces the storage requirements for solving symmetric and linear linear symmetric conservation laws in curvilinear domains when compared to the standard discontinuous galerkin method we perform a semidiscrete a priori convergence analysis of lsc dg and determine sufficient conditions on sequences of meshes to guarantee the same rate of convergence as the original upwind dg method on curvilinear domains computational results confirm the optimal order convergence on sample mesh sequences that satisfy the sufficiency conditions additional results show that the sufficient conditions for optimal order convergence of lsc dg may also be necessary conditions

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

time exponential splitting integrator for the klein gordon equation with free parameters in the hagstrom warburton absorbing boundary conditions abstract the klein gordon equation on an infinite two dimensional strip is considered numerical computation is reduced to a finite domain by using the hagstrom warburton h w absorbing boundary conditions abcs with free parameters in the formulation of the auxiliary variables the spatial discretization is achieved by using fourth order finite differences and the time integration is made by means of an efficient and easy to implement fourth order exponential splitting scheme which was used in alonso mallo and portillo 2016 considering the fixed pade parameters in the formulation of the abcs here we generalize the splitting time technique to other choices of the parameters to check the time integrator we consider on one hand four types of fixed parameters the newmann s parameters the chebyshev s parameters the pade s parameters and optimal parameters proposed in hagstrom et al 2007 and on the other hand an adaptive scheme for the dynamic control of the order of absorption and the parameters we study the efficiency of the splitting scheme by comparing with the fourth order four stage runge kutta method

on discretely entropy stable weight adjusted discontinuous galerkin methods curvilinear meshes abstract we construct entropy conservative and entropy stable discontinuous galerkin dg discretizations for time dependent nonlinear hyperbolic conservation laws on curvilinear meshes the resulting schemes preserve a semi discrete quadrature approximation of a continuous global entropy inequality the proof requires the satisfaction of a discrete geometric conservation law which we enforce through an appropriate polynomial approximation we extend the construction of entropy conservative and entropy stable dg schemes to the case when high order curvilinear mass matrices are approximated using low storage weight adjusted approximations and describe how to retain global conservation properties under such an approximation for certain types of curvilinear meshes these weight adjusted approximations deliver optimal rates of convergence finally the high order accuracy local conservation and discrete conservation or dissipation of entropy for under resolved solutions are verified through numerical experiments for the compressible euler equations on triangular and tetrahedral meshes

high order full discretization for anisotropic wave equations abstract two dimensional linear wave equation in anisotropic media on a rectangular domain with initial conditions and periodic boundary conditions is considered the energy of the problem is contemplated the space discretization is reached by means of finite differences on a uniform grid paying attention to the mixed derivative of the equation the discrete energy of the semi discrete problem is introduced for the time integration of the system of ordinary differential equations obtained a fourth order exponential splitting method which is a geometric integrator is proposed this time integrator is efficient and easy to implement the stability condition for time step and space step ratio is deduced numerical experiments displaying the good behavior in the long time integration and the efficiency of the numerical solution are provided

weight adjusted discontinuous galerkin methods curvilinear meshes traditional time domain discontinuous galerkin dg methods result in large storage costs at high orders of approximation due to the storage of dense elemental matrices in this companion paper to j chan r j hewett and t warburton weight adjusted discontinuous galerkin methods wave propagation in heterogeneous media preprint https arxiv org abs 1608 01944 2016 we propose weight adjusted dg wadg methods for curvilinear meshes which reduce storage costs while retaining energy stability a priori error estimates show that high order accuracy is preserved under sufficient conditions on the mesh which are illustrated through convergence tests with different sequences of meshes numerical and computational experiments verify the accuracy and performance of wadg for a model problem on curved domains

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

a comparison of the dispersion and dissipation errors of gauss and gauss lobatto discontinuous galerkin spectral element methods we examine the dispersion and dissipation properties of the gauss and gauss lobatto discontinuous galerkin spectral element methods dgsems for linear wave propagation problems we show that the inherent underintegration in the gauss lobatto variant can be interpreted as a modal filtering of the highest polynomial mode this in turn has a drastic impact on the dispersion and dissipation relations of the gauss lobatto dgsem compared to the gauss variant we show that the gauss dgsem is typically more accurate than the gauss lobatto variant needing fewer points per wavelength for a given accuracy while on the other hand being more restricted in the explicit time step choice we show that the spectra of the dgsem operators depend on the boundary conditions applied and that the ratio of the time step restrictions of the two schemes depends on the choice of boundary conditions

low storage implicit explicit runge kutta schemes for the simulation of stiff high dimensional ode systems implicit explicit imex runge kutta rk schemes are effective for time marching ode systems with both stiff and nonstiff terms on the rhs such schemes implement an often a stable or better implicit rk scheme for the stiff part of the ode which is often linear and simultaneously a more convenient explicit rk scheme for the nonstiff part of the ode which is often nonlinear low storage rk schemes are especially effective for time marching high dimensional ode discretizations of pde systems on modern cache based computational hardware in which memory management is often the most significant computational bottleneck in this paper we develop and characterize eight new low storage implicit explicit rk schemes which have higher accuracy and better stability properties than the only low storage implicit explicit rk scheme available previously the venerable second order crank nicolson runge kutta wray cn rkw3 algorithm that has dominated the dns les literature for the last 25 years while requiring similar storage two three or four registers of length n and comparable floating point operations per timestep

dispersive and dissipative behavior of the spectral element method if the nodes for the spectral element method are chosen to be the gauss legendre lobatto points and a lagrange basis is used then the resulting mass matrix is diagonal and the method is sometimes then described as the gauss point mass lumped finite element scheme we study the dispersive behavior of the scheme in detail and provide both a qualitative description of the nature of the dispersive and dissipative behavior of the scheme along with precise quantitative statements of the accuracy in terms of the mesh size and the order of the scheme we prove that a the gauss point mass lumped scheme i e spectral element method tends to exhibit phase lag whereas the consistent finite element scheme tends to exhibit phase lead b the absolute accuracy of the spectral element scheme is 1 p times better than that of the finite element scheme despite the use of numerical integration c when the order p the mesh size h and the frequency of the wave omega satisfy 2p 1 approx omega h the true wave is essentially fully resolved as a consequence one obtains a proof of the general rule of thumb sometimes quoted in the context of spectral element methods pi modes per wavelength are needed to resolve a wave

discretely conservative finite difference formulations for nonlinear conservation laws in split form theory and boundary conditions the lax wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities the discrete operator however need not be derived from the divergence form of the continuous equations indeed conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal norm skew symmetric summation by parts sbp spatial operator yield discrete operators that are conservative furthermore split form discretely conservation operators can be derived for periodic or finite domain sbp spatial operators of any order examples are presented of a fourth order sbp finite difference operator with second order boundary closures sixth and eighth order constructions are derived and are supplied in an accompanying text file

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

an energy stable discontinuous galerkin spectral element discretization for variable coefficient advection problems we develop a stable split form discontinuous galerkin spectral element method for the solution of variable coefficient linear hyperbolic systems of conservation laws in our presentation we start with the simplest problem and introduce complexity as we progress we begin with the approximation of scalar conservation laws in one space dimension we then extend the derivation to hyperbolic systems finally we approximate the two dimensional problem numerical experiments are performed to compare the approximations

near conserving energy numerical schemes for two dimensional coupled seismic wave equations abstract two dimensional coupled seismic waves satisfying the equations of linear isotropic elasticity on a rectangular domain with initial conditions and periodic boundary conditions are considered a quantity conserved by the solution of the continuous problem is used to check the numerical solution of the problem second order spatial derivatives in the x direction in the y direction and mixed derivative are approximated by finite differences on a uniform grid the ordinary second order in time system obtained is transformed into a first order in time system in the displacement and velocity vectors for the time integration of this system second order and fourth order exponential splitting methods which are geometric integrators are proposed these explicit splitting methods are not unconditionally stable and the stability condition for time step and space step ratio is deduced numerical experiments displaying the good behavior in the long time integration and the efficiency of the numerical solution are provided

a comparison of time discretization linearization approaches for the incompressible navier stokes equations abstract this paper presents a numerical study of two ways for discretizing and linearizing the time dependent incompressible navier stokes equations one approach consists in first applying a semi discretization in time by a fully implicit scheme then in each discrete time the equations are linearized by a fixed point iteration the number of iterations to reach a given stopping criterion is a priori unknown in this approach in the second approach rosenbrock schemes with s stages are used as temporal discretization the non linearity of the navier stokes equations is treated internally in the rosenbrock methods in each discrete time exactly s linear systems of equations have to be solved the numerical study considers five two dimensional problems with distinct features four implicit time stepping schemes and five rosenbrock methods are involved

order reduction and how to avoid it when explicit runge kutta nystrom methods are used to solve linear partial differential equations in this paper we study the order reduction which turns up when explicit runge kutta nystrom methods are used to discretize linear second order hyperbolic equations by means of the method of lines the order observed in practice including its fractional part is obtained it is also proved that the order reduction can be completely avoided taking the boundary values of the intermediate stages of the time semidiscretization the numerical experiments confirm that the optimal order can be reached

a spectral deferred correction method for incompressible flow with variable viscosity abstract this paper presents a semi implicit spectral deferred correction sdc method for incompressible navier stokes problems with variable viscosity and time dependent boundary conditions the proposed method integrates elements of velocity and pressure correction schemes which yields a simpler pressure handling and a smaller splitting error than the sdpc method of minion and saye j comput phys 375 797 822 2018 combined with the discontinuous galerkin spectral element method for spatial discretization it can in theory reach arbitrary order of accuracy in time and space numerical experiments in three space dimensions demonstrate up to order 12 in time and 17 in space for constant as well as varying solution dependent viscosity compared to sdpc the present method yields a substantial improvement of accuracy and robustness against order reduction caused by time dependent boundary conditions

multi patch discontinuous galerkin isogeometric analysis for wave propagation explicit time stepping and efficient mass matrix inversion abstract we present a class of spline finite element methods for time domain wave propagation which are particularly amenable to explicit time stepping the proposed methods utilize a discontinuous galerkin discretization to enforce continuity of the solution field across geometric patches in a multi patch setting which yields a mass matrix with convenient block diagonal structure over each patch we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight adjusted approximation of the mass matrix inverse this approximation restores a tensor product structure while retaining provable high order accuracy and semi discrete energy stability we also estimate the maximum stable timestep for spline based finite elements and show that the use of spline spaces results in less stringent cfl restrictions than equivalent c 0 or discontinuous finite element spaces finally we explore the use of optimal knot vectors based on l 2 n widths we show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep and present a simple heuristic method for approximating optimal knot positions numerical experiments confirm the accuracy and stability of the proposed methods

the theoretical accuracy of runge kutta time discretizations for the initial boundary value problem a study of the boundary error the conventional method of imposing time dependent boundary conditions for runge kutta rk time advancement reduces the formal accuracy of the space time method to first order locally and second order globally independently of the spatial operator this counter intuitive result is analyzed in this paper two methods of eliminating this problem are proposed for the linear constant coefficient case 1 impose the exact boundary condition only at the end of the complete rk cycle 2 impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives the first method while retaining the rk accuracy in all cases results in a scheme with much reduced cfl condition rendering the rk scheme less attractive the second method retains the same allowable time step as the periodic problem however it is a general remedy only for the linear case for non linear hyperbolic equations the second method is effective only for for rk schemes of third order accuracy or less numerical studies are presented to verify the efficacy of each approach

integral deferred correction methods constructed with high order runge kutta integrators spectral deferred correction sdc methods for solving ordinary differential equations odes were introduced by dutt greengard and rokhlin 2000 it was shown in that paper that sdc methods can achieve arbitrary high order accuracy and possess nice stability properties their sdc methods are constructed with low order integrators such as forward euler or backward euler and are able to handle stiff and non stiff terms in the odes in this paper we use high order runge kutta rk integrators to construct a family of related methods which we refer to as integral deferred correction idc methods the distribution of quadrature nodes is assumed to be uniform and the corresponding local error analysis is given the smoothness of the error vector associated with an idc method measured by the discrete sobolev norm is a crucial tool in our analysis the expected order of accuracy is demonstrated through several numerical examples superior numerical stability and accuracy regions are observed when high order rk integrators are used to construct idc methods

on the use of higher order projection methods for incompressible turbulent flow an important issue in the development of higher order methods for incompressible flow is how they perform when the flow is turbulent a useful diagnostic of a method for turbulent flow is the minimum resolution that is required to adequately resolve the turbulent energy cascade at a given reynolds number in this paper we present careful numerical experiments to assess the utility of higher order numerical methods based on this metric we first introduce a numerical method for the incompressible navier stokes equations based on fourth order discretizations in both space and time the method is based on an auxiliary variable formulation and combines fourth order finite volume differencing with a semi implicit spectral deferred correction temporal integration scheme we also introduce for comparison purposes versions based on second order spatial and or temporal discretizations we demonstrate that for smooth problems each of the methods exhibits the expected order of convergence in time and space we

extrapolated implicit explicit time stepping this paper constructs extrapolated implicit explicit time stepping methods that allow one to efficiently solve problems with both stiff and nonstiff components the proposed methods are based on euler steps and can provide very high order discretizations of odes index 1 daes and pdes in the method of lines framework implicit explicit schemes based on extrapolation are simple to construct easy to implement and straightforward to parallelize this work establishes the existence of perturbed asymptotic expansions of global errors explains the convergence orders of these methods and studies their linear stability properties numerical results with stiff ode dae and pde test problems confirm the theoretical findings and illustrate the potential of these methods to solve multiphysics multiscale problems

on galerkin difference methods energy stable difference methods for hyperbolic initial boundary value problems are constructed using a galerkin framework the underlying basis functions are lagrange functions associated with continuous piecewise polynomial approximation on a computational grid both theoretical and computational evidence shows that the resulting methods possess excellent dispersion properties in the absence of boundaries the spectral radii of the operators for the first and second derivative matrices are bounded independent of discretization order with boundaries the spectral radius of the first order derivative matrix appears to be bounded independent of discretization order and grows only slowly with discretization order for problems in second order form

numerical comparison of high order absorbing boundary conditions and perfectly matched layers for a dispersive one dimensional medium abstract high order absorbing boundary conditions abc and perfectly matched layers pml are two powerful methods to numerically solve wave problems in unbounded domains the aim of the proposed study is to analyze and compare the performance of these methods in the one dimensional problem governed by the dispersive wave equation the pmls proposed in literature for time harmonic dynamics are applied to time dependent wave problems and linear quadratic and cubic polynomial stretching functions are considered the resulting pmls exhibit a double absorbing action i they reduce the amplitude of the incident waves and ii they delay the wave propagation then the abc proposed by givoli and neta and those proposed by hagstrom mar or and givoli are considered the former are a reformulation of the higdon high order non reflecting boundary conditions the latter improve the higdon conditions and extend them to take into account evanescent waves the accuracy of the pmls and abcs implemented in a finite element code is first investigated with respect to the frequency of the incident wave being it progressive or evanescent then the response to a wave train characterized by a broad frequency spectrum resulting from an impulsive force is studied a detailed analysis is performed to detect the influence of the parameters of both the abcs and the pmls on the absorption of waves the performances of pmls and abcs are compared and merits and drawbacks of the two methods are pointed out