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

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

an algebraic interpretation of the continuous big q hermite polynomials the continuous big q hermite polynomials are shown to realize a basis for a representation space of an extended q oscillator algebra an expansion formula is algebraically derived using this model

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

the bivariate rogers szego polynomials we present an operator approach to deriving mehler s formula and the rogers formula for the bivariate rogers szego polynomials hn x y q the proof of mehler s formula can be considered as a new approach to the nonsymmetric poisson kernel formula for the continuous big q hermite polynomials hn x a q due to askey rahman and suslov mehler s formula for hn x y q involves a 32 sum and the rogers formula involves a 21 sum the proofs of these results are based on parameter augmentation with respect to the q exponential operator and the homogeneous q shift operator in two variables by extending recent results on the rogers szego polynomials hn x q due to hou lascoux and mu we obtain another rogers type formula for hn x y q finally we give a change of base formula for hn x a q which can be used to evaluate some integrals by using the askey wilson integral

ramanujan type continuous measures for biorthogonal q rational functions it is shown that ramanujan type continuous measures for a hierarchy of biorthogonal rational functions can be systematically built from simple cases of the rogers szego for 0 1 polynomials by using the berg ismail procedure of attaching generating functions to measures

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

mean first passage times of processes driven by white shot noise we consider mean first passage times in systems driven by white shot noise with exponentially distributed jump heights simple interpretable results are obtained and the linkage between those results and the steady state probability density function of the process is presented the virtual waiting time or takacs process constant losses and the shot noise process with linear losses are analyzed in depth along with a more complex process with useful implications for the modeling of the soil moisture dynamics in hydrology

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

correlation anti correlation transition by state dependent poisson noise noise may induce some degrees of order in many biological and complex systems we present an example of a noise dependent transition from correlation to anti correlation of the states of a bivariate system coupled by state dependent poisson noises by varying the degree of dependence of the rate of jumps on the other variable the system undergoes different degrees of correlation from independence to perfect correlation which in this case coincides with perfect synchronization in the limit of the deterministic periodic case

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

higher order temporal integration for the incompressible navier stokes equations in bounded domains author s minion ml saye ri abstract 2018 elsevier inc this paper compares and contrasts higher order semi implicit temporal integration strategies for the incompressible navier stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations particular focus is placed on the imposition of boundary conditions in the semi implicit formulation the accurate treatment of the pressure term and the smoothness of the numerical solution for different formulations the main result presented here is the formulation and numerical validation of a single step semi implicit method called spectral deferred pressure corrections which can in theory obtain arbitrary formal order of accuracy in time numerical results demonstrate up to eighth order accuracy scenarios where optimal temporal accuracy is attained and scenarios where order reduction is observed due to time dependent boundary conditions

on lifting q difference operators in the askey scheme of basic hypergeometric polynomials we construct a q difference operator that lifts the continuous q hermite polynomials hn x q of rogers up to the continuous big q hermite polynomials hn x a q on the next level in the askey scheme of basic hypergeometric polynomials this operator is defined as exton s q exponential function eq aqdq in terms of the askey wilson divided q difference operator dq and it represents a particular q extension of the standard shift operator we next show that one can move two steps more upwards in order first to reach the al salam chihara family of polynomials qn x a b q and then the continuous dual q hahn polynomials pn x a b c q in both these cases lifting operators respectively turn out to be convolution type products of two and three one parameter q difference operators of the same type eq aqdq at the initial step at each step we also determine q difference operators that lift the weight function for the continuous q hermite polynomials hn x q successively up to the weight functions for hn x a q qn x a b q and pn x a b c q

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

interweaving pfasst and parallel multigrid the parallel full approximation scheme in space and time pfasst introduced by emmett and minion in 2012 is an iterative strategy for the temporal parallelization of odes and discretized pdes as the name suggests pfasst is similar in spirit to a space time full approximation scheme multigrid method performed over multiple time steps in parallel however since the original focus of pfasst was on the performance of the method in terms of time parallelism the solution of any spatial system arising from the use of implicit or semi implicit temporal methods within pfasst have simply been assumed to be solved to some desired accuracy completely at each substep and each iteration by some unspecified procedure it hence is natural to investigate how iterative solvers in the spatial dimensions can be interwoven with the pfasst iterations and whether this strategy leads to a more efficient overall approach this paper presents an initial investigation on the relative performance of different strategies for cou

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

integrate and fire neurons with threshold noise a tractable model of how interspike interval correlations affect neuronal signal transmission many neurons exhibit interval correlations in the absence of input signals we study the influence of these intrinsic interval correlations of model neurons on their signal transmission properties for this purpose we employ two simple firing models one of which generates a renewal process while the other leads to a nonrenewal process with negative interval correlations different methods to solve for spectral statistics in the presence of a weak stimulus spike train power spectra cross spectra and coherence functions are presented and their range of validity is discussed using these analytical results we explore a lower bound on the mutual information rate between output spike train and input stimulus as a function of the system s parameters we demonstrate that negative correlations in the baseline activity can lead to enhanced information transfer of a weak signal by means of noise shaping of the background noise spectrum we also show that an enhancement is not compulsory char22 for a stimulus with power exclusively at high frequencies the renewal model can transfer more information than the nonrenewal model does we discuss the application of our analytical results to other problems in neuroscience our results are also relevant to the general problem of how a signal affects the power spectrum of a nonlinear stochastic system

the analogue of grad div stabilization in dg methods for incompressible flows limiting behavior and extension to tensor product meshes abstract grad div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems for mitigating velocity errors that are sometimes called poor mass conservation such errors arise due to the relaxation of the divergence constraint in classical mixed methods and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures in this contribution an analogue of grad div stabilization for discontinuous galerkin methods is studied here the key is the penalization of the jumps of the normal velocities over facets of the triangulation which controls the measure valued part of the distributional divergence of the discrete velocity solution our contribution is twofold first we characterize the limit for arbitrarily large penalization parameters which shows that the stabilized nonconforming discontinuous galerkin methods remain robust and accurate in this limit second we extend these ideas to the case of non simplicial meshes here broken grad div stabilization must be used in addition to the normal velocity jump penalization in order to get the desired pressure robustness effect the analysis is performed for the stokes equations and more complex flows and crouzeix raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings

fourier gauss transforms of the al salam chihara polynomials we discuss classical fourier gauss transforms of a three parameter family of the continuous al salam chihara polynomials it is shown that they are related to both the continuous big q hermite and the q hermite polynomials

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

higher order discontinuous galerkin time stepping and local projection stabilization techniques for the transient stokes problem abstract we introduce and analyze discontinuous galerkin time discretizations coupled with continuous finite element methods based on equal order interpolation in space for velocity and pressure in transient stokes problems spatial stability of the pressure is ensured by adding a stabilization term based on local projection we present error estimates for the semi discrete problem after discretization in space only and for the fully discrete problem the fully discrete pressure shows an instability in the limit of small time step length numerical tests are presented which confirm our theoretical results including the pressure instability

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

on the stability of projection methods for the incompressible navier stokes equations based on high order discontinuous galerkin discretizations abstract the present paper deals with the numerical solution of the incompressible navier stokes equations using high order discontinuous galerkin dg methods for discretization in space for dg methods applied to the dual splitting projection method instabilities have recently been reported that occur for small time step sizes since the critical time step size depends on the viscosity and the spatial resolution these instabilities limit the robustness of the navier stokes solver in case of complex engineering applications characterized by coarse spatial resolutions and small viscosities by means of numerical investigation we give evidence that these instabilities are related to the discontinuous galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high order accuracy with respect to the temporal discretization this new formulation is stable in the limit of small time steps for both equal order and mixed order polynomial approximations although the dual splitting scheme itself includes inf sup stabilizing contributions we demonstrate that spurious pressure oscillations appear for equal order polynomials and small time steps highlighting the necessity to consider inf sup stability explicitly

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

state dependent fire models and related renewal processes we introduce a general class of stochastic processes forced by instantaneous random fires i e jumps that reset the state variable x to a given value since in many physical systems the fire activity is often dependent on the actual value of the state variable as in the case of natural fires in ecosystems and firing dynamics in neuronal activity the frequency of fire occurrence is assumed to be state dependent such dynamics leads to independent interfire statistics i e to renewal point processes various functions relating the frequency of fire occurrence to x t are analyzed and compared the relation between the probabilistic dynamics of x t and the interfire statistics is derived and some exact probability distribution of both x t and the interfire times are obtained for systems with different degrees of complexity after studying processes in which the fire activity is coupled only to a deterministic drift we also analyze processes forced by either additive or multiplicative gaussian white noise

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

symmetry techniques for the al salam chihara polynomials an algebraic interpretation for the al salam chihara q orthogonal polynomials is presented using an explicit realization in terms of divided difference operators their symmetry q algebra is determined and studied this algebraic structure allows one to derive a new expansion formula for these polynomials

noise induced transitions in state dependent dichotomous processes in a number of stochastic systems the random forcing is represented as a dichotomous markov noise a common characteristic of these models is that the noise is usually supposed to be independent of the state of the forced dynamical system however there are several situations in which positive or negative feedback exist between the system and the random driver this paper investigates a class of systems characterized by feedback between dichotomous markov noise and the system s dynamics the effect of the feedback is accounted for through a state dependency in the transition rates of the dichotomous noise we study noise induced transitions in these systems with special attention to the delicate problem of correctly defining the deterministic counterpart of the stochastic system we find that i if in the absence of any feedback the dynamical system has a single deterministic stable point the deterministic dynamics remain monostable when a negative feedback is introduced while they may become bistable in the presence of a positive feedback ii noise may induce bistability in the presence of a null or negative feedback iii bistable deterministic dynamics induced by the positive feedback may be destroyed by the noise which tends to stabilize the system around a new intermediate stable state between those of the deterministic dynamics

the covariant linear oscillator and generalized realization of the dynamical su 1 1 symmetry algebra an exactly solvable problem for the finite difference schrodinger equation in the relativistic configurational space is considered the appropriate finite difference generalization of the factorization method is developed the theory of new special functions the relativistic hermite polynomials in which the solutions are expressed is constructed

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

probabilistic dynamics of some jump diffusion systems some exact solutions to the forward chapman kolmogorov equation are derived for processes driven by both gaussian and compound poisson shot noise the combined action of these two forms of white noise is analyzed in transient and equilibrium conditions for different jump distributions and additive gaussian noise steady state distributions with power law tails are obtained for exponentially distributed jumps and multiplicative linear gaussian noise two applications are discussed namely the virtual waiting time or takacs process including gaussian oscillations and a simplified model of soil moisture dynamics in which rainfall is modeled as a compound poisson process and fluctuations in potential evapotranspiration are gaussian

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 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

elements of q harmonic analysis abstract measures that act as q analogs of lebesgue measure are introduced some q analogs of fourier and laplace transforms are studied last the connection between certain spaces of q exponential type and discrete q fourier transforms is discussed

more on the q oscillator algebra and q orthogonal polynomials properties of certain q orthogonal polynomials are connected to the q oscillator algebra the wall and q laguerre polynomials are shown to arise as matrix elements of q exponentials of the generators in a representation of this algebra a realization is presented where the continuous q hermite polynomials form a basis of the representation space various identities are interpreted within this model in particular the connection formula between the continuous big q hermite and the continuous q hermite polynomials is thus obtained and two generating functions for these last polynomials are derived algebraically

fourier gauss transforms of the continuous big q hermite polynomials we examine the fourier gauss transformation properties of a two parameter family of the continuous big q hermite polynomials

fourier gauss transforms of the askey wilson polynomials we discuss classical fourier gauss integral transforms for a five parameter family of the askey wilson polynomials

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

velocity correction projection methods for incompressible flows we introduce and study a new class of projection methods namely the velocity correction methods in standard form and in rotational form for solving the unsteady incompressible navier stokes equations we show that the rotational form provides improved error estimates in terms of the h1 norm for the velocity and of the l2 norm for the pressure we also show that the class of fractional step methods introduced in s a orsag m israeli and m deville j sci comput 1 1986 pp 75 111 and k e karniadakis m israeli and s a orsag j comput phys 97 1991 pp 414 443 can be interpreted as the rotational form of our velocity correction methods thus to the best of our knowledge our results provide the first rigorous proof of stability and convergence of the methods in those papers we also emphasize that contrary to those of the above groups our formulations are set in the standard l2 setting and consequently they can be easily implemented by means of any variational approximation techniques in particular the finite element methods

on the rogers szego polynomials an orthogonality relation on the full real line for the rogers szego polynomials is discussed it is argued that fourier transformation with the standard exponential kernel exp ixy relates the rogers szego and stieltjes wigert functions

robust multigrid for high order discontinuous galerkin methods we present a polynomial multigrid method for nodal interior penalty and local discontinuous galerkin formulations of the poisson equation on cartesian grids for smoothing we propose two classes of overlapping schwarz methods the first class comprises element centered and the second face centered methods within both classes we identify methods that achieve superior convergence rates prove robust with respect to the mesh spacing and the polynomial order at least up to p 32 consequent structure exploitation yields a computational complexity of o pn where n is the number of unknowns further we demonstrate the suitability of the face centered method for element aspect ratios up to 32 p multigrid for discontinuous formulations of poisson problems on cartesian grids robust with respect to the mesh size and polynomial order at least up to p 32 convergence rates between 0 02 and 0 003 improving with increasing p computational complexity of pn for n unknowns linear complexity in runtime excellent performance up to aspect ratios of 1 16

solving the chapman kolmogorov equation for a jumping process a general solution to the chapman kolmogorov equation for a jumping process called the kangaroo process is derived a special case of algebraic dependences is discussed in detail in particular simple asymptotic formulas for probability distribution are presented it is demonstrated that there are two different classes of limiting stationary distributions an expression for the covariance is also derived

continuous hahn polynomials a slightly more general orthogonality relation for the hahn polynomials of a continuous variable than the recent one given by atakishiyev and suslov 1985 is given here

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

intertime jump statistics of state dependent poisson processes a method to obtain the probability distribution of the interarrival times of jump occurrences in systems driven by state dependent poisson noise is proposed such a method uses the survivor function obtained by a modified version of the master equation associated to the stochastic process under analysis a model for the timing of human activities shows the capability of state dependent poisson noise to generate power law distributions the application of the method to a model for neuron dynamics and to a hydrological model accounting for land atmosphere interaction elucidates the origin of characteristic recurrence intervals and possible persistence in state dependent poisson models

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 a one parameter family of q exponential functions we examine the properties of a family of q exponential functions which depend on an extra parameter these functions have a well defined meaning for both the 0 1 cases if only it is shown that any two members of this family with different values of the parameter are related to each other by a fourier gauss transformation

a high order semi explicit discontinuous galerkin solver for 3d incompressible flow with application to dns and les of turbulent channel flow abstract we present an efficient discontinuous galerkin scheme for simulation of the incompressible navier stokes equations including laminar and turbulent flow we consider a semi explicit high order velocity correction method for time integration as well as nodal equal order discretizations for velocity and pressure the non linear convective term is treated explicitly while a linear system is solved for the pressure poisson equation and the viscous term the key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under resolved high reynolds number flows as well as small time steps this penalty method is similar to the grad div stabilization widely used in continuous finite elements we further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations the solver is specifically designed for large scale computations through matrix free linear solvers including efficient preconditioning strategies and tensor product elements which have allowed us to scale this code up to 34 4 billion degrees of freedom and 147 456 cpu cores we validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large eddy simulation of turbulent channel flow at r e 180 as well as 590

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