a semi implicit numerical scheme for reacting flow an additive semi implicit projection scheme for the simulation of unsteady combustion in two dimensions is constructed the scheme relies on a zero mach number formulation of the compressible conservation equations with detailed chemistry the governing equations are discretized in space using second order differences and integrated in time using a semi implicit approach time integration of the evolution equations for species mass fraction thermodynamic pressure and density is performed using a semi implicit nonsplit scheme that combines a second order predictor corrector treatment of convection and diffusion terms and a stiff integrator for the reaction source terms meanwhile the momentum equations are integrated using a second order projection scheme the projection scheme is based on a predictor corrector approach that couples the evolution of the velocity and density fields in order to stabilize computations of reacting flows with large density variations a pressure poisson equation is inverted following both the predictor and corrector steps using a fast solver the advantages of the stiff integration of reaction source terms are analyzed by comparing the performance of the scheme to that of a predictor corrector scheme in which reaction and diffusion are integrated in a similar nonstiff fashion the comparison in based on both one dimensional 1d unsteady tests of a premixed methane air flame and unsteady two dimensional tests of the same flame interacting with a counterrotating vortex pair in both cases the grimech1 2 reaction mechanism with 32 species and 177 elementary reactions is used computed results show that the stiff reaction scheme enables selection of larger time steps and thus leads to substantial improvement in the performance of the computations for the present reaction mechanism and flame conditions speedup factors of about 10 are achieved in the 1d tests and about five in two dimensions possible extensions of the present scheme to further improve efficiency are also discussed

a stochastic projection method for fluid flow ii random process an uncertainty quantification scheme is developed for the simulation of stochastic thermofluid processes the scheme relies on spectral representation of uncertainty using the polynomial chaos pc system the solver combines a galerkin procedure for the determination of pc coefficients with a projection method for efficiently simulating the resulting system of coupled transport equations implementation of the numerical scheme is illustrated through simulations of natural convection in a 2d square cavity with stochastic temperature distribution at the cold wall the properties of the uncertainty representation scheme are analyzed and the predictions are contrasted with results obtained using a monte carlo approach

fast fourier transforms for direct solution of poisson s equation with staggered boundary conditions abstract this paper describes pre and postprocessing algorithms used to incorporate the fast fourier transform fft into the solution of finite difference approximations to multi dimensional poisson s equation on a staggered grid where the boundary is located midway between two grid points all frequently occurring boundary conditions neumann dirichlet or cyclic are considered including the combination of staggered neumann boundary condition on one side with nonstaggered dirichlet boundary condition on the other side experiences from implementing these algorithms in vectorized coding in fortran subroutines are reported

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

on the bootstrap for moran s i test for spatial dependence this paper is concerned with the use of the bootstrap for statistics in spatial econometric models with a focus on the test statistic for moran s i test for spatial dependence we show that for many statistics in spatial econometric models the bootstrap can be studied based on linear quadratic lq forms of disturbances by proving the uniform convergence of the cumulative distribution function for lq forms to that of a normal distribution we show that the bootstrap is generally consistent for test statistics that can be approximated by lq forms including moran s i possible asymptotic refinements of the bootstrap are most commonly studied using edgeworth expansions for spatial econometric models we may establish asymptotic refinements of the bootstrap based on asymptotic expansions of lq forms when the disturbances are normal we prove the existence of the usual edgeworth expansions for lq forms when the disturbances are not normal we establish an asymptotic expansion of lq forms based on martingales these results are applied to show the second order correctness of the bootstrap for moran s i test

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

a robust numerical model for premixed flames with high density ratios based on new pressure correction and imex schemes in this study we present a model for the interaction of premixed flames with obstacles in a channel flow although the flow equations are solved with direct numerical simulation using a low mach number approximation the resolution used in the computation is limited 1mm hence the inner structure of the flame and the chemical scales are not solved the species equations are substituted with a source term in the energy equation that simulates a one step global reaction a level set method is applied to track the position of the flame and its zero level is used to activate the source term in the energy equation only at the flame front an immersed boundary method reproduces the geometry of the obstacles the main contribution of the paper is represented by the proposed numerical approach an imex implicit explicit runge kutta scheme is used for the time integration of the energy equation and a new pressure correction algorithm is introduced for the time integration of the momentum equations the approach presented here allows to calculate flames which produce high density ratios between burnt and unburnt regions the model is verified by simulating first simple solutions for one and two dimensional flames at last the experiments performed by masri and ibrahim with square and rectangular bodies are calculated

solution of the time domain inverse resistivity problem in the model reduction framework part i one dimensional problem with siso data many time domain problems in engineering applications can be described by means of parameter dependent time invariant dynamic systems we are interested in parameter estimation by fitting available transient measurements using the nonlinear least square method as the main application we consider the control source electromagnetic csem method of geophysical exploration governed by the diffusion maxwell system where the unknown parameters describe the spatial distribution of electrical resistivity we propose a novel model reduction approach for constructing an efficient approximation of the jacobian the reduction is based on subspace projection of the state variable and it allows us to split the time and space dependence of the derivative we examine several popular rational krylov subspaces and single out the h 2 optimal subspace that not only minimizes the approximation error but completely annuls its influence on the inversion result preliminary numerical experiments with a simplified one dim

a parallel dirk method for stiff initial value problems abstract in this note we propose a fast parallel iteration process for solving a low order implicit runge kutta method the resulting scheme can be regarded as a parallel singly diagonally implicit runge kutta pdirk method on a two processor computer this method requires effectively the solution of two implicit relations per step by two numerical experiments we compare this method with some sequential methods from the literature and show its efficient behaviour

regular article a semi implicit numerical scheme for reacting flow a stiff 1 operator split projection scheme is constructed for the simulation of unsteady two dimensional reacting flow with detailed kinetics the scheme is based on the compressible conservation equations for an ideal gas mixture in the zero mach number limit the equations of motion are spatially discretized using second order centered differences and are advanced in time using a new stiff predictor corrector approach the new scheme is a modified version of the additive stiff scheme introduced in a previous effort by h n najm p s wyckoff and o m knio 1998 j comput phys 143 381 the predictor updates the scalar fields using a strang type operator split integration step which combines several explicit diffusion sub steps with a single stiff step for the reaction terms such that the global time step may significantly exceed the critical diffusion stability limit convection terms are explicitly handled using a second order multi step scheme the velocity field is advanced using a projection scheme which consists of a partial convection diffusion update followed by a pressure correction step a split approach is also adopted for the convection diffusion step in the momentum update this splitting combines an explicit treatment of the convective terms at the global time step with several explicit fractional steps for diffusion finally a corrector step is implemented in order to couple the evolution of the density and velocity fields and to stabilize the computations the corrector acts only on the convective terms and the pressure field while the predicted updates due to diffusion and reaction are left unchanged the correction of the scalar fields is implemented using a single step non split non stiff second order time integration a similar procedure is used for the velocity field which is followed by a pressure projection step the performance and behavior of the operator split scheme are first analyzed based on tests for a nonlinear reaction diffusion equation in one space dimension followed by computations with a detailed c 1c 2 methane air mechanism in one and two dimensions the tests are used to verify that the scheme is effectively second order in time and to suggest guidelines for selecting integration parameters including the number of fractional diffusion steps and tolerance levels in the stiff integration for two dimensional simulations with the present reaction mechanism flame conditions and resolution parameters speedup factors of about 5 are achieved over the previous additive scheme and about 25 over the original explicit scheme

numerical instability due to varying time steps in explicit wave propagation and mechanics calculations explicit central difference time integration is frequently used to solve the wave equation and the classical criterion for numerical stability is the courant friedrichs lewy condition similarly explicit integration of a spring mass mechanical system has a stability condition these conditions are derived under the assumption of constant time steps this paper demonstrates the new and perhaps surprising result that numerical instability may occur when time steps vary even though all steps are substantially less than the constant step criterion

a functional optimization approach to an inverse magneto convection problem a formulation and an fem implementation for the solution of an ill posed inverse design magneto convection problem is proposed in particular an incompressible viscous electrically conducting liquid material occupying a given domain is considered convection is driven by buoyancy effects as well as a lorentz force generated due to the applied magnetic field thermal boundary conditions are only prescribed in the part h0 of the boundary in addition the temperature distribution is also prescribed in the part i of the boundary h1 where the heat flux is known i e i is a boundary with overspecified thermal boundary conditions the inverse magneto convection problem is posed as an optimization problem in l2 i 0 tmax for the calculation of the boundary heat flux qo x t with x t h0 0 tmax the optimization scheme minimizes the discrepancy m x t x t qo l2 i 0 tmax between the temperature x t qo calculated from the solution of a direct problem for each flux qo and the desired or measured temperature m x t on the boundary i the exact l2 gradient of the cost functional is obtained from appropriately defined adjoint fields the adjoint problem is defined from the sensitivity operators obtained by linearization of the equations governing the direct problem the standard supg pspg stabilized fem formulation for incompressible fluid flow simulation is here extended to thermo magnetically driven flows the proposed fem formulation is used for the calculation of the direct adjoint and sensitivity thermal fluid flow and electric potential fields the entire optimization algorithm is solved using the conjugate gradient method finally the method is demonstrated through the solution of a few inverse problems with known results the need for regularization is identified in one of the example problems with uniformly distributed random errors in the temperature data m x t and a h1 regularized formulation is introduced to obtain stable solutions finally the numerical results are elucidated and potential applications are addressed

time accurate calculation of variable density flows with strong temperature gradients and combustion a time accurate algorithm is proposed for low mach number variable density flows with or without chemical reactions the algorithm is based on a predictor corrector time integration scheme that employs a projection method for the momentum equation a constant coefficient poisson equation is solved for the pressure following both the predictor and corrector steps to fully satisfy the continuity equation at each time step spatial discretization is performed on a collocated grid system that offers computational simplicity and straightforward extension to curvilinear coordinate systems to avoid the pressure odd even decoupling that is typically encountered in such grids a flux interpolation technique is introduced for the equations governing variable density flows an important characteristic of the proposed algorithm is that it can be applied to flows in both open and closed domains its robustness and accuracy are illustrated with a series of numerical experiments in particular we present simulations of non isothermal turbulent channel flow as well as simulations of a premixed flame vortex interaction

positive solutions of an infinite boundary value problem for nth order nonlinear impulsive singular integro differential equations in banach spaces abstract in this paper the cone theory and monch fixed point theorem combined with a monotone iterative technique are used to investigate the positive solutions of a class of boundary problems for n th order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in banach spaces the conditions for the existence of a positive solution are established in addition an explicit iterative approximation of the solution for the boundary value problem is derived

malliavin calculus and asymptotic expansion for martingales we present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale the emphasis is put on the use of the malliavin calculus the uniform nondegeneracy of the malliavin covariance under certain truncation plays an essential role as the cramer condition did in the case of independent observations applications to statistics are presented

a conservative discrete compatibility constraint low mach pressure correction algorithm for time accurate simulations of variable density flows in this paper we develop the discrete compatibility constraint pressure correction algorithm for transient simulations of variable density flows at low mach numbers the constraint for the velocity field is constructed from a combination of the discrete equations of continuity and scalar e g energy transport imposing that the newly predicted state must be compatible in agreement with the equation of state this way mass and scalar conservation are guaranteed and the equation of state is exactly fulfilled at every time step for comparison reasons two other types of well known pressure correction algorithms are also used the first class denoted as continuity constraint pressure correction is based on a constraint for the velocity field that is derived solely from the continuity equation the second class denoted as analytical compatibility constraint pressure correction constructs the constraint from an analytical combination of the material derivative of the equation of state and the continuity and scalar equations the algorithms are tested for three example fluid configurations a single fluid ideal gas a two fluid inert mixture and a two fluid reacting mixture the latter is special in the sense that the equation of state is non linear and not everywhere differentiable the continuity constraint pressure correction algorithm yields unstable solutions if density ratios are high the analytical compatibility constraint pressure correction algorithm yields stable results but the predicted states do not correspond to the equation of state the discrete compatibility constraint pressure correction algorithm performs well on all test cases the simulation results are stable and exactly match the equation of state

positive solutions for nth order nonlinear impulsive singular integro differential equations on infinite intervals in banach spaces abstract in this paper the cone theory and monch fixed point theorem combined with a monotone iterative technique are used to investigate the positive solutions of a class of boundary problems for n th order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in banach spaces the conditions for the existence of a positive solution are established in addition an explicit iterative approximation of the solution for the boundary value problem is derived

existence of positive solutions for nth order nonlinear impulsive singular integro differential equations in banach spaces abstract this paper obtains the existence of positive solutions for a boundary value problem of n th order nonlinear impulsive singular integro differential equations on an infinite interval in a banach space by means of the fixed point theory of completely continuous operators

conservative high order finite difference schemes for low mach number flows three finite difference algorithms are proposed to solve a low mach number approximation for the navier stokes equations these algorithms exhibit fourth order spatial and second order temporal accuracy they are dissipation free and thus well suited for dns and les of turbulent flows the key ingredient common to each of the methods presented is a poisson equation with variable coefficient that is solved for the hydrodynamic pressure this feature ensures that the velocity field is constrained correctly it is shown that this approach is needed to avoid violation of the conservation of kinetic energy in the inviscid limit which would otherwise arise through the pressure term in the momentum equation an existing set of finite difference formulae for incompressible flow is generalized to handle arbitrary large density fluctuations with no violation of conservation through the non linear convective terms an algorithm which conserves mass momentum and kinetic energy fully is obtained when an approximate equation of state is used instead of the exact one results from a model problem are used to show both spatial and temporal convergence rates and several test cases are presented to illustrate the performance of the algorithms

explicit time scale splitting algorithm for stiff problems auto ignition of gaseous mixtures behind a steady shock a new explicit algorithm based on the computational singular perturbation csp method is presented this algorithm is specifically designed to solve stiff problems and its performance increases with stiffness the key concept in its structure is the splitting of the fast from the slow time scales in the problem realized by embedding csp concepts into an explicit scheme in simple terms the algorithm marches in time with only the terms producing the slow time scales while the contribution of the terms producing the fast time scales is taken into account at the end of each integration step as a correction the new algorithm is designed for the integration of stiff systems of pdes by means of explicit schemes for simplicity in the presentation and discussion of the different features of the new algorithm a simple test case is considered involving the auto ignition of a methane air mixture behind a normal shock wave which is described by a system of odes the performance of the new algorithm accuracy and computational efficiency is then compared with the well known lsode package its merits when used for the solution of systems of pdes are discussed although when dealing with a stiff system of odes the new algorithm is shown to provide equal accuracy with that delivered by lsode at the cost of higher execution time the results indicate that its performance could be superior when facing a stiff system of pdes

transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations abstract using a new method for dealing with the impulse term of second impulsive differential equations we are concerned with determining values of for which there exist positive solutions of the following boundary value problems in the form of x t f t x t 0 t j t t k x t k x t k c k x t k k 1 2 n a x 0 b x 0 a x 1 b x 1 0 1 h s x t d t where 0 is a positive parameter c k is a real sequence with c k 1 k 1 2 n the dependence of positive solution x t on the parameter is also studied i e lim x or lim x 0 the proof of our main result is based upon transformation techniques and fixed point theories this is probably the first time the existence of positive solutions of second order impulsive differential equations has been studied by applying this technique

accurate bootstrap confidence limits for the cumulative hazard and survivor functions under random censoring abstract in clinical trials and other settings confidence intervals for survival probabilities at a fixed point in time are often required typically such intervals are constructed on the cumulative hazard scale the quantiles required for computing these intervals are generally obtained using a first order normal theory approximation the appropriateness of such intervals for small sample sizes or under moderate to heavy censoring is suspect we demonstrate that significant improvement is possible for randomly censored data using the nonparametric bootstrap in particular using edgeworth expansion it is shown that the percentile t and bc a bootstrap methodologies based on the studentized nelson aalen estimator yield second order correct confidence limits for the cumulative hazard function at a fixed time point an explicit formula for the correction term of the edgeworth expansion is found easy to compute estimators for the components of the correction term are also found and lead to simple analytic

adjoint consistency analysis of discontinuous galerkin discretizations this paper is concerned with the adjoint consistency of discontinuous galerkin dg discretizations adjoint consistency in addition to consistency is the key requirement for dg discretizations to be of optimal order in l 2 as well as measured in terms of target functionals we provide a general framework for analyzing the adjoint consistency of dg discretizations which is also useful for the derivation of adjoint consistent methods this analysis will be performed for the dg discretizations of the linear advection equation the interior penalty dg method for elliptic problems and the dg discretization of the compressible euler equations this framework is then used to derive an adjoint consistent dg discretization of the compressible navier stokes equations numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions

a unique positive solution for nth order nonlinear impulsive singular integro differential equations on unbounded domains in banach spaces in this paper the fixed point theory combined with a monotone iterative technique is used to investigate the unique positive solution of a boundary problem for nth order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in a banach space the conditions for the existence of a unique positive solution are established in addition an explicit iterative sequence for approximating the solution of the boundary value problem is derived together with an error estimate furthermore the conditions of the theorems can be easily verified

asymptotic expansions and bootstrapping distributions for dependent variables a martingale approach the paper develops a one step triangular array asymptotic expansion for continuous martingales which are asymptotically normal mixing conditions are not required but the quadratic variations of the martingales must satisfy a law of large numbers and a central limit type condition from this result we derive expansions for the distributions of estimators in asymptotically ergodic differential equation models and also for the bootstrapping estimators of these distributions

partial mixing and edgeworth expansion introducing a conditional mixing property gotze and hipp s theory is generalized to a continuous time conditional markov process satisfying this property the malliavin calculus for jump processes applies to random coefficient stochastic differential equations with jumps with the aid of the support theorem to verify the non degeneracy condition i e a conditional type cramer condition

a conservative level set method suitable for variable order approximations and unstructured meshes this paper presents a formulation for free surface computations capable of handling complex phenomena such as wave breaking without excessive mass loss or smearing of the interface the formulation is suitable for discretizations using finite elements of any topology and order or other approaches such as isogeometric and finite volume methods furthermore the approach builds on standard level set tools and can therefore be used to augment existing implementations of level set methods with discrete conservation properties implementations of the method are tested on several difficult two and three dimensional problems including two incompressible air water flow problems with available experimental results linear and quadratic approximations on unstructured tetrahedral and trilinear approximations on hexahedral meshes were tested global conservation and agreement with experiments as well as computations by other researchers are obtained

algorithm developments for discrete adjoint methods this paper presents a number of algorithm developments for adjoint methods using the discrete approach in which the discretisation of the non linear equations is linearised and the resulting matrix is then transposed with a new iterative procedure for solving the adjoint equations exact numerical equivalence is maintained between the linear and adjoint discretisations the incorporation of strong boundary conditions within the discrete approach is discussed as well as a new application of adjoint methods to linear unsteady flow in turbomachinery

krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes application to discontinuous galerkin methods abstract integration factor methods are a class of exactly linear part time discretization methods in q nie y t zhang r zhao efficient semi implicit schemes for stiff systems journal of computational physics 214 2006 521 537 a class of efficient implicit integration factor iif methods were developed for solving systems with both stiff linear and nonlinear terms arising from spatial discretization of time dependent partial differential equations pdes with linear high order terms and stiff lower order nonlinear terms the tremendous challenge in applying iif temporal discretization for pdes on high spatial dimensions is how to evaluate the matrix exponential operator efficiently for spatial discretization on unstructured meshes to solve pdes on complex geometrical domains how to efficiently apply the iif temporal discretization was open in this paper we solve this problem by applying the krylov subspace approximations to the matrix exponential operator then we apply this novel time discretization technique to discontinuous galerkin dg methods on unstructured meshes for solving reaction diffusion equations numerical examples are shown to demonstrate the accuracy efficiency and robustness of the method in resolving the stiffness of the dg spatial operator for reaction diffusion pdes application of the method to a mathematical model in pattern formation during zebrafish embryo development shall be shown

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

extremal solutions for nth order impulsive integro differential equations on the half line in banach spaces abstract by establishing a comparison result and using the upper and lower solutions the author obtains the existence of maximal and minimal solutions for a boundary value problem of n th order nonlinear impulsive integro differential equations on the half line in banach spaces

multiple positive solutions of a boundary value problem for nth order impulsive integro differential equations in a banach space abstract in this paper the author obtains the existence of multiple positive solutions for a boundary value problem of a class of nth order nonlinear impulsive integro differential equations on an infinite interval in a banach space by means of the fixed point index theory of completely continuous operators

an operator splitting algorithm for coupled one dimensional advection diffusion reaction equations abstract an operator splitting algorithm for a system of one dimensional advection diffusion reaction equations describing the transport of non conservative pollutants is presented in this paper the algorithm is a strang type splitting procedure incorporating contributions from the inhomogeneous terms by the duhamel s principle the associated homogeneous equations are split into advection diffusion and reaction equations and solved by a backward method of characteristic a finite element method and an explicit runge kutta method respectively the boundary conditions applicable to the split equations are derived numerical analyses of the algorithm consisting of the stability the accuracy and the convergence of the solution procedure are presented the composite algorithm is second order accurate in time and space and conditionally stable the numerical characteristics of the algorithm are demonstrated by several examples

on mass conservation and solvability of the discretized variable density zero mach navier stokes equations abstract this paper presents a study of a variable density zero mach navier stokes discretization and subtle difficulties associated with the pressure a consistent compatibility condition between the temperature and velocity is first derived it is shown that one can accurately and efficiently solve all primitive variables with the uniform order of accuracy of the specified discretization and time marching schemes it is shown that higher density ratios than those commonly reported are achievable and it is in part traced to the solvability of the pressure poisson equation numerical properties are assessed with the method of manufactured solutions and several classical testing cases are carried out to validate the code ranging from variable density mixing problems to premixed flames with forcing

embedding and asymptotic expansions for martingales the paper develops a way of embedding general martingales in continuous ones in such a way that the quadratic variation of the continuous martingale has conditional cumulants given the original martingale that are explicitly given in terms of optional and predictable variations of the original process bartlett identities for the conditional cumulants are also found a main corollary to these results is the establishment of second and in some cases higher order asymptotic expansions for martingales

the random projection method for stiff multispecies detonation capturing in this paper we extend the random projection method proposed for general hyperbolic systems with stiff reaction terms for underresolved numerical simulation of stiff inviscid multispecies detonation waves the key idea in this method is to randomize the ignition temperatures in suitable domains several numerical experiments in both one and two dimensions demonstrate the reliability and robustness of this novel method

estimating jump diffusions using closed form likelihood expansions the indispensable role of likelihood expansions in financial econometrics for continuous time models has been established since the ground breaking work of ait sahalia 1999 2002a 2008 jump diffusions play an important role in modeling a variety of economic and financial variables as a significant generalization of li 2013 we propose a new closed form expansion for transition density of poisson driven jump diffusion models and its application in maximum likelihood estimation based on discretely sampled data technically speaking our expansion is obtained by perturbing paths of the underlying model correction terms can be calculated explicitly using any symbolic software numerical examples and monte carlo evidence for illustrating the performance of density expansion and the resulting approximate mle are provided in order to demonstrate the practical applicability of the method using the theoretical results in hayashi and ishikawa 2012 some convergence properties related to the density expansion and the approximate mle method can be justified under some standard sufficient but not necessary conditions

additive semi implicit runge kutta methods for computing high speed nonequilibrium reactive flows this paper is concerned with time stepping numerical methods for computing stiff semi discrete systems of ordinary differential equations for transient hypersonic flows with thermo chemical nonequilibrium the stiffness of the equations is mainly caused by the viscous flux terms across the boundary layers and by the source terms modeling finite rate thermo chemical processes implicit methods are needed to treat the stiff terms while more efficient explicit methods can still be used for the nonstiff terms in the equations this paper studies three different semi implicit runge kutta methods for additively split differential equations in the form ofu f u g u wherefis treated by explicit runge kutta methods andgis simultaneously treated by three implicit runge kutta methods a diagonally implicit runge kutta method and two linearized implicit runge kutta methods the coefficients of up to third order accurate additive semi implicit runge kutta methods have been derived such that the methods are both high order accurate and strongly a stable for the implicit terms the results of two numerical tests on the stability and accuracy properties of these methods are also presented in the paper

a highly accurate technique for the treatment of flow equations at the polar axis in cylindrical coordinates using series expansions numerical methods for solving the flow equations in cylindrical or spherical coordinates should be able to capture the behavior of the exact solution near the regions where the particular form of the governing equations is singular in this work we focus on the treatment of these numerical singularities for finite difference methods by reinterpreting the regularity conditions developed in the context of pseudospectral methods a generally applicable numerical method for treating the singularities present at the polar axis when nonaxisymmetric flows are solved in cylindrical coordinates using highly accurate finite difference schemes e g pade schemes on nonstaggered grids is presented governing equations for the flow at the polar axis are derived using series expansions near r 0 the only information needed to calculate the coefficients in these equations are the values of the flow variables and their radial derivatives at the previous iteration or time level these derivatives which are multivalued at the polar axis are calculated without dropping the accuracy of the numerical method using a mapping of the flow domain from 0 r 0 2 to r r 0 where r is the radius of the computational domain this allows the radial derivatives to be evaluated using high order differencing schemes e g compact schemes at points located on the polar axis the accuracy of the method is checked by comparison with the theoretical solution corresponding to a circular compressible forced jet in the regime of linear growth the proposed technique is illustrated by results from simulations of laminar forced jets and turbulent compressible jets using large eddy simulation les methods in terms of the general robustness of the numerical method and smoothness of the solution close to the polar axis the present results compare very favorably to similar calculations in which the equations are solved in cartesian coordinates at the polar axis or in which the singularity is removed by employing a staggered mesh in the radial direction without a mesh point at r 0 following the method proposed recently by mohseni and colonius 1 extension of the method described here for incompressible flows or for any other set of equations that is solved on a nonstaggered mesh in cylindrical or spherical coordinates with finite difference schemes of various levels of accuracy is immediate

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

edgeworth expansion for ergodic diffusions the edgeworth expansion for an additive functional of an ergodic diffusion is validated under fairly weak conditions the validation procedure does not depend on the stationarity or the geometric mixing property but exploits the strong markov property of the process in particular for an ito diffusion of dimension one verifiable conditions for the validity of the expansion are given in terms of the coefficients of the corresponding stochastic differential equation the maximum likelihood estimator for the cir process is treated as example

extremal solutions of nonlinear impulsive integrodifferential equations in banach spaces abstract in this paper we combine the monotone iterative technique with fixed point theory to investigate the extremal solutions of nonlinear impulsive integrodifferential equations in banach spaces existence criteria are obtained and several examples are worked out

direct numerical simulation of a turbulent reactive plume on a parallel computer a computational algorithm is described for direct numerical simulation dns of a reactive plume in spatially evolving grid turbulence the algorithm uses sixth order compact differencing in conjunction with a fifth order compact boundary scheme which has been developed and found to be stable a compact filtering method is discussed as a means of stabilizing calculations where the viscous diffusive terms are differenced in their conservative form this approach serves as an alternative to nonconservative differencing previously advocated as a means of damping the 2 waves in numerically solving the low mach number equations the time derivative of the density field in the pressure poisson equation was found to be the most destabilizing part of the calculation even ordered finite difference approximations to this derivative were found to be more stable allow for larger density gradients than odd ordered approximations turbulence at the inlet boundary is generated by scanning through an existing three dimensional field of fully developed turbulence in scanning through the inlet field it was found that a high order interpolation e g cubic spline interpolation is necessary in order to provide continuous velocity derivatives regarding pressure a neumann inlet condition combined with a dirichlet outlet condition was found to work well the chemistry follows the single step irreversible global reaction fuel r oxidizer 1 r product heat with parameters chosen to match experimental data as far as allowed by resolution constraints simulation results are presented for four different cases in order to examine the effects of heat release damkohler number and arrhenius kinetics on the flow physics statistical data from the dns are compared to theory and wind tunnel data and found in reasonable agreement with regard to growth of turbulent length scales decay of turbulent kinetic energy decay of centerline scalar concentration decrease in scalar rms and spread of plume profile reactive scalar statistics are consistent with expected behavior

a local anisotropic adaptive algorithm for the solution of low mach transient combustion problems a novel numerical algorithm for the simulation of transient combustion problems at low mach and moderately high reynolds numbers is presented these problems are often characterized by the existence of a large disparity of length and time scales resulting in the development of directional flow features such as slender jets boundary layers mixing layers or flame fronts this makes local anisotropic adaptive techniques quite advantageous computationally in this work we propose a local anisotropic refinement algorithm using for the spatial discretization unstructured triangular elements in a finite element framework for the time integration the problem is formulated in the context of semi lagrangian schemes introducing the semi lagrange galerkin slg technique as a better alternative to the classical semi lagrangian sl interpolation the good performance of the numerical algorithm is illustrated by solving a canonical laminar combustion problem the flame vortex interaction first a premixed methane air flame vortex interaction with simplified transport and chemistry description test i is considered results are found to be in excellent agreement with those in the literature proving the superior performance of the slg scheme when compared with the classical sl technique and the advantage of using anisotropic adaptation instead of uniform meshes or isotropic mesh refinement as a more realistic example we then conduct simulations of non premixed hydrogen air flame vortex interactions test ii using a more complex combustion model which involves state of the art transport and chemical kinetics in addition to the analysis of the numerical features this second example allows us to perform a satisfactory comparison with experimental visualizations taken from the literature

a computational framework for the regularization of adjoint analysis in multiscale pde systems this paper examines the regularization opportunities available in the adjoint analysis and optimization of multiscale pde systems regularization may be introduced into such optimization problems by modifying the form of the evolution equation and the forms of the norms and inner products used to frame the adjoint analysis typically l2 brackets are used in the definition of the cost functional the adjoint operator and the cost functional gradient if instead we adopt the more general sobolev brackets the various fields involved in the adjoint analysis may be made smoother and therefore easier to resolve numerically the present paper identifies several relationships which illustrate how the different regularization options fit together to form a general framework the regularization strategies proposed are exemplified using a id kuramoto sivashinsky forecasting problem and computational examples are provided which exhibit their utility a multiscale preconditioning algorithm is also proposed that noticeably accelerates convergence of the optimization procedure application of the proposed regularization strategies to more complex optimization problems of physical and engineering relevance is also discussed

small time edgeworth type expansions for weakly convergent nonhomogeneous markov chains we consider triangular arrays of markov chains that converge weakly to a diffusion process second order edgeworth type expansions for transition densities are proved the paper differs from recent results in two respects we allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points

accurate confidence limits for quantiles under random censoring in survival analysis estimates of median survival times in homogeneous samples are often based on the kaplan meier estimator of the survivor function confidence intervals for quantiles such as median survival are typically constructed via large sample theory or the bootstrap the former has suspect accuracy for small sample sizes under moderate censoring and the latter is computationally intensive in this paper improvements on so called test based intervals and reflected intervals cf slud byar and green 1984 biometrics 40 587 600 are sought using the edgeworth expansion for the distribution of the studentized nelson aalen estimator derived in strawderman and wells 1997 journal of the american statistical association 92 we propose a method for producing more accurate confidence intervals for quantiles with randomly censored data the intervals are very simple to compute and numerical results using simulated data show that our new test based interval outperforms commonly used methods for computing confidence intervals for small sample sizes and or heavy censoring especially with regard to maintaining specified coverage

analysis of costate discretizations in parameter estimation for linear evolution equations a widely used approach to parameter identification is the output least squares formulation numerical methods for solving the resulting minimization problem almost invariably require the computation of the gradient of the output least squares functional when the identification problem involves time dependent distributed parameter systems or approximations thereof numerical evaluation of the gradient can be extremely time consuming the costate method can greatly reduce the cost of computing these gradients however questions have been raised concerning the accuracy and convergence of costate approximations even when the numerical methods being used are known to converge rapidly on the forward problem in this paper it is shown that the use of time marching schemes that yield high order accuracy on the forward problem does not necessarily lead to high order accurate costate approximations in fact in some cases these approximations do not converge at all however under certain circumstances rapidly converging gradient approximations do result because of rapid weak star type convergence of the costate approximations these issues are treated both theoretically and numerically

gauss seidel iteration for stiff odes from chemical kinetics a simple gauss seidel technique is proposed that exploits the special form of the chemical kinetics equations classical aitken extrapolation is applied to accelerate convergence the technique is meant for implementation in stiff solvers that are used in long range transport air pollution codes using operator splitting splitting necessarily gives rise to a great deal of integration restarts because the gauss seidel iteration works matrix free it has much less overhead than the modified newton method start up costs therefore can be kept low with this technique preliminary promising numerical results are presented for a prototype of a second order backward differentiation formula bdf solver applied to a stiff ordinary differential equation ode from atmospheric chemistry a favourable comparison with the general purpose bdf code dassl is included the matrix free technique may also be of interest for other chemically reacting fluid flow problems

method of upper and lower solutions for second order impulsive integro differential equations in a banach space abstract this paper discusses the initial value problem for second order nonlinear impulsive integro differential equations of volterra type in a banach space by means of upper and lower solutions by establishing a comparison result criteria on the existence of maximal and minimal solutions are obtained

a stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

numerical appraisal of three low mach number algorithms for radiative convective flows in enclosures abstract the present study investigates three different algorithms for the numerical simulation of non boussinesq convection with thermal radiative heat transfer based on a low mach number formulation the solution methodology employs a fractional step approach based on the finite volume method on arbitrary polyhedral meshes the three algorithms compute the coupled governing equations in a segregated manner using the conservative form of momentum equations in conjunction with a variable coefficient pressure poisson equation the first algorithm algorithm a uses conservation of mass and energy equation to compute density and temperature the other two algorithms algorithm b and algorithm c calculates temperature and density from the equation of state respectively and solves a conservative form of the continuity and energy equation to obtain density and temperature respectively the energy and mass conservation errors arising due to the use of algorithms b and c are derived concerning various non dimensional parameters governing the flow and heat transfer the significance of these errors is highlighted by performing investigations over a range of rayleigh prandtl and planck numbers for various two and three dimensional natural convection problems with radiative heat transfer finally the role of balancing of the pressure and buoyancy terms is emphasized for robust calculations of large temperature difference thermo buoyant convection with radiative heat transfer

adjoint algorithms for the navier stokes equations in the low mach number limit this paper describes a derivation of the adjoint low mach number equations and their implementation and validation within a global mode solver the advantage of using the low mach number equations and their adjoints is that they are appropriate for flows with variable density such as flames but do not require resolution of acoustic waves two versions of the adjoint are implemented and assessed a discrete adjoint and a continuous adjoint the most unstable global mode calculated with the discrete adjoint has exactly the same eigenvalue as the corresponding direct global mode but contains numerical artifacts near the inlet the most unstable global mode calculated with the continuous adjoint has no numerical artifacts but a slightly different eigenvalue the eigenvalues converge however as the timestep reduces apart from the numerical artifacts the mode shapes are very similar which supports the expectation that they are otherwise equivalent the continuous adjoint requires less resolution and usually converges more quickly than the discrete adjoint but is more challenging to implement finally the direct and adjoint global modes are combined in order to calculate the wavemaker region of a low density jet

a computationally efficient semi implicit iterative method for the time integration of reacting flows with stiff chemistry a semi implicit preconditioned iterative method is proposed for the time integration of the stiff chemistry in simulations of unsteady reacting flows such as turbulent flames using detailed chemical kinetic mechanisms emphasis is placed on the simultaneous treatment of convection diffusion and chemistry without using operator splitting techniques the preconditioner corresponds to an approximation of the diagonal of the chemical jacobian upon convergence of the sub iterations the fully implicit second order time accurate crank nicolson formulation is recovered performance of the proposed method is tested theoretically and numerically on one dimensional laminar and three dimensional high karlovitz turbulent premixed n heptane air flames the species lifetimes contained in the diagonal preconditioner are found to capture all critical small chemical timescales such that the largest stable time step size for the simulation of the turbulent flame with the proposed method is limited by the convective cfl rather than chemistry the theoretical and numerical stability limits are in good agreement and are independent of the number of sub iterations the results indicate that the overall procedure is second order accurate in time free of lagging errors and the cost per iteration is similar to that of an explicit time integration the theoretical analysis is extended to a wide range of flames premixed and non premixed unburnt conditions fuels and chemical mechanisms in all cases the proposed method is found theoretically to be stable and to provide good convergence rate for the sub iterations up to a time step size larger than 1 s this makes the proposed method ideal for the simulation of turbulent flames

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

high order multi implicit spectral deferred correction methods for problems of reactive flow models for reacting flow are typically based on advection diffusion reaction a d r partial differential equations many practical cases correspond to situations where the relevant time scales associated with each of the three sub processes can be widely different leading to disparate time step requirements for robust and accurate time integration in particular interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection the numerical strategy introduced in this paper is a general procedure to account for this time scale disparity the proposed methods are high order multi implicit generalizations of spectral deferred correction methods misdc methods constructed for the temporal integration of a d r equations spectral deferred correction methods compute a high order approximation to the solution of a differential equation by using a simple low order numerical method to solve a series of correction equations each of which increases the order of accuracy of the approximation the key feature of misdc methods is their flexibility in handling several sub processes implicitly but independently while avoiding the splitting errors present in traditional operator splitting methods and also allowing for different time steps for each process the stability accuracy and efficiency of misdc methods are first analyzed using a linear model problem and the results are compared to semi implicit spectral deferred correction methods furthermore numerical tests on simplified reacting flows demonstrate the expected convergence rates for misdc methods of orders three four and five the gain in efficiency by independently controlling the sub process time steps is illustrated for nonlinear problems where reaction and diffusion are much stiffer than advection although the paper focuses on this specific time scales ordering the generalization to any ordering combination is straightforward