the method of regularized stokeslets a numerical method for computing stokes flows in the presence of immersed boundaries and obstacles is presented the method is based on the smoothing of the forces leading to regularized stokeslets the resulting expressions provide the pressure and velocity field as functions of the forcing the latter expression can also be inverted to find the forces that impose a given velocity boundary condition the numerical examples presented demonstrate the wide applicability of the method and its properties solutions converge with second order accuracy when forces are exerted along smooth boundaries examples of segmented boundaries and forcing at random points are also presented

approximate analysis of two mass spring systems and buckling of a column max min approach mma is applied to obtain an approximate solution of three practical cases in terms of a nonlinear oscillation system after finding maximal and minimal solution thresholds of a nonlinear problem an approximate solution of the nonlinear equation can be easily achieved using he chengtian s interpolation numerical results indicate the effectiveness of the proposed method both in respect of the whole range of involved parameters as well as the excellent agreement with the approximate frequencies and periodic solutions with the exact ones it is predicted that mma can be found widely applicable in engineering

applications of a generalized galerkin s method to non linear oscillations of two degree of freedom systems abstract a generalized galerkin s method is formulated for multi degree of freedom holonomic systems two autonomous non linear two d o f oscillators are used as the examples for the applications of the method i free oscillations of a non linear mass spring system with two d o f ii two weakly non linearly coupled identical van der pol oscillators the accuracy of the approximate solutions is discussed the effect of different time intervals of integration on the results is also investigated

quick and accurate q parameterization in viscoelastic wave modeling we introduce a procedure for including the attenuation factor q in a consistent manner in seismic modeling and show 3d examples the q fitting over a chosen frequency band involves two algorithms the first creates starting values of relaxation times and the second does nonlinear inversion using the results of the first as initial values the resulting q function gives a good approximation to a constant q over the chosen frequency band the algorithm is combined with a finite difference f d code that includes topographies in 3d seismic media the velocity stress formulation for viscoelastic wave modeling is used with an arbitrary number of relaxation mechanisms to model a desired q behavior these equations are discretized by high order f ds in the interior of the medium and we gradually reduce the f d order to two at the stress free surface where we implement our free surface boundary conditions the seismic f d algorithm is applied to a marine seismic experiment with and without viscoelasticity to emphasize the importance of including physical attenuation and dispersion in seismic modeling their inclusion even for marine surveys is clearly important for lossy ocean bottoms our procedure for more accurate modeling of physical dispersion and attenuation may increase future motivation to include viscoelasticity in seismic inversion

application of he chengtian s interpolation to bethe equation in this work he chengtian s interpolation which has millennial history is applied to the bethe equation the result shows that the present technology is very convenient and effective

gaussian spectral rules for the three point second differences i a two point positive definite problem in a semi infinite domain we suggest an approach to grid optimization for a second order finite difference scheme for elliptic equations a model problem corresponding to the three point finite difference semidiscretization of the laplace equation on a semi infinite strip is considered we relate the approximate boundary neumann to dirichlet map to a rational function and calculate steps of our finite difference grid using the pade chebyshev approximation of the inverse square root it increases the convergence order of the neumann to dirichlet map from second to exponential without increasing the stencil of the finite difference scheme and losing stability

implicit second order immersed boundary methods with boundary mass abstract the immersed boundary method is a computational framework for problems involving the interaction of a fluid and immersed elastic structures immersed boundary computations typically evaluate the elastic forces explicitly in the configuration of the immersed elastic structure in many applications this results in a severe restriction on the time step we present a semi implicit and a fully implicit second order accurate immersed boundary method the methods provide a natural way to handle mass on the immersed elastic structures we demonstrate their performance for a prototypical fluid structure interaction problem the methods are shown to possess superior stability properties that significantly alleviate the typically severe time step restriction of explicit computations

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

adaptive time stepping for vesicle suspensions we present an adaptive high order accurate time stepping numerical scheme for the flow of vesicles suspended in stokesian fluids our scheme can be summarized as an approximate implicit spectral deferred correction sdc method applying a fully implicit sdc scheme to vesicle flows is prohibitively expensive for this reason we introduce several approximations our scheme is based on a semi implicit linearized low order time stepping method our discretization is spectrally accurate in space we expect that the accuracy can be arbitrary order but our examples suffer from order reduction which limits the observed accuracy to third order we also use invariant properties of vesicle flows constant area and boundary length in two dimensions to reduce the computational cost of error estimation for adaptive time stepping we present results in two dimensions for single vesicle flows constricted geometry flows converging flows and flows in a couette apparatus we experimentally demonstrate that the proposed scheme enables automatic selection of the time step size and high order accuracy

semi implicit krylov deferred correction methods for differential algebraic equations in the recently developed krylov deferred correction kdc methods for differential algebraic equation initial value problems 33 a picardtype collocation formulation is preconditioned using low order time integration schemes based on spectral deferred correction sdc and the resulting system is solved efficiently using newton krylov methods kdc methods have the advantage that methods with arbitrarily high order of accuracy can be easily constructed which have similar computational complexity as lower order methods in this paper we investigate semi implicit kdc si kdc methods in which the stiff component of the preconditioner is treated implicitly and the non stiff parts explicitly for certain types of problems such a semi implicit treatment can significantly reduce the computational cost of the preconditioner compared to fully implicit kdc fi kdc methods preliminary analysis and numerical experiments show that the convergence of newton krylov iterations in the si kdc methods is similar to that in fi kdc and hence the si kdc methods offer a reduction in overall computational cost for such problems

vibrations of a coupled two degree of freedom system abstract in this paper the motion of a two mass system with two degrees of freedom is discussed the masses are connected with three springs the motion of the system is described with a system of two coupled strong non linear differential equations for the case when the non linearity is of a cubic type the analytical solution of the system is obtained it is a combination of a jacobi elliptic function and a trigonometric function an approximate analytical method based on the krylov bogolubov procedure is developed for the system which contains small non linearities two examples are considered the case when all the three stiffnesses are non linear and the case when small damping acts the analytical solutions are compared with numerical ones they show a good agreement

best uniform rational approximation of on 0 1 a strong error estimate for the uniform rational approximation of x on 0 1 is given and its proof is sketched let e nn x 0 1 denote the minimal approximation error in the uniform norm then it is shown that formule holds true for each 0

krylov approximations for matrix square roots in stiff boundary value problems recently we proposed an algebraic difference scheme with extended stability properties for linear boundary value problems involving stiff differential equations of first order here an efficient approximation scheme is presented for matrix square roots which provides the stabilization of that scheme in case of stiffness it combines the use of low rank matrix approximations from projections onto krylov subspaces with an accelerated sign iteration for the matrix square root the krylov approximation being accurate in eigenspaces with large eigenvalues preserves the stability of the scheme and the 0 n3 square root computation need be performed only in lower dimension operation counts and numerical results show that the effort for the numerical scheme is essentially proportional to the number of stiff components but not to the norm of the coefficient matrix approximation properties of low rank krylov matrices which may be of independent interest are analyzed 1 the sqrt one step difference scheme a standard approximation for the differential equation in the linear boundary value problem bvp u x a x u x g x x e 0 1 u x er bou o b u l 8 is the trapezoidal rule on a suitable grid 0 xo 1 this restriction leads to unacceptable small stepsizes in regions where the solution is smooth this situation is usually referred to as stiffness in certain cases however a relaxation received march 27 1990 revised december 5 1990 1991 mathematics subject classification primary 65l10 65l20 65f30

a three dimensional computational method for blood flow in the heart 1 immersed elastic fibers in a viscous incompressible fluid abstract this paper describes the numerical solution of the 3 dimensional equations of motion of a viscous incompressible fluid that contains an immersed system of elastic fibers implementation details such as vectorization and the efficient use of external memory are discussed the method is applied to the damped vibrations of a fiber wound toroidal tube and empirical evidence of convergence is presented

a stochastic immersed boundary method for fluid structure dynamics at microscopic length scales in modeling many biological systems it is important to take into account flexible structures which interact with a fluid at the length scale of cells and cell organelles thermal fluctuations of the aqueous environment become significant in this work it is shown how the immersed boundary method of c s peskin the immersed boundary method acta num 11 2002 1 39 for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations a stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps an important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved partially resolved or fully resolved while retaining a good level of accuracy error estimates in each of these regimes are given for the method a number of theoretical and numerical checks are furthermore performed to assess its physical fidelity for a conservative force the method is found to simulate particles with the correct boltzmann equilibrium statistics it is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters the method is also shown to reproduce a well known hydrodynamic effect of a brownian particle in which the velocity autocorrelation function exhibits an algebraic 3 2 decay for long times b j alder t e wainwright decay of the velocity autocorrelation function phys rev a 1 1 1970 18 21 a few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method specifically we present simulations of osmotic effects of molecular dimers worm like chain polymer knots and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force the theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales

finite difference modeling of viscoelastic materials with quality factors of arbitrary magnitude to minimize acoustic noise designers of sonic logging tools often consider coatings of viscoelastic materials with very high attenuation properties efficient finite difference modeling of viscoelastic materials is a topic of current research to model viscoelastic materials in the time domain through finite differences efficiently one needs to replace the time convolution which enters in the stress strain relations by a set of first order differential equations this procedure is equivalent to computing a rational approximation of a certain form to the frequency dependent complex modulus of viscoelasticity known schemes for computing such approximations are designed to treat materials with low attenuation such as underground formations but fail to produce accurate or even physically meaningful results for highly attenuative materials we propose a novel scheme that allows one to construct for a given frequency range a uniformly optimal rational approximation for the most widely used model of mater

a general system of images for regularized stokeslets and other elements near a plane wall we derive a general system of images for regularized sources stokeslets and other related elements starting from an arbitrary regularization kernel blob used in the simulation of stokes flows in three dimensions bounded by a plane this generalizes previous work in which the image system for a stokeslet had been derived for one specific blob the significance of this generalization is that recent work on regularization methods requires the use of blobs designed to satisfy certain properties such as zero moment conditions and fast decay and thus it is absolutely necessary to have the system of images starting from an arbitrary blob the system of images for a regularized element consists of a set of several elements usually of higher order that produce a flow that is zero at the bounding plane in order for the resultant flow to vanish analytically at the wall two different but related blobs must be used for any given blob we provide the formula for the companion blob that accomplishes the cancellation and we derive a systematic way to compute the image system of regularized stokeslets sources and dipoles other elements can be derived from these by taking the limit as the regularization parameter approaches zero the system of images for the corresponding singular elements is found

validation of a simple method for representing spheres and slender bodies in an immersed boundary method for stokes flow on an unbounded domain we test the efficacy of using a single lagrangian point to represent a sphere and a one dimensional array of such points to represent a slender body in a new immersed boundary method for stokes flow a numerical parameter the spacing of the eulerian grid is used to determine the effective radius of the immersed sphere or slender body such representations are much less expensive computationally than those with two or three dimensional meshes of lagrangian points to perform this test we develop a numerical method to solve the discretized stokes equations on an unbounded eulerian grid which contains an arbitrary configuration of lagrangian points that apply force to the fluid and that move with the fluid we compare results computed with this new immersed boundary method to known results for spheres and rigid cylinders in stokes flow in r 3 we find that for certain choices of parameters the interactions with the fluid of a single lagrangian point accurately replicate those of a sphere of some particular radius independent of the location of the point with respect to the eulerian grid the interactions of a linear array of lagrangian points for certain choices of parameters accurately replicate those of a cylinder of some particular radius independent of the position and orientation of the array with respect to the eulerian grid the effective radius of the sphere and the effective radius of the cylinder turn out to be related in a simple and natural way our results suggest recipes for choosing parameters that should be useful to practitioners one surprising result is that one must not use too many lagrangian points in an array another is that the approximate delta functions traditionally used in the immersed boundary method perform much better than higher order delta functions with the same support

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

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

modeling of a constant q methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique linear anelastic phenomena in wave propagation problems can be well modeled through a viscoelastic mechanical model consisting of standard linear solids in this paper we present a method for modeling of constant q as a function of frequency based on an explicit closed formula for calculation of the parameter fields several standard linear solids connected in parallel can be tuned through a single parameter to yield an excellent constant q approximation the proposed method enables substantial savings in computations and memory requirements experiments show that the new method also yields higher accuracy in the modeling of q than e g the pade approximant method

application of the difference gaussian rules to solution of hyperbolic problems two of the authors earlier suggested a method of calculating special grid steps for three point finite difference schemes which yielded exponential superconvergence of the neumann to dirichlet map we apply this approach to solve the two dimensional time domain wave problem and the 2 5 d elasticity system in cylindrical coordinates our numerical experiments exhibit exponential convergence at prescribed points with the cost per grid node close to that of the standard second order finite difference scheme the scheme demonstrates high accuracy with slightly more than two grid points per wavelength the reduction of the grid size by one order compared to the standard scheme with the equidistant grids is observed

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

optimal finite difference grids and rational approximations of the square root i elliptic problems the main objective of this paper is optimization of second order finite difference schemes for elliptic equations in particular for equations with singular solutions and exterior problems a model problem corresponding to the laplace equation on a semi infinite strip is considered the boundary impedance neumann to dirichlet map is computed as the square root of an operator using the standard three point finite difference scheme with optimally chosen variable steps the finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator s spectrum we have implemented zolotarev s optimal rational approx imant obtained in terms of elliptic functions we have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant for bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number for the case of unbounded operators in sobolev spaces associated with elliptic equations the error decays as the exponential of the square root of the mesh dimension as an example we numerically compute the green function on the boundary for the laplace equation some features of the optimal grid obtained for the laplace equation remain valid for more general elliptic problems with variable coefficients 2000 john wiley sons inc

incorporation of attenuation into time domain computations of seismic wave fields the only numerically tractable way yet found to incorporate attenuation into numerical time domain computations of seismic wave fields is to approximate the viscoelastic modulus by a low order rational function of frequency the coefficients of this function can be determined by the pade approximation our test computations show however that this approximation generally is of poor quality therefore we suggest a new approach which is based on the rheological model of the generalized maxwell body which has a modulus of the desired rational form we choose the relaxation frequencies logarithmically equidistant in the frequency band of interest and determine the weight factors by simple numerical curve fitting to an arbitrary q law this approach is superior to the method above both in accuracy and in computational efficiency for most practical applications approximations of orders 2 or 3 are sufficient the computing time and memory requirements for a finite difference calculation are then approximately twice those of a purely elastic calculation as a first application of the method we compute sh channel waves in discontinuous coal seams with q 50 within the coal the results show that the high frequency airy phase is strongly attenuated this indicates that care has to be taken in comparing the results of purely elastic model calculations of the propagation of seam waves with experimental data

modeling slender bodies with the method of regularized stokeslets the motion and flow generated by immersed structures in a fluid in the stokes regime can be modeled with a variety of different numerical methods the mathematical structure of the stokes equations allows one to describe the flow around a three dimensional object using only information regarding its geometry this leads to computational techniques such as boundary integral methods or the method of regularized stokeslets that discretize the surface of the immersed object in the flow however when the body in question is slender a more computationally efficient alternative is to represent the flow by a one dimensional discretization along the centerline of the object rather than a discretization of the boundary using an exact and an asymptotic solution describing the nontrivial three dimensional fluid flow generated by a slender precessing spheroid we present a careful analysis of the approximation of the flow using the method of regularized stokeslets where the regularized stokeslets are placed along the centerline of the spheroid guidance is presented on how best to choose the numerical parameters within the method of regularized stokeslets to minimize the error for a given application

2 d parachute simulation by the immersed boundary method parachute aerodynamics involves an interaction between the flexible elastic porous parachute canopy and the high speed airflow relative to the parachute through which the parachute falls computer simulation of parachute dynamics typically simplifies the problem in various ways e g by considering the parachute as a rigid bluff body here we avoid such simplification by using the immersed boundary ib method to study the full fluid structure interaction the ib method is generalized to handle porous immersed boundaries and the generalized method is used to study the influence of porosity on parachute stability

numerical modeling of transient two dimensional viscoelastic waves this paper deals with the numerical modeling of transient mechanical waves in linear viscoelastic solids dissipation mechanisms are described using the generalized zener model no time convolutions are required thanks to the introduction of memory variables that satisfy local in time differential equations by appropriately choosing the relaxation parameters it is possible to accurately describe a large range of materials such as solids with constant quality factors the evolution equations satisfied by the velocity the stress and the memory variables are written in the form of a first order system of pdes with a source term this system is solved by splitting it into two parts the propagative part is discretized explicitly using a fourth order ader scheme on a cartesian grid and the diffusive part is then solved exactly jump conditions along the interfaces are discretized by applying an immersed interface method numerical experiments of wave propagation in viscoelastic and fluid media show the efficiency of this numerical modeling for dealing with challenging problems such as multiple scattering configurations

unconditionally stable discretizations of the immersed boundary equations the immersed boundary ib method is known to require small timesteps to maintain stability when solved with an explicit or approximately implicit method many implicit methods have been proposed to try to mitigate this timestep restriction but none are known to be unconditionally stable and the observed instability of even some of the fully implicit methods is not well understood in this paper we prove that particular backward euler and crank nicolson like discretizations of the nonlinear immersed boundary terms of the ib equations in conjunction with unsteady stokes flow can yield unconditionally stable methods we also show that the position at which the spreading and interpolation operators are evaluated is not relevant to stability so as long as both operators are evaluated at the same location in time and space we further demonstrate through computational tests that approximate projection methods which do not provide a discretely divergence free velocity field appear to have a stabilizing influence for these problems and that the implicit methods of this paper when used with the full navier stokes equations are no longer subject to such a strict timestep restriction and can be run up to the cfl constraint of the advection terms

multirate runge kutta schemes for advection equations explicit time integration methods can be employed to simulate a broad spectrum of physical phenomena the wide range of scales encountered lead to the problem that the fastest cell of the simulation dictates the global time step multirate time integration methods can be employed to alter the time step locally so that slower components take longer and fewer time steps resulting in a moderate to substantial reduction of the computational cost depending on the scenario to simulate s osher r sanders numerical approximations to nonlinear conservation laws with locally varying time and space grids math comput 41 1983 321 336 h tang g warnecke a class of high resolution schemes for hyperbolic conservation laws and convection diffusion equations with varying time and pace grids siam j sci comput 26 4 2005 1415 1431 e constantinescu a sandu multirate timestepping methods for hyperbolic conservation laws siam j sci comput 33 3 2007 239 278 in air pollution modeling the advection part is usually integrated explicitly in time where the time step is constrained by a locally varying courant friedrichs lewy cfl number multirate schemes are a useful tool to decouple different physical regions so that this constraint becomes a local instead of a global restriction therefore it is of major interest to apply multirate schemes to the advection equation we introduce a generic recursive multirate runge kutta scheme that can be easily adapted to an arbitrary number of refinement levels it preserves the linear invariants of the system and is of third order accuracy when applied to certain explicit runge kutta methods as base method

analysis of stiffness in the immersed boundary method and implications for time stepping schemes the immersed boundary method is known to exhibit a high degree of numerical stiffness associated with the interaction of immersed elastic fibres with the surrounding fluid we perform a linear analysis of the underlying equations of motion for immersed fibres and identify a discrete set of fibre modes which are associated solely with the presence of the fibre this work generalises our results in a previous paper 1995 siam j appl math 55 1577 by incorporating the effect of spreading the singular fibre force over a finite smoothing radius corresponding to the approximate delta function used in the immersed boundary method we investigate the stability of the fibre modes their stiffness and their dependence on the problem parameters focusing on the influence of smoothing we then extend the analytical results to include the effect of time discretisation and draw conclusions about the time step restrictions on various explicit schemes as well as the convergence rates for an iterative semi implicit method we draw comparisons with computations and show how the results can be applied to help in choosing alternate time stepping schemes that are specially tailored to handle the stiffness in immersed fibres in particular we present numerical results that show how fully explicit runge kutta schemes perform in comparison with the best of the semi implicit schemes currently in use

error analysis of imex runge kutta methods derived from differential algebraic systems in this paper we present an error analysis of the imex runge kutta methods when applied to stiff problems containing a nonstiff term and a stiff term characterized by a small stiffness parameter varepsilon in this analysis we expand the global error in powers of varepsilon and show that the coefficients of the error are the global errors of the imex runge kutta method applied to a differential algebraic system interesting convergence results of these errors and of the remainder of the expansion allow us to determine sharp error bounds for stiff problems as a representative example of stiff problems we have chosen the van der pol equation we illustrate that the theoretical prediction is confirmed by the numerical test specifically an order reduction phenomenon is observed when the problem becomes increasingly stiff in particular making several assumptions we try to improve global error estimates of several imex runge kutta methods existing in the literature

spectral elements on infinite domain a spectral element technique to approximate partial differential equations pdes on an infinite domain is examined the method is based on boyd s mapping of a semiinfinite interval to a finite interval and it is extended to a variational setting which allows for an implementation using a spectral element method by extending the method to a variational form a straightforward implementation allows for high order approximations over an infinite computational domain methods to find a high order approximation have been proposed by funaro maday mavriplis and many others the proposed methods all have drawbacks including restrictions on the behavior of the approximation at infinity because the proposed method is based on a mapping to a finite domain a much wider variety of boundary conditions can be enforced on the approximation

stability of a multi rate method for numerical integration of ode s abstract this paper derives sufficient conditions for the absolute stability of a certain multi rate method of numerical integration of systems of first order ode s which have been separated into two subsystems the second system being made up of the faster response equation it is assumed that the subsystems are integrated with fourth order and third order runge kutta methods respectively it is shown that there will be regions of stability provided the original system is sufficiently diagonally dominant if the subsystems are weakly coupled the regions of stability are nearly as large as the classical regions

zolotarev quadrature rules and load balancing for the feast eigensolver the feast method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization this relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function we propose improved rational approximants leading to feast variants with faster convergence in particular when using rational approximants based on the work of zolotarev numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues the new approach improves both convergence robustness and load balancing when feast runs on multiple search intervals in parallel

implicit explicit multistep methods for quasilinear parabolic equations efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in 1 in this note we present a refined analysis to allow more general nonlinearities the abstract theory is applied to a quasilinear parabolic equation

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

a multirate time integrator for regularized stokeslets the method of regularized stokeslets is a numerical approach to approximating solutions of fluid structure interaction problems in the stokes regime regularized stokeslets are fundamental solutions to the stokes equations with a regularized point force term that are used to represent forces generated by a rigid or elastic object interacting with the fluid due to the linearity of the stokes equations the velocity at any point in the fluid can be computed by summing the contributions of regularized stokeslets and the time evolution of positions can be computed using standard methods for ordinary differential equations rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid for problems with boundaries modeled by springs with large spring constants the resulting ordinary differential equations become stiff and hence the time step for explicit time integration methods is severely constrained unfortunately the use of standard implicit time integration methods for the method of regularized stokeslets requires the solution of dense nonlinear systems of equations for many relevant problems here an alternate strategy using an explicit multirate time integration scheme based on spectral deferred corrections is incorporated that in many cases can significantly decrease the computational cost of the method the multirate methods are higher order methods that treat different portions of the ode explicitly with different time steps depending on the stiffness of each component numerical examples on two nontrivial three dimensional problems demonstrate the increased efficiency of the multi explicit approach with no significant increase in numerical error

conservative multi implicit spectral deferred correction methods for reacting gas dynamics in most models of reacting gas dynamics the characteristic time scales of chemical reactions are much shorter than the hydrodynamic and diffusive time scales rendering the reaction part of the model equations stiff moreover non linear forcings may introduce into the solutions sharp gradients or shocks the robust behavior and correct propagation of which require the use of specialized spatial discretization procedures this study presents high order conservative methods for the temporal integration of model equations of reacting flows by means of a method of lines discretization on the flux difference form of the equations these methods compute approximations to the cell averaged or finite volume solution the temporal discretization is based on a multi implicit generalization of spectral deferred correction methods the advection term is integrated explicitly and the diffusion and reaction terms are treated implicitly but independently with the splitting errors reduced via the spectral deferred correction procedure to reduce computational cost different time steps may be used to integrate processes with widely differing time scales numerical results show that the conservative nature of the methods allows a robust representation of discontinuities and sharp gradients the results also demonstrate the expected convergence rates for the methods of orders three four and five for smooth problems