analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

adaptive runge kutta methods for nonlinear two point boundary value problems with mild boundary layers abstract a variable mesh deferred correction algorithm based on implicit runge kutta formulae is described for the approximate numerical solution of nonlinear two point boundary value problems a strategy for automatically choosing the variable mesh spacing is described and this seeks to equidistribute an approximation to the global truncation error of the runge kutta formula the facility of being able to use a variable mesh extends considerably the range of applicability of runge kutta methods and in particular allows the possibility of solving rather more difficult problems such as those with mild boundary layers or turning points in an efficient manner an extensive set of numerical results is given to illustrate the various algorithms described and a comparison is made with the deferred correction code of lentini and pereyra appearing in the nag library

a highly stable deferred correction scheme with interpolant for systems of nonlinear two point boundary value problems iterated deferred correction is a very popular approach to the numerical solution of general first order systems of nonlinear two point boundary value problems typically the order of accuracy of the solution is increased by 2 each time that a deferred correction is applied and this allows inexpensive local error estimates to be computed using embedding recent research has shown how deferred correction schemes having the very desirable property of superconvergence can be derived this marks a significant step forward in the design of deferred correction algorithms although the price to be paid for superconvergence is that it is difficult to derive realistic embedded local error estimates in the present paper we show that richardson extrapolation is a viable alternative for error estimation and we consider in some detail the relative merits of embedding and extrapolation guided by this analysis we are able to derive for the first time a lobatto deferred correction code which is very efficient for the solution of stiff problems particularly in cases where the mirk code twpbvp f is unstable furthermore we are able to derive efficient interpolants for this lobatto code which are also applicable to mirk formulae and we consider the problem of estimating the error in these interpolants this completely solves the interpolation problem for our deferred correction codes apart perhaps for deriving interpolants for extremely stiff problems this new lobatto code is available on the web page of one of the present authors and fills a large gap in currently available deferred correction software

real zeros of quadratic dirichlet l functions a small part of the generalized riemann hypothesis asserts that l functions do not have zeros on the line segment 2 1 the question of vanishing at s 2 often has deep arithmetical significance and has been investigated extensively a persuasive view is that l functions vanish at 2 either for trivial reasons the sign of the functional equation being negative or for deep arithmetical reasons such as the l function of an elliptic curve of positive rank and that the latter case happens very rarely n katz and p sarnak 7 have formulated precise conjectures on the low lying zeros in families of l functions which support this view in the case of dirichlet l functions it is expected that l 2 is never zero and so l 0 for all 2 1 this conjecture appears to have been first enunciated by s d chowla 2 in the special case of quadratic characters progress towards these non vanishing questions has been in two directions zero density type results which establish that very few l functions have a zero in 2 1 see for example a selberg 10 m jutila 6 and d r heath brown 4 and a growing body of work on non vanishing at 2 see for example r balasubramanian and v k murty 1 h iwaniec and sarnak 5 and k soundararajan 11 further much numerical evidence for the grh has been accumulated and these calculations support chowla s conjecture see 8 and 9 however the state of knowledge could not exclude the possibility that every dirichlet l function of sufficiently large conductor has a non trivial real zero in this

the order of accuracy for deferred corrections using uncentered end formulas a result of christiansen and russell siam j numer math 35 1980 pp 21 33 on the order of accuracy of an iterated deferred corrections algorithm of lentini and pereyra siam j numer math 14 1977 pp 91 111 is improved at the same time providing a realistic example for a theoretical framework developed by the author for proving accuracy results for deferred corrections

a variable order deferred correction algorithm for the numerical solution of nonlinear two point boundary value problems abstract we develop two variable order deferred correction algorithms based on finite difference methods for the approximate numerical solution of nonlinear two point boundary value problems the deferred corrections provide a powerful local error estimator and this estimate can be used to refine the mesh if need be in order to obtain a prescribed tolerance mesh refinement is particularly straightforward since our algorithms are truly one step in nature the approach described is particularly appropriate for problems where only low or engineering precision is required and for very large systems of equations where storage space is an extremely important consideration some numerical results are given and a comparison with the method developed by lentini and pereyra and with richardson extrapolation is made

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

large greatest common divisor sums and extreme values of the riemann zeta function it is shown that the maximum of 1 2 it on the interval t1 2 t t is at least exp 1 2 o 1 logtlogloglogt loglogt our proof uses soundararajan s resonance method and a certain large greatest common divisor sum the method of proof shows that the absolute constant a in the inequality sup 1 n1 nn k l 1ngcd nk nl nknl nexp alognlogloglognloglogn established in a recent paper of ours cannot be taken smaller than 1

lower bounds for the maximum of the riemann zeta function along vertical lines let alpha in 1 2 1 be fixed we prove that begin aligned max 0 le t le t zeta alpha it ge exp left frac c alpha log t 1 alpha log log t alpha right end aligned for all sufficiently large t where we can choose c alpha 0 18 2 alpha 1 1 alpha the same result has already been obtained by montgomery with a smaller value for c alpha however our proof which uses a modified version of soundararajan s resonance method together with ideas of hilberdink is completely different from montgomery s this new proof also allows us to obtain lower bounds for the measure of those t in 0 t for which zeta alpha it is of the order mentioned above

demography of cirsium vulgare in a grazing experiment the complete demographies of 16 populations of cirsium vulgare were followed in a replicated experiment the experiment was a factorial combination of two intensities of sheep grazing in each of three seasons winter grazed or ungrazed spring grazed or ungrazed and summer light or heavy grazing giving eight treatments in two blocks for 6 years from 1987 to 1992 the population sizes of c vulgare were monitored in each of the 16 paddocks after 1989 grazing in spring or winter or increased grazing in summer all increased population sizes population sizes fluctuated widely between years the effects of the grazing treatments and plant sizes on the transitions between nine life history stages were determined

a theoretical framework for proving accuracy results for deferred corrections general techniques are described for proving accuracy results for deferred correction solutions to differential equations these techniques apply also to computational estimates of the local discretization error the proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter

maximum of the riemann zeta function on a short interval of the critical line we prove the leading order of a conjecture by fyodorov hiary and keating about the maximum of the riemann zeta function on random intervals along the critical line more precisely as t for a set of t t 2t of measure 1 o 1 t we have max t u 1log 12 iu 1 o 1 loglogt

diagonally implicit runge kutta formulae for the numerical integration of nonlinear two point boundary value problems abstract fully implicit runge kutta formulae based on interpolatory quadrature schemes for the approximate numerical solution of nonlinear two point boundary value problems have been investigated by weiss such formulae are not used very often in practice however because they generally require such a large computational effort as to make them uncompetitive with for example integration schemes based on the trapezoidal or implicit mid point rules in this paper we consider an alternative class of runge kutta formulae namely diagonally implicit runge kutta dirk formulae which can be implemented more efficiently than the fully implicit formulae considered by weiss we also consider how these dirk formulae can be implemented in a defect or deferred correction framework and we give some numerical results to illustrate the algorithms derived one particular formula belonging to the dirk class is the implicit mid point rule in this paper we derive an efficient implementation of this formula which is applicable when the given boundary conditions are non separated

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

extreme values of the riemann zeta function and its argument we combine our version of the resonance method with certain convolution formulas for zeta s and log zeta s this leads to a new omega result for zeta 1 2 it the maximum of zeta 1 2 it on the interval 1 le t le t is at least exp left 1 o 1 sqrt log t log log log t log log t right we also obtain conditional results for s t 1 pi times the argument of zeta 1 2 it and s 1 t int 0 t s tau d tau on the riemann hypothesis the maximum of s t is at least c sqrt log t log log log t log log t and the maximum of s 1 t is at least c 1 sqrt log t log log log t log log t 3 on the interval t beta le t le t whenever 0 le beta 1

the maximum size of l functions we conjecture the true rate of growth of the maximum size of the riemann zeta function and other l functions we support our conjecture using arguments from random matrix theory conjectures for moments of l functions and also by assuming a random model for the primes

high order convergent deferred correction schemes based on parametrerized runge kutta nystrom methods for second order boundary value problems abstract iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two point boundary value problems normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2 and as a direct result each time deferred correction is used the order of the overall scheme is increased by a maximum of 2 in 16 however it has been shown that there exist schemes based on parameterized runge kutta methods which allow a higher increase of the overall order a first example of such a high order convergent scheme which allows an increase of 4 orders per deferred correction was based on two mono implicit runge kutta methods in the present paper we will investigate the possibility for high order convergence of schemes for the numerical solution of second order nonlinear two point boundary value problems not containing the first derivative two examples of such high order convergent schemes based on parameterized runge kutta nystrom methods of orders 4 and 8 are analysed and discussed

deferred correction with mono implicit runge kutta methods for first order ivps abstract to reach a high order of accuracy for numerical solutions of ivps with mono implicit runge kutta mirk methods the technique of deferred correction is used special attention is paid to the possible increase of the order and the stability of such schemes several schemes are given

value distribution of l ρ abstract let l be an automorphic l function assuming the riemann hypothesis for l s and the selberg normality conjecture we obtain a lower bound for the second negative moment and extreme small values of l where is a zero of l s

superconvergent deferred correction methods for first order systems of nonlinear two point boundary value problems iterated deferred correction is a widely used approach to the numerical solution of first order systems of nonlinear two point boundary value problems normally the orders of accuracy of the various methods used in a deferred correction scheme differ by 2 and as a direct result each time a deferred correction is applied the order of the overall scheme is increased by a maximum of 2 in this paper we consider the construction of mono implicit runge kutta mirk methods where an increase of four orders of accuracy is obtained for each deferred correction we develop a very powerful yet rather straightforward theory which allows us to identify the appropriate runge kutta formulae for inclusion in such schemes in particular we will focus on the construction of pairs of mirk formulae of order 4 and 8 which will allow this superconvergence to be realized we will further show that it is possible to derive formulae of this type for which high order interpolants and accurate error estimates are readily available

arithmetic zeros and nodal domains on the sphere we obtain lower bounds for the number of nodal domains of hecke eigenfunctions on the sphere assuming the generalized lindelof hypothesis we prove that the number of nodal domains of any hecke eigenfunction grows with the eigenvalue of the laplacian by a very different method we show unconditionally that the average number of nodal domains of degree l hecke eigenfunctions grows significantly faster than the uniform growth obtained under lindelof

moments of the riemann zeta function assuming the riemann hypothesis we obtain an upper bound for the moments of the riemann zeta function on the critical line our bound is nearly as sharp as the conjectured asymptotic formulae for these moments the method extends to moments in other families of l functions

on the method of modified equations vi asymptotic analysis of and asymptotic successive corrections techniques for two point boundary value problems in ode s the modified equation technique is extended to two point boundary value problems and a second order accurate implicit centered finite difference scheme for non homogeneous second order ordinary differential equations with linear boundary conditions is analyzed the first second and third modified equations or equivalent second equivalent and simply modified equations respectively for this scheme and its boundary conditions are presented it is shown that the three kinds of modified equations are asymptotically equivalent when the equivalent equation is used for the boundary conditions since an asymptotic analysis of these equations with the grid size as small parameter yields exactly the same results for a linear problem multiple scales and summed up asymptotic techniques are used and the resulting uniform asymptotic expansions are shown to be equivalent to the solution of the original finite difference scheme asymptotic successive corrections techniques are also applied to the three kinds of modified equations to obtain higher order schemes higher order boundary conditions are easily treated in the asymptotic successive corrections technique although these boundary conditions must be obtained by using the equivalent equation in order to obtain a correct estimate of the global error near the domain boundaries the methods introduced in this paper are applied to homogeneous and non homogeneous second order linear and non linear ordinary differential equations and yield very accurate results

defect correction for two point boundary value problems on nonequidistant meshes new finite difference formulae of arbitrary order are derived for the speciai class of second order two point boundary value problems y f x y x a x b variable mesh spacing is possible and the required accuracy is achieved under a very mild mesh condition a naturai defect correction framework is set up to compute the higher order approximations

extreme values of zeta and l functions we introduce a resonance method to produce large values of the riemann zeta function on the critical line and large and small central values of l functions

lobatto deferred correction for stiff two point boundary value problems abstract an iterated deferred correction algorithm based on lobatto runge kutta formulae is developed for the efficient numerical solution of nonlinear stiff two point boundary value problems an analysis of the stability properties of general deferred correction schemes which are based on implicit runge kutta methods is given and results which are analogous to those obtained for initial value problems are derived a revised definition of symmetry is presented and this ensures that each deferred correction produces an optimal increase in order finally some numerical results are given to demonstrate the superior performance of lobatto formulae compared with mono implicit formulae on stiff two point boundary value problems

large values of l functions from the selberg class abstract in the present paper we prove lower bounds for l functions from the selberg class by this means improving earlier results obtained by the second author together with jorn steuding we formulate two theorems which use slightly different technical assumptions and give two totally different proofs the first proof uses the resonance method which was introduced by soundararajan while the second proof uses methods from diophantine approximation which resemble those used by montgomery interestingly both methods lead to roughly the same lower bounds which fall short of those known for the riemann zeta function and seem to be difficult to be improved additionally to these results we also prove upper bounds for l functions in the selberg class and present a further application of a theorem of chen which is used in the diophantine approximation method mentioned above

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

grid adaptation and non iterative defect correction for improved accuracy of numerical solutions of pdes in this work we present a computational approach for improving the order of accuracy of a given finite difference method for solution of linear and nonlinear hyperbolic partial differential equations the methodology consists of analysis of leading order terms in the discretization error of any given finite difference method leading to a modified version of the original partial differential equation singular perturbations of this modified equation are regularized using an adaptive grid distribution and a non iterative defect correction method is used to eliminate the leading order regular perturbation terms in the modified equation implementation of this approach on a low order finite difference scheme not only results in an increase in its order of accuracy but also results in an improvement in its numerical stability due to the regularization of singular perturbations the proposed approach is applied to four different canonical problems including the numerical solution of 1 liouville equation 2 inviscid burgers equation 3 nonlinear reaction advection equation and 4 a system of hyperbolic pdes when compared to exact solutions the numerical results demonstrate the ability of this method in both boosting the accuracy of finite difference schemes up to the desired order and also providing a fully stable numerical solution

lower bounds for moments of l functions the moments of central values of families of l functions have recently attracted much attention and with the work of keating and snaith 2000 commun math phys 214 57 89 and 91 110 there are now precise conjectures for their limiting values we develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of l functions as an example we work out the case of the family of all dirichlet l functions to a prime modulus

thirteen ways to estimate global error various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations particularly ordinary differential equations are classified described and compared for most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated

on the method of modified equations iii numerical techniques based on the second equivalent equation for the euler forward difference method new numerical successive correction techniques for ordinary differential equations based on the euler forward explicit method and the first modified or equivalent equation are presented these techniques are similar to iterative updating deferred methods and are based on the application of asymptotic methods to modified equations which do not require initial conditions for the high order derivatives in the truncation terms and which yield stable numerical methods it is shown that depending on the discretization of the high order derivatives in the high order correction equations different methods of as high order of consistency as required can be developed in this paper backward and centered formulas are used but the resulting numerical methods are not self starting it is shown that if the starting procedure is not adequate the numerical order of the method can be smaller than the theoretical one in order to avoid this loss of numerical order a method for consistently starting the asymptotic successive correction technique based on the use of fictitious times is presented and applied to autonomous and non autonomous ordinary differential equations and compared with the results of second and fourth order runge kutta methods it is shown that the fourth order runge kutta method is more accurate than the successive correction techniques for large time steps due to the higher order derivatives in the successive correction but for sufficiently small time steps these techniques have almost the same accuracy as the fourth order runge kutta method