the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

stability of runge kutta methods for linear impulsive delay differential equations with piecewise constant arguments abstract this paper is concerned with a class of linear impulsive delay differential equations with piecewise continuous argument the conditions for stability of the exact solution are obtained under these conditions it is proven that the method preserves stability of the equations furthermore stability of runge kutta methods for this kind of equations is studied some numerical examples are given to confirm the theoretical results

stability of runge kutta methods in the numerical solution of equation u t au t a 0 u t this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a0u t the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

oscillation analysis of numerical solutions for nonlinear delay differential equations of hematopoiesis with unimodal production rate this paper is concerned with oscillation and non oscillation of numerical solutions for nonlinear delay differential equation of hematopoiesis with unimodal production rate by investigating the roots of characteristic equation some conditions under which the numerical solution is oscillatory and non oscillatory are obtained the properties of non oscillatory numerical solutions are investigated to verify our results we give numerical experiments

logarithmic stabilization of the euler bernoulli transmission plate equation with locally distributed kelvin voigt damping abstract in this work we consider a transmission problem for a plate equation where one small part of the domain is made of a viscoelastic material with kelvin voigt constitutive relation we apply the general results due to burq s 10 in the study of asymptotic behavior of solutions and prove that the semigroup associated to the system is logarithmically stable the main ingredient of the proof is the carleman estimates the method consist to use carleman estimates to obtain information on the resolvent for high frequency

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

numerical stability analysis of differential equations with piecewise constant arguments with complex coefficients abstract in this paper we consider the analytical and numerical stability regions of runge kutta methods for differential equations with piecewise continuous arguments with complex coefficients it is shown that the analytical stability region contained in the numerical one is violated for a r by the geometric technique and we give the conditions under which the analytical stability region is contained in the union of the numerical stability regions of two runge kutta methods at last some experiments are given

stability analysis for linear discretisations of the advection equation with runge kutta time integration for the 1 dim linear advection problem stability limits of runge kutta rk methods from 1st to 7th order in combination with upwind or centered difference schemes from 1st to 6th order are presented the analysis can be carried out in a rather general way by introduction of a broad class of runge kutta methods here called linear case runge kutta lc rk methods which behave completely similar for linear time independent and homogeneous ode systems and contain the classical order stage rk methods the set of conditions for the coefficients of these lc rk schemes could be derived explicitly for arbitrary order n from an efficiency viewpoint the lc rk 3rd order methods in combination with upwind 3rd or 5th order or the lc rk 4th order scheme with 4th order centered difference advection are a good choice the analysis can be extended easily to multidimensional splited advection for which a necessary stability condition is presented

numerical oscillation of neutral logistic delay differential equation the equation is transformed by an invariant oscillation transformation some conditions under which the numerical solution oscillates are obtained we consider the asymptotic property of non oscillatory numerical solution this paper mainly deals with the oscillation of numerical solution for the neutral logistic delay differential equation applying the methods to the simplified type of the original equation which is got through an invariant oscillation transformation it is proven that every non oscillatory numerical solution tends to the steady state of the model moreover some conditions under which the numerical solution oscillates are obtained finally to demonstrate our theoretical results some numerical examples are adopted

a sufficient condition for gpn stability for delay differential equations in this paper the author considers a linear test delay differential equation with non constant coefficients related to the definitions of pn and gpn stability for numerical methods he defines stability properties for an ordinary differential equation together with stability properties of interpolants for numerical methods and in this way he gives sufficient conditions for gpn stability

time point relaxation runge kutta methods for ordinary differential equations bellen a z jackiewicz and m zennaro time point relaxation runge kutta methods for ordinary differential equations journal of computational and applied mathematics 45 1993 121 137 we investigate convergence order and stability properties of time point relaxation runge kutta methods for systems of ordinary differential equations these methods can be implemented in gauss jacobi or gaussseidel modes denoted by trgjrk k or trgsrk k respectively where k stands for the number of picard lindelof iterations as k f 0 these modes tend to the same one step method for ordinary differential equations called diagonal split runge kutta dsrk method it is proved that these methods are convergent with order min k p where p is the order of the underlying runge kutta method recurrence relations resulting from application of trgjrk k trgsrk k and dsrk methods to the two dimensional test system u au pu u pu au t 3 0 where a and p are real parameters are derived and stability regions in the h p plane are plotted for some methods using a variant of the boundary locus method in most cases stability regions increase as the number of picard lindeldf iterations k becomes larger

integrating finite rotations abstract we study the integration of problems of evolution in the rotation group instead of attacking the problem in the nonlinear differential manifold so 3 pure rotational dynamics as is usually done we derive equations for the complete problem of motion translational and rotational dynamics on an extended manifold we develop a generalization of runge kutta methods that by design ensures that the solution will remain on the manifold for any choice of the tableau this is obtained through configuration updates performed via the exponential map we show how certain terms can be approximated while retaining the order of accuracy of the scheme and how the method conserves the total momentum of the system within this framework we develop two nonlinearly unconditionally stable time integration schemes that are associated with discrete laws of conservation dissipation of the total energy the dissipating algorithm generalizes to the nonlinear case the high frequency damping characteristics provided by some well known conventional methods we present numerical results to support our analysis and we develop a complete application of this methodology to the nonlinear dynamics of three dimensional rods undergoing large displacements and finite rotations under the assumption of small strains

runge kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics in this paper a new runge kutta method with minimal dispersion and dissipation error is developed the chebyshev pseudospectral method is utilized using spatial discretization and a new fourth order six stage runge kutta scheme is used for time advancing the proposed scheme is more efficient than the existing ones for acoustic computations

runge kutta methods for first order periodic boundary value differential equations with piecewise constant arguments in this paper we deal with the numerical solutions of runge kutta methods for first order periodic boundary value differential equations with piecewise constant arguments the numerical solution is given by the numerical green s function it is shown that runge kutta methods preserve their original order for first order periodic boundary value differential equations with piecewise constant arguments we give the conditions under which the numerical solutions preserve some properties of the analytic solutions e g uniqueness and comparison theorems finally some experiments are given to illustrate our results

implementation of two step runge kutta methods for ordinary differential equations we investigate the potential for efficient implementation of two step runge kutta methods tsrk a new class of methods introduced recently by jackiewicz and tracogna for numerical integration of ordinary differential equations the implementation issues addressed are the local error estimation changing stepsize using nordsieck technique and construction of interpolants the numerical experiments indicate that the constructed error estimates are very reliable in a fixed and variable stepsize environment

high accuracy large step explicit runge kutta hale rk schemes for computational aeroacoustics in many realistic calculations the computational grid spacing required to resolve the mean flow gradients is much smaller than the grid spacing required to resolve the unsteady propagating waves of interest because of this the high temporal resolution provided by existing optimized time marching schemes can be excessive due to the small time step required for stability in regions of clustered grid in this work explicit fourth order accurate runge kutta time marching schemes are optimized to increase the inviscid stability limit rather than the accuracy at large time steps single and multiple step optimized schemes are developed and analyzed the resulting schemes are validated on several realistic benchmark problems

convergence and asymptotic stability of galerkin methods for a partial differential equation with piecewise constant argument the paper deals with the convergence and asymptotic stability of galerkin methods for a partial differential equation with piecewise constant argument the optimal convergence orders are obtained for the semidiscrete and full discrete backward euler methods respectively both the discrete solutions are proved to be asymptotically stable under the condition that the analytical solution is asymptotically stable

stability of runge kutta methods for the alternately advanced and retarded differential equations with piecewise continuous arguments this paper deals with the numerical properties of runge kutta methods for the solution of u t au t a 0u t 12 it is shown that the runge kutta method can preserve the convergence order the necessary and sufficient conditions under which the analytical stability region is contained in the numerical stability region are obtained it is interesting that the q methods with 0 q 12 are asymptotically stable some numerical experiments are given

comparison of monolithic and splitting solution schemes for dynamic porous media problems 5 summary proceeding from the governing equations describing a saturated poroelastic material with intrinsically 7 incompressible solid and fluid constituents we compare the monolithic and splitting solution of the different multi field formulations feasible in porous media dynamics because of the inherent solid fluid momentum 9 interactions one is concerned with the class of volumetrically coupled problems involving a potentially strong coupling of the momentum equations and the algebraic incompressibility constraint here the 11 resulting set of differential algebraic equations dae is solved by the finite element method fem following two different strategies 1 an implicit monolithic approach where the equations are first 13 discretized in space using stable mixed finite elements and second in time using stiffly accurate implicit time integrators 2 a semi explicit implicit splitting scheme in the sense of a fractional step method 15 where the dae are first discretized in time split using intermediate variables and then discretized in space using linear equal order approximations for all primary unknowns finally a one and a two dimensional 17 wave propagation example serve to reveal the pros and cons in regard to accuracy and stability of both solution strategies therefore several test cases differing in the used multi field formulation the 19 monolithic time stepping method and the approximation order of the individual unknowns are analyzed for varying degrees of coupling controlled by the permeability parameter in the end we provide a 21 reliable recommendation which of the presented strategies and formulations is the most suitable for which particular dynamic porous media problem copyright q 2009 john wiley sons ltd 23

automatic differentiation of numerical integration algorithms automatic differentiation ad is a technique for automatically augmenting computer programs with statements for the computation of derivatives this article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations odes in particular the ramifications of the fact that ad is applied not only to the solution of such an algorithm but to the solution procedure itself this subtle issue can lead to surprising results when ad tools are applied to variable stepsize variable order ode integrators the computation of the final time step plays a special role in determining the computed derivatives we investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives

a three stage vsvo hermite birkhoff taylor ode solver the new variable step variable order ode solver hbt p of order p presented in this paper combines a three stage runge kutta method of order 3 with a taylor series method of order p 2 to solve initial value problems y f t y y t 0 y 0 where y r r d and f rxr d r d the order conditions satisfied by hbt p are formulated and they lead to vandermonde type linear algebraic systems whose solutions are the coefficients in the formulae for hbt p a detailed formulation of variable step hbt p in both fixed order and variable order modes is presented the new method and the taylor series method have similar regions of absolute stability to obtain high accuracy results at high order this method has been implemented in multiple precision

a note on stability of pseudospectral methods for wave propagation in this paper we deal with the effects on stability of subtle differences in formulations of pseudospectral methods for solution of the acoustic wave equation we suppose that spatial derivatives are approximated by chebyshev pseudospectral discretizations through reformulation of the equations as first order hyperbolic systems any appropriate ordinary differential equation solver can be used to integrate in time however the resulting stability and hence efficiency properties of the numerical algorithms are drastically impacted by the manner in which the absorbing boundary conditions are incorporated specifically mathematically equivalent well posed approaches are not equivalent numerically an analysis of the spectrum of the resultant system operator predicts these properties

parallel iteration across the steps of high order runge kutta methods for nonstiff initial value problems abstract for the parallel integration of nonstiff initial value problems ivps three main approaches can be distinguished approaches based on parallelism across the problem on parallelism across the method and on parallelism across the steps the first type of parallelism does not require special integration methods and can be exploited within any available ivp solver the method parallelism approach received much attention particularly within the class of explicit runge kutta methods originating from fixed point iteration of implicit runge kutta methods of gaussian type the construction and implementation on a parallel machine of such methods is extremely simple since the computational work per processor is modest with respect to the number of data to be exchanged between the various processors this type of parallelism is most suitable for shared memory systems the required number of processors is roughly half the order of the generating runge kutta method and the speed up with respect to a good sequential ivp solver is about a factor 2 the third type of parallelism step parallelism can be achieved in any ivp solver based on predictor corrector iteration and requires the processors to communicate after each full iteration if the iterations have sufficient computational volume then the step parallel approach may be suitable for implementation on distributed memory systems most step parallel methods proposed so far employ a large number of processors but lack the property of robustness due to a poor convergence behaviour in the iteration process hence the effective speed up is rather poor the dynamic step parallel iteration process proposed in the present paper is less massively parallel but turns out to be sufficiently robust to achieve speed up factors up to 15

order conditions for integrators and mappings with applications abstract we regard the elementary weight function as a mapping from input to output so that the mappings allow us to derive the order conditions for a given method more easily and concisely this paper presents the order conditions of one step runge kutta methods runge kutta type methods with derivatives a class of two step runge kutta methods and three step runge kutta methods based on the elementary weight mappings the composition formula of two mappings as well as the mappings i d r e defined on the set t of all rooted trees from an input to an output it is pointed out that the order conditions can be easily achieved for linear multistep methods and second derivative linear multistep methods of order p and it suffices to consider only bushy trees of order up to p together with the empty tree 0 for a linear multistep method or a second derivative linear multistep method of order p

exponential decay of the energy of a one dimensional nonhomogeneous medium the energy method based on wave propagation is used to show that locally distributed damping can stabilize the motion of a composite bar exponentially fast

stability analysis of runge kutta methods for unbounded retarded differential equations with piecewise continuous arguments abstract this paper deals with the asymptotical stability of the analytic solutions for the unbounded retarded differential equations with piecewise continuous arguments epca u t f t u t u t n t 0 u 0 u 0 where n z a sufficient condition that the differential equations are asymptotically stable is derived this paper is also concerned with the stability analysis of the runge kutta methods for equation u t au t bu t n t 0 u 0 u 0 where n z the conditions that the numerical solutions preserve the stability of the analytic solutions are obtained and some numerical experiments are given

asymptotic stability of runge kutta methods for nonlinear differential equations with piecewise continuous arguments this paper deals with the asymptotic stability of numerical solutions for differential equations with piecewise continuous arguments epcas the necessary and sufficient condition is given for non confluent runge kutta methods to preserve the stability of nonlinear scalar epcas as for systems we prove that some algebraically stable methods can preserve the asymptotic stability

stability of θ methods for advanced differential equations with piecewise continuous arguments abstract this paper deals with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments we focus on the behaviour of the one leg method and the linear method in the solution of the equation x t ax t a 0 x t a 1 x t 1 with real a a 0 a 1 and designates the greatest integer function the stability regions of two methods are determined the conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

stability of generalized runge kutta methods for stiff kinetics coupled differential equations a stability and efficiency improved class of generalized runge kutta methods of order 4 are developed for the numerical solution of stiff system kinetics equations for linear and or nonlinear coupled differential equations the determination of the coefficients required by the method is precisely obtained from the so called equations of condition which in turn are derived by an approach based on butcher series since the equations of condition are fewer in number free parameters can be chosen for optimizing any desired feature of the process a further related coefficient set with different values of these parameters and the region of absolute stability of the method have been introduced in addition the a stability properties of the method are investigated implementing the method in a personal computer estimated the accuracy and speed of calculations and verified the good performances of the proposed new schemes for several sample problems of the stiff system point kinetics equations with reactivity feedback

stability of runge kutta methods in the numerical solution of equation u t au t a0u t a1u t 1 this paper is concerned with the stability analysis of the runge kutta methods for the equation u t au t a 0u t a 1u t 1 the stability regions for the runge kutta methods are determined the conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

general linear methods for volterra integro differential equations with memory a new class of numerical methods for volterra integro differential equations with memory is developed the methods are based on the combination of general linear methods with compound quadrature rules sufficient conditions that guarantee global and asymptotic stability of the solution of the differential equation and its numerical approximation are established numerical examples illustrate the convergence and effectiveness of the numerical methods

the improved linear multistep methods for differential equations with piecewise continuous arguments abstract this paper deals with the convergence of the linear multistep methods for the equation x t ax t a 0 x t numerical experiments demonstrate that the 2 step adams bashforth method is only of order p 0 when applied to the given equation an improved linear multistep methods is constructed it is proved that these methods preserve their original convergence order for ordinary differential equations odes and some numerical experiments are given

estimating the error of the classic runge kutta formula abstract a high quality error estimator is presented for the classic four stage fourth order runge kutta formula that is free when using this estimator the solution can be approximated accurately throughout the step at no additional cost

theoretical analysis for blow up behaviors of differential equations with piecewise constant arguments in this paper we discuss the blow up behaviors of differential equations with piecewise constant arguments epcas some fundamental results on the local existence and uniqueness of solutions of epcas are reviewed and some conditions are given under which the unique solution exists globally sufficient conditions for the finite blowup are presented and some examples illustrate that the blow up behaviors of epcas are quite different from those of the corresponding ordinary differential equations

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

cluster expansion for abstract polymer models new bounds from an old approach we revisit the classical approach to cluster expansions based on tree graphs and establish a new convergence condition that improves those by koteck preiss and dobrushin as we show in some examples the two ingredients of our approach are i a careful consideration of the penrose identity for truncated functions and ii the use of iterated transformations to bound tree graph expansions

oscillation analysis of advertising capital model analytical and numerical studies abstract this paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model analytically and numerically firstly the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic secondly the asymptotic behavior of non oscillatory analytic solution is verified thirdly the methods are applied to the mentioned equation some conditions under which the numerical solution oscillates are obtained moreover it is proved that every non oscillatory numerical solution tends to the steady state of the model finally some numerical simulations for verifying the theoretical findings are provided

a one step 7 stage hermite birkhoff taylor ode solver of order 11 a one step 7 stage hermite birkhoff taylor method of order 11 denoted by hbt 11 7 is constructed for solving nonstiff first order initial value problems y f t y y t 0 y 0 the method adds the derivatives y to y 6 used in taylor methods to a 7 stage runge kutta method of order 6 forcing an expansion of the numerical solution to agree with a taylor expansion of the true solution to order 11 leads to taylor and runge kutta type order conditions these conditions are reorganized into vandermonde type linear systems whose solutions are the coefficients of the method the new method has a larger scaled interval of absolute stability than the dormand prince dp87 and a larger unscaled interval of absolute stability than the taylor method t11 of order 11 hbt 11 7 is superior to dp87 and t11 in solving several problems often used to test higher order ode solvers on the basis of the number of steps cpu time and maximum global error numerical results show the benefit of adding high order derivatives to runge kutta methods

validation of first order diffraction theory for the traveltimes and amplitudes of propagating waves ultrasonic measurements of acoustic wavefields scattered by single spheres placed in a homogenous background medium water are presented the dimensions of the spheres are comparable to the wavelength and the wavelength and represent both positive rubber and negative teflon velocity anomalies with respect to the background medium the sensitivity of the recorded wavefield to scattering in terms of traveltime delay and amplitude variation is investigated the results validate a linear first order diffraction theory for wavefields propagating in heterogeneous media with anomaly contrasts on the order of 15 the diffraction theory is compared further with the exact results known from literature for scattering from an elastic sphere formulated in terms of legendre polynomials to investigate the 2d case the first order scattering theory is tested against 2d elastic finite difference calculations as the presented theory involves a volume integral it is applicable to any geometric shape and the scattering object does not need to be spherical or any other specific symmetrical shape furthermore it can be implemented easily in seismic data inversion schemes which is illustrated with examples from seismic crosswell tomography and a reflection experiment

dynamic properties of the local linearization method for initial value problems some dynamic properties of the local linearization ll scheme for the numerical integration of initial value problems in ordinary differential equations odes are investigated specifically the general conditions under which this scheme preserves the stationary points and periodic orbits of the odes and the local stability at these steady states are studied these dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes in addition a brief review of the computational implementations of the ll scheme is also presented

asymptotical stability of runge kutta methods for advanced linear impulsive differential equations with piecewise constant arguments this paper is concerned with a class of advanced linear impulsive differential equations with piecewise continuous argument the sufficient and necessary condition for asymptotical stability of the exact solution is obtained under this condition asymptotical stability of runge kutta methods for this kind of equations is studied some numerical examples are given to confirm the theoretical results

preservation of oscillations of the runge kutta method for equation x t ax t a1x t 1 0 the paper deals with the preservation of oscillations of the runge kutta method for equation x t ax t a 1x t 1 0 it is proved that oscillations of the analytic solution are preserved by the runge kutta method special interpolation functions of the numerical solutions are given it turns out that zeros of the interpolation function of the numerical solution converge to ones of the analytic solution with the same order of accuracy as that of the corresponding runge kutta method some numerical experiments are presented

exponential decay of energy of the euler bernoulli beam with locally distributed kelvin voigt damping in this paper we consider the longitudinal and transversal vibrations of the euler bernoulli beam with kelvin voigt damping distributed locally on any subinterval of the region occupied by the beam we prove that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable although the semigroup associated with the equation for the longitudinal motion of the beam is not exponentially stable due to the locally distributed and unbounded nature of the damping we use a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent we also show that the associated semigroups are not analytic

partitioned krylov subspace iteration in implicit runge kutta methods abstract application of the method of lines to partial differential equation leads to very large sparse systems of ordinary differential equations which are usually stiff we consider high order implicit runge kutta methods for the time integration of these systems and propose a partitioned krylov method for the solution of the resulting linear systems it is shown that this method allows for parallel computation in the construction of krylov subspaces and in a few examples it converges much faster than gmres

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

parallel iterated runge kutta methods for stiff ordinary differential equations abstract for the numerical integration of a stiff ordinary differential equation fully implicit runge kutta methods offer nice properties like a high classical order and high stage order as well as an excellent stability behaviour however such methods need the solution of a set of highly coupled equations for the stage values and this is a considerable computational task this paper discusses an iteration scheme to tackle this problem by means of a suitable choice of the iteration parameters the implicit relations for the stage values as they occur in each iteration can be uncoupled so that they can be solved in parallel the resulting scheme can be cast into the class of diagonally implicit runge kutta dirk methods and similar to these methods requires only one lu factorization per step per processor the stability as well as the computational efficiency of the process strongly depends on the particular choice of the iteration parameters and on the number of iterations performed we discuss several choices to obtain good stability and fast convergence based on these approaches we wrote two codes possessing local error control and stepsize variation we have implemented both codes on an alliant fx 4 machine four parallel vector processors and shared memory and measured their speedup factors for a number of test problems furthermore the performance of these codes is compared with the performance of the best stiff ode codes for sequential computers like simple lsode and radau5

prestack gaussian beam depth migration kirchhoff migration is the most popular method of three dimensional prestack depth migration because of its flexibility and efficiency its effectiveness can become limited however when complex velocity structure causes multipathing of seismic energy an alternative is gaussian beam migration which is an extension of kirchhoff migration that overcomes many of the problems caused by multipathing unlike first arrival and most energetic arrival methods which retain only one traveltime this alternative method retains most arrivals by the superposition of gaussian beams this paper presents a prestack gaussian beam migration method that operates on common offset gathers the method is efficient because the computation of beam superposition isolates summations that do not depend on the seismic data and evaluates these integrals by considering their saddle points gaussian beam migration of the two dimensional marmousi test data set demonstrates the method s effectiveness for structural imaging in a case w

persistence contractivity and global stability in logistic equations with piecewise constant delays abstract we establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium n 1 a i 0 m b i of the following differential equation with piecewise constant arguments dn t dt n t r t 1 an t i 0 m b i n n i n t n 0 1 2 n 0 n 0 0 and n j n j 0 j 1 2 m where r t is a nonnegative continuous function on 0 r t 0 i 0mbi 0 bi 0 i 0 1 2 m and a i 0mbi 0 these new conditions depend on a b0 and i 1mbi and hence these are other type conditions than those given by so and yu hokkaido math j 24 1995 269 286 and others in particular in the case m 0 and r t r 0 we offer necessary and sufficient conditions for the persistence and contractivity of solutions we also investigate the following differential equation with nonlinear delay terms dx t dt x t r t 1 ax t g x t x t 1 x t m t 0 x k k 0 0 k m and 0 0 where r t is a nonnegative continuous function on 0 r t 0 1 ax g x x x 0 has a unique solution x 0 and g x0 x1 xm c1 0 0 0

a general strategy for the optimization of runge kutta schemes for wave propagation phenomena we analyze optimized explicit runge kutta schemes rk for computational aeroacoustics and wave propagation phenomena in general exploiting the analysis developed in s pirozzoli performance analysis and optimization of finite difference schemes for wave propagation problems j comput phys 222 2007 809 831 we rigorously evaluate the performance of several time integration schemes in terms of appropriate error and cost metrics and provide a general strategy to design runge kutta methods tailored for specific applications we present families of optimized second and third order runge kutta schemes with up to seven stages and describe their implementation in the framework of williamson s 2n storage formulation j h williamson low storage runge kutta schemes j comput phys 35 1980 48 56 numerical simulations of the 1d linear advection equation and of the 2d linearized euler equations are performed to demonstrate the validity of the theory and to quantify the improvement provided by optimized schemes

wave equation tomography the relation between ray trace and diffraction tomography is usually obscured by formulation of the two methods in different domains the former in space the latter in wavenumber here diffraction tomography is reformulated in the space domain under the title of wave equation tomography with this transformation wave equation tomography projects monochromatic scattered wavefields back over source receiver wavepaths just as ray trace tomography projects traveltime delays back over source receiver raypaths derived under the born approximation these wavepaths are wave theoretic back projection patterns for reflected energy derived under the rytov approximation they are wave theoretic back projection patterns for transmitted energy differences between ray trace and wave equation tomography are examined through comparison of wavepaths and raypaths followed by their application to a transmission geometry synthetic data set rytov wave equation tomography proves superior to ray trace tomography in de

piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments abstract in this paper we introduce a piecewise reproducing kernel method for impulsive delay differential equations with piecewise constant arguments the method is an improved reproducing kernel method compared with the classical reproducing kernel method the solutions obtained using the present method can give good approximations for a larger time interval some numerical examples are used to show the effectiveness and simplicity of the method

oscillation analysis of numerical solution in the θ methods for equation x t ax t a1x t 1 0 abstract the paper deals with the oscillation analysis of numerical solution in the methods for equation x t ax t a 1 x t 1 0 the conditions of the oscillation for the method are obtained it is proved that the oscillation of the analytic solution is preserved by the method it turns out that the zeros of the linear interpolation function of the numerical solution can converge to the zeros of the analytic solution with the order of accuracy 1 1 2 and 2 1 2 some numerical experiments are given

locally distributed control and damping for the conservative systems in this paper we note the equivalence between exact controllability and exponential stabilizability for an abstract conservative system with bounded control this enables us to establish a frequency domain characterization for the exact controllability uniform exponential decay property of second order elastic systems such as the wave equation and the petrovsky equation with locally distributed control damping a piecewise multiplier method for frequency domain is introduced for several classes of pdes on regions which are not necessarily smooth we obtain a sufficient condition for the subregion on which the application of control damping will yield the exact controllability uniform exponential decay property this result provides useful information for designing the location of controllers dampers for distributed systems with a law of conservation

impulsive continuous runge kutta methods for impulsive delay differential equations abstract the classical continuous runge kutta methods are widely applied to compute the numerical solutions of delay differential equations without impulsive perturbations however the classical continuous runge kutta methods cannot be applied directly to impulsive delay differential equations because the exact solutions of the impulsive delay differential equations are not continuous in this paper impulsive continuous runge kutta methods are constructed for impulsive delay differential equations with the variable delay based on the theory of continuous runge kutta methods convergence of the constructed numerical methods is studied and some numerical examples are given to confirm the theoretical results

on the unique solvability of the runge kutta equations we consider the existence of a unique solution to the systems of equations that arise when we apply a runge kutta method to a stiff nonlinear system of differential equationsu f t u withf satisfying a one sided lipschitz condition with constant for any given product h whereh denotes the step size we present algebraic conditions on the runge kutta matrixa which are necessary and sufficient for unique solvability of the equations as a second topic we consider the question whether the solution to the system of equations is stable with respect to perturbations known as bsi stability for this property also necessary and sufficient algebraic conditions ona are presented

the finite frequency sensitivity kernel for migration residual moveout and its applications in migration velocity analysis we have derived a broadband sensitivity kernel that relates the residual moveout rmo in prestack depth migration psdm to velocity perturbations in the migration velocity model we have compared the kernel with the rmo directly measured from the migration image the consistency between the sensitivity kernel and the measured sensitivity map validates the theory and the numerical implementation based on this broadband sensitivity kernel we propose a new tomography method for migration velocity analysis and updating specifically for the shot record psdm and shot index common image gather as a result time consuming angle domain analysis is not required we use a fast one way propagator and multiple forward scattering and single backscattering approximations to calculate the sensitivity kernel using synthetic data sets we can successfully invert velocity perturbations from the migration rmo this wave equation based method naturally incorporates the wave phenomena and is best teamed with the wave

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

sensitivity kernels for seismic fresnel volume tomography fresnel volume tomography fvt offers higher resolution and better accuracy than conventional seismic raypath tomography a key problem in fvt is the sensitivity kernel we propose amplitude and traveltime sensitivity kernels expressed directly with green s functions for transmitted waves for 2d 3d homogeneous heterogeneous media the green s functions are calculated with a finite difference operator of the full wave equation in the frequency space domain in the special case of homogeneous media we analyze the properties of the sensitivity kernels extensively and gain new insight into these properties according to the constructive interference of waves the spatial distribution ranges of the monochromatic sensitivity kernels in fvt differ from each other greatly and are 1 8 2 8 3 8 and 4 8 periods of seismic waves respectively for 2d amplitude 3d amplitude 2d traveltime and 3d traveltime conditions we also have a new understanding of the relationship between raypath tomography and fvt within the first fresnel volume of the dominant frequency the band limited sensitivity kernels of fvt in homogeneous media or smoothly heterogeneous media are very close to those of the dominant frequency thus it is practical to replace the band limited sensitivity kernel with a few selected frequencies or even the single dominant frequency to save computation when performing band limited fvt the numerical experiment proves that fvt using our sensitivity kernels can achieve more accurate results than traditional raypath tomography

explicit methods based on a class of four stage fourth order runge kutta methods for preserving quadratic laws this paper concerns the study of explicit 4 stage fourth order runge kutta methods which preserve quadratic conservation laws when their weights are made solution dependent application to orthogonal and hamiltonian linear systems and numerical tests are also reported

discretizations of nonlinear differential equations using explicit finite order methods abstract in this work we study the appearance of spurious solutions when first order differential equations with unimodal right hand sides are discretized using runge kutta schemes these spurious solutions are explained in terms of the iteration functions schemes that produce good approximating solutions for much longer times are given

p stable general nystrom methods for y f y t we focus our attention on the family of general linear methods glms for the numerical solution of second order ordinary differential equations odes these are multivalue methods introduced in d ambrosio et al 2012 3 with the aim to provide an unifying approach for the analysis of the properties of accuracy of numerical methods for second order odes our investigation is addressed to providing the building blocks useful to analyze the linear stability properties of glms for second order odes thus we present the extension of the classical notions of stability matrix stability polynomial stability and periodicity interval a stability and p stability to the family of glms special attention will be given to the derivation of highly stable glms whose stability properties depend on the stability polynomial of indirect runge kutta nystrom methods based on gauss legendre collocation points which are known to be p stable in this way we are able to provide p stable glms whose order of convergence is greater than that of the corresponding rkn method without heightening the computational cost we finally provide and discuss examples of p stable irreducible glms satisfying the mentioned features

the fresnel volume and transmitted waves in seismic imaging experiments it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high frequency limit consequently it is assumed that waves propagate along infinitely narrow lines through space called rays that join the source and receiver in reality recorded waves have a finite frequency content the band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path this volume is called the fresnel volume in this tutorial we introduce the physics of the fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves the finite frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band limited transmitted and reflected waves to slowness variations in the earth the fresnel zone and the finite frequency sensitivity kernels are closely connected through the concept of constructive interference of waves the finite frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield also it turns out that fermat s theorem in the context of geometric ray theory is a special case of the finite frequency wave theory in the limit of infinite frequency last we address the misconception that the width of the fresnel volume limits the resolution in imaging experiments

stability of the euler maclaurin methods for neutral differential equations with piecewise continuous arguments abstract this paper deals with the stability analysis of the euler maclaurin methods for neutral differential equations with piecewise continuous arguments u t au t i 0 n a i u i t the stability regions of the euler maclaurin methods are determined the conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given

stability analysis of runge kutta methods for systems u t lu t mu t this paper deals with the stability of runge kutta methods applied to the complex linear system u t lu t mu t the condition under which the numerical solution is asymptotically stable is presented which is stronger than a stability and weaker than a f stability furthermore in the case of 2 norm and l being a real symmetric matrix by using email protected approximation and order star theory it is proved that for a stable runge kutta methods suppose whose stability function is given by the r s email protected approximation to e x the numerical solution is asymptotically stable if and only if r is even

stability and oscillations of numerical solutions for differential equations with piecewise continuous arguments of alternately advanced and retarded type this paper is concerned with the numerical properties of q methods for the solution of alternately advanced and retarded differential equations with piecewise continuous arguments using two q methods namely the one leg q method and the linear q method the necessary and sufficient conditions under which the analytic stability region is contained in the numerical stability region are obtained and the conditions of oscillations for the q methods are also obtained it is proved that oscillations of the analytic solution are preserved by the q methods furthermore the relationships between stability and oscillations are revealed some numerical experiments are presented to illustrate our results

existence computability and stability for solutions of the diffusion equation with general piecewise constant argument abstract we study the solution of a class of pde with piecewise constant argument of generalized type separation of variables leads to a solution formed by a series of products in previous works the convergence and bounds of the solution could be obtained from the study of the solution on the first constancy interval only in the general case however each term of the series may be unbounded at every interval implying that the solution is not computable we establish conditions where the convergence of the solution can be verified computing a finite number of terms of the series in each constancy interval without requiring any regularity on the initial condition moreover we combine asymptotic properties for each variable of the equation to obtain an exponential bound for the solution

high order taylor galerkin methods for linear hyperbolic systems a new family of high order taylor galerkin schemes is presented for the analysis of first order linear hyperbolic systems the schemes are unconditionally stable which makes them very attractive to use in conjunction with adaptive hp finite element methods for spatial approximation

first born and rytov approximations modeling and inversion conditions in a canonical example first born and more recently first rytov approximations are widely used for solving complex forward and inverse seismic scattering problems the exact pressure field response from a one dimensional velocity slab included in an infinite constant velocity medium is expressed analytically this solution is used for testing these two approximations in modeling and in inversion conditions of applicability and parameter estimates lead to the following results the first rytov approximation is best suited for modeling and inverting the transmitted or forward scattered part of the wave field whereas the first born approximation is best suited for modeling and inverting the primary reflected or backscattered part of the wave field if the long wavelength condition holds linearization of the field with respect to slowness squared is preferable over linearization with respect to slowness allowing for a very large velocity contrast between the slab and the reference medium however if the weak heterogeneity condition holds i e small velocity contrast the linearization choice is reversed numerical examples show and display these differences quantitatively