adaptivity in space and time for reaction diffusion systems in electrocardiology the paper introduces and studies numerical methods that are fully adaptive in both three dimensional 3d space and time to challenging multiscale cardiac reaction diffusion models in these methods temporal adaptivity comes via stepsize control in function space oriented linearly implicit time integration while spatial adaptivity is realized within multilevel finite element methods controlled by a posteriori local error estimators in contrast to other recent adaptivity approaches to cardiac modeling that discretize first in space and then in time so called method of lines our method discretizes first in time and then in space so called rothe method an approach that has already proven to be highly efficient in a number of challenging multiscale problems in science and technology kardos code library with this method the evolution of a complete heartbeat from the excitation to the recovery phase is simulated both in the frame of the anisotropic monodomain models and in the more realistic anisotropic bidomain models coupled with either a variant of the simple fitzhugh nagumo model or the more complex phase i luo rudy ionic model the numerical results exhibit a rather satisfactory performance of our adaptive method for complex cardiac reaction diffusion models on 3d domains up to moderate sizes in particular the method accurately resolves the evolution of the intra and extracellular potentials gating variables and ion concentrations during the excitation plateau and recovery phases

dense output for extrapolation methods this paper is concerned with dense output formulas for extrapolation methods for ordinary differential equations in particular the extrapolated explicit euler method the gbs method for non stiff equations and the extrapolated linearly implicit euler method for stiff and differential algebraic equations are considered existence and uniqueness questions for dense output formulas are discussed and an algorithmic description for their construction is given several numerical experiments illustrate the theoretical results

coefficients for studying one step rational schemes for ivps in odes ii abstract in 1 we derived variable order variable stepsize one step nonlinear methods in which global formulation is facilitated by defining some certain coefficients the analysis has a broader application and facilitates the derivation of new formulae the methods are based on a variant self adjusting rational interpolant of luke et al 2 and otunta and ikhile 3 4 in this paper we motivate an extension of this analysis to obtain a more generalized class of the rational methods in 1 for an arbitrarily desired order

practical improvements to simpr codes for stiff odes extrapolation of a semi implicit midpoint rule simpr is an effective way to solve stiff initial value problems for a system of ordinary differential equations odes two issues peculiar to simpr are investigated it is shown how to improve subtantially the robustness and efficiency of simpr codes

a linearly implicit predictor corrector method for reaction diffusion equations abstract a novel linearly implicit predictor corrector scheme is developed for the numerical solution of reaction diffusion equations iterative processes are avoided by treating the nonlinear reaction terms explicitly while maintaining superior accuracy and stability properties compared to the well known methods and linearly implicit runge kutta methods the proposed method allows the opportunity of solving large systems of reaction diffusion equations by alleviating the necessity of solving the accompanying large linear systems of algebraic equations due to the natural parallelism which surfaces across the system numerical results confirm the enhanced stability accuracy and efficiency of the method when applied to reaction diffusion equations arising in biochemistry and population ecology

normalized sph with stress points smoothed particle hydrodynamics is extended to a normalized staggered particle formulation with boundary conditions a companion set of interpolation points is introduced that carry the stress velocity gradient and other derived field variables the method is stable linearly consistent and has an explicit treatment of boundary conditions also a new method for finding neighbours is introduced which selects a minimal and robust set and is insensitive to anisotropy in the particle arrangement test problems show that these improvements lead to increased accuracy and stability published in 2000 by john wiley sons ltd

locally linearized runge kutta method of dormand and prince in this paper the effect that produces the local linearization of the embedded runge kutta formulas of dormand and prince for initial value problems is studied for this embedded locally linearized runge kutta formulas are defined and their performance is analyzed by means of exhaustive numerical simulations for a variety of well known test equations with different dynamics the simulation results show that the locally linearized formulas exhibit significant higher accuracy than the original ones which implies a substantial reduction of the number of time steps and consequently a sensitive reduction of the overall computational cost of their adaptive implementation

higher derivative explicit one step methods for non linear dynamic problems part i design and theory a new higher order explicit one step scheme is presented for use in the solution of a certain class of non linear problems in structural dynamics the generalized scheme covers some recently developed multistep algorithms as well as some new one step methods with improved numerical properties a weighted residual approach is used which contains new aspects for design of single step algorithms accuracy and stability are investigated for linear and non linear systems algorithms with higher accurary than the central difference method are developed and optimized to minimize the reduction in the stability limit

an adaptive extrapolation discontinuous galerkin method for the valuation of asian options we consider the approximation of the optimal stopping problem associated with ultradiffusion processes in the context of mathematical finance and the valuation of asian options in particular the value function is characterized as the solution of an ultraparabolic variational inequality employing the penalty method and a regularization of the state space we develop higher order adaptive approximation schemes which utilize the extrapolation discontinuous galerkin method in temporal space numerical examples are provided in order to demonstrate the approach

partitioning ode systems with an application to air pollution models abstract the numerical treatment of stiff ode systems is carried out by using implicit methods a long sequence of nonlinear systems has to be treated when implicit methods are used the newton iterative method is often used in the solution of these systems this leads to the calculation of jacobian matrices and to the inversion of these matrices the computational work can in some cases be reduced considerably when some kind of partitioning is used the conditions under which the partitioning procedures can successfully be used will be studied in this paper an example taken from a large air pollution model will be given to illustrate the usefulness of the theoretical results

coefficients for studying one step rational schemes for ivps in odes iii extrapolation methods abstract this paper considers extrapolation of rational methods for the numerical solution of initial value problems ivps in ordinary differential equations odes the extrapolation code difex2 of fatunla 1 which is a modification of the automatic extrapolation code difexi of deuflhard 2 is further modified to accommodate the basic integrators the extrapolation method which we refer to as difex2 for reference purposes is compared with results from difexi and difex2 and gbs extrapolation method

stability and accuracy of time extrapolated adi fdtd methods for solving wave equations some recent work on the adi fdtd method for solving maxwell s equations in 3 d have brought out the importance of extrapolation methods for the time stepping of wave equations such extrapolation methods have previously been used for the solution of odes the present context of wave equations brings up two main questions which have not been addressed previously 1 when will extrapolation in time of an unconditionally stable scheme for a wave equation again feature unconditional stability and 2 how will the accuracy and computational efficiency depend on how frequently in time the extrapolations are carried out we analyze these issues here

relationships between emission sources and excess ozone concentrations while the ozone layer in the stratosphere is shielding the earth s surface from radioactivity high ozone concentrations in the troposphere have damaging effects on plants animals and humans when they exceed certain critical levels this is why it is necessary to resolve successfully three tasks 1 i to determine in a sufficiently accurate way the critical levels of the ozone concentrations 2 ii to reduce the ozone concentrations to the critical levels and 3 iii to develop reliable and robust control strategies by which the ozone concentrations can be kept under the critical levels both the second task and the third task are very expensive therefore the high ozone concentrations must be reduced to the critical levels but no more furthermore it is necessary to reduce as much as possible the economical costs of these actions mathematical models can be used in the search of an optimal solution of the tasks listed above a mathematical model for studying air pollution levels in europe the danish eulerian model will be shortly discussed the model consists of a system of 35 partial differential equations and leads to huge computational tasks when it is handled numerically therefore it is necessary to use big modern high speed computers with vector and or parallel capabilities the danish eulerian model has been used to run various scenarios in order to study some of the relationships between emission sources in europe and high ozone concentrations modern visualization techniques have been applied to illustrate better the effects of different emission reductions used in the scenarios on the ozone concentrations and to draw some conclusions

numerical treatment of singular initial value problems abstract an efficient extrapolation scheme whose basic integrator is the inverse euler scheme cf fatunla 1982 is proposed for nonlinear singular initial value problems y x y y 0 y 0 the automatic polynomial rational extrapolation code difexi deufhard 1983 1985 is modified to accomodate the basic integrator the new algorithm was implemented in variable step variable order mode and compares favourably with the earlier works

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

recent progress in extrapolation methods for ordinary differential equations the paper is intended to give a survey on the state of the art of extrapolation methods for initial value problems in ordinary differential equation systems all basic discretizations which are su

theoretical aspects of the smoothed finite element method sfem this paper examines the theoretical bases for the smoothed finite element method sfem which was formulated by incorporating cell wise strain smoothing operation into standard compatible finite element method fem the weak form of sfem can be derived from the hu washizu three field variational principle for elastic problems it is proved that 1d linear element and 2d linear triangle element in sfem are identical to their counterparts in fem while 2d bilinear quadrilateral elements in sfem are different from that of fem when the number of smoothing cells scs of the elements equals 1 the sfem solution is proved to be variationally consistent and has the same properties with those of fem using reduced integration when sc approaches infinity the sfem solution will approach the solution of the standard displacement compatible fem model when sc is a finite number larger than 1 the sfem solutions are not variationally consistent but energy consistent and will change monotonously from the solution of sfem sc 1 to that of sfem sc it is suggested that there exists an optimal number of sc such that the sfem solution is closest to the exact solution the properties of sfem are confirmed by numerical examples copyright 2006 john wiley sons ltd

operator splitting for chemical reaction systems with fast chemistry abstract splitting methods are frequently used for large scale chemical reaction systems the main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem however a splitting error is introduced the most popular splitting schemes are the first order lie trotter splitting and the second order strang splitting however in case of stiff differential equations the strang splitting suffers from order reduction and both schemes have order one for stiff differential equations therefore the step size restrictions due to the low order can result in a prohibitive small step size hence splitting schemes with order larger than one are necessary for stiff differential equations the richardson extrapolation of the lie trotter splitting is a second order scheme in this paper we examine differential equations with a fast chemical source term and a slow transport term thus all stiffness of the system is related to the chemical source term we show that the richardson extrapolation does not suffer from order reduction in this case furthermore we perform a stability analysis of the extrapolated splitting scheme for a linear test problem thereby the operators of the test problem do not commute

consistent pseudo derivatives in meshless methods a meshless petrov galerkin formulation is developed in which derivatives of the trial functions are obtained as a linear combination of derivatives of shepard functions a key contribution is the development of conditions on test functions and trial functions for nonintegrable pseudo derivatives for petrov galerkin method which pass the patch test numerical results show that the resulting method is substantially more accurate than the galerkin method with shepard approximants and exceeds the rate of convergence of linear finite elements

one step and extrapolation methods for differential algebraic systems the paper analyzes one step methods for differential algebraic equations dae in terms of convergence order in view of extrapolation methods certain perturbed asymptotic expansions are shown to hold for the special dae extrapolation solver based on the semi implicit euler discretization the perturbed order pattern of the extrapolation tableau is derived in detail the theoretical results lead to modifications of the known code the efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion

ill conditioned matrices and the integration of stiff odes abstract popular methods for the integration of a stiff initial value problem for a system of ordinary differential equations odes require the solution of systems of linear equations it is shown that the matrices are very ill conditioned implicit linear multistep methods lmms can be evaluated accurately by iteration even when the matrices are very ill conditioned although semi implicit methods do not involve iteration it is observed that codes based on these methods cope with ill conditioned matrices about as well as codes based on lmms an explanation is provided for this fact

a family of anadromic numerical methods for matrix riccati differential equations matrix riccati differential equations mrdes x a21 xa11 a22x xa12x x 0 x0 where aij aij t appear frequently throughout applied mathematics science and engineering naturally the existing conventional runge kutta methods and linear multi step methods can be adapted to solve mrdes numerically indeed they have been adapted there are a few unconventional numerical methods too but they are suited more for time invariant mrdes than time varying ones for stiff mrdes existing implicit methods which are preferred to explicit ones require solving nonlinear systems of equations of possibly much higher dimensions than the original problem itself of for example implicit runge kutta methods and thus they can pose implementation difficulties and also be expensive in the past the property of an mrde which has been most preserved is the symmetry property for a symmetric mrde many other crucial properties have been discarded besides the symmetry property our proposed methods also preserve two other important properties bilinear rational dependence on the initial value and a generalized inverse relation between an mrde and its complementary mrde by preserving the generalized inverse relation our methods are accurately able to integrate an mrde whose solution has singularities by preserving the property of bilinear dependence on the initial value our methods also conserve the rank of change to the initial value and a solution s monotonicity property our methods are anadromic 1meaning if an mrde is integrated by one of our methods from t to and then integrated backward from t to using the same method the value at t is recovered in the absence of rounding errors this implies that our methods are necessarily of even order of convergence for time invariant mrdes methods of any even order of convergence are established while for time varying mrdes methods of order as high as 10 are established but only methods of order up to 6 are stated in detail our methods are semi implicit in the sense that there are no nonlinear systems of matrix equations to solve only linear ones unlike any pre existing implicit method given the availability of high quality codes for linear matrix equations our methods can easily be implemented and embedded into any application software package that needs a robust mrde solver numerical examples are presented to support our claims 1we coined anadromic from the greek roots back up and o o act of running of anadromous which describes fish that must return from the ocean to spawn in the same streams whence they hatched after we had tried to describe our numerical methods with words such as reflexive 27 28 symmetric which is still used today in the literature e g 22 and reversible which turned out to be overused received by the editor received august 25 2009 and in revised form october 30 2010 2010 mathematics subject classification primary 65l05

conservation laws and the numerical solution of odes ii abstract many physical systems described by an initial value problem for a system of ordinary differential equations odes conserve physical quantities such as the net charge or total energy as the system evolves typical codes for the numerical solution of the odes will not conserve these quantities and this can lead to solutions which are not even qualitatively correct one way to impose conservation laws is to perturb the numerical solution at each step of the integration a simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one step methods it is also easy to interpret the effect on the accuracy of the perturbations

reconstructing blockages in a symmetric duct via quasi isospectral horn operators abstract this paper proposes a new method for the reconstruction of the blockage area function in a symmetric duct by resonant frequencies under a given set of end conditions i e open open or closed closed ends the analysis is based on the explicit determination of quasi isospectral ducts that is duct profiles which have the same spectrum as a given duct with the exception of a single eigenfrequency which is free to move in a prescribed interval the analytical reconstruction was numerically implemented and tested for the detection of blockages numerical results show that the accuracy of identification increases with the number of eigenfrequencies used and that the reconstruction is rather stable with respect to the shape the size and the position of the blockages

the extrapolated explicit midpoint scheme for variable order and step size controlled integration of the landau lifschitz gilbert equation abstract a practical and efficient scheme for the higher order integration of the landau lifschitz gilbert llg equation is presented the method is based on extrapolation of the two step explicit midpoint rule and incorporates adaptive time step and order selection we make use of a piecewise time linear stray field approximation to reduce the necessary work per time step the approximation to the interpolated operator is embedded into the extrapolation process to keep in step with the hierarchic order structure of the scheme we verify the approach by means of numerical experiments on a standardized nist problem and compare with a higher order embedded runge kutta formula the efficiency of the presented approach increases when the stray field computation takes a larger portion of the costs for the effective field evaluation

pricing options in jump diffusion models an extrapolation approach we propose a new computational method for the valuation of options in jump diffusion models the option value function for european and barrier options satisfies a partial integrodifferential equation pide this pide is commonly integrated in time by implicit explicit imex time discretization schemes where the differential diffusion term is treated implicitly while the integral jump term is treated explicitly in particular the popular imex euler scheme is first order accurate in time second order accuracy in time can be achieved by using the imex midpoint scheme in contrast to the above approaches we propose a new high order time discretization scheme for the pide based on the extrapolation approach to the solution of odes that also treats the diffusion term implicitly and the jump term explicitly the scheme is simple to implement can be added to any pide solver based on the imex euler scheme and is remarkably fast and accurate we demonstrate our approach on the examples of merton s and kou s jump diffusion models the diffusion extended variance gamma model as well as the two dimensional duffie pan singleton model with correlated and contemporaneous jumps in the stock price and its volatility by way of example pricing a one year double barrier option in kou s jump diffusion model our scheme attains accuracy of 10 5 in 72 time steps in 0 05 seconds in contrast it takes the first order imex euler scheme more than 1 3 million time steps in 873 seconds and the second order imex midpoint scheme 768 time steps in 0 49 seconds to attain the same accuracy our scheme is also well suited for bermudan options combining simplicity of implementation and remarkable gains in computational efficiency we expect this method to be very attractive to financial engineering modelers

control of step size and order in extrapolation codes abstract extrapolation of the semi implicit midpoint rule is an effective way to solve stiff initial value problems for a system of ordinary differential equations the theory of the control of step size and order is advanced by investigating questions not taken up before providing additional justification for some algorithms and proposing an alternative to the information theory approach of deuflhard an experimental code simp implementing the algorithms proposed is shown to be as good as and in some respects better than the research code metan1 of bader and deuflhard

interpolation for variable order runge kutta methods abstract runge kutta methods approximate the solution of an initial value problem for a system of ordinary differential equations only on a mesh recent work considers how to approximate the solution everywhere variable order runge kutta methods are developed that have this capability one is about as efficient as a popular variable order method without this capability

extrapolation at stiff differential equations asymptotic expansions of the global error of numerical methods are well understood if the differential equation is non stiff this paper is concerned with such expansions for the implicit euler method the linearly implicit euler method and the linearly implicit mid point rule when they are applied tostiff differential equations in this case perturbation terms are present whose dominant one is given explicitly this permits us to better understand the behaviour ofextrapolation methods at stiff differential equations numerical examples supporting the theoretical results are included

improved quasi steady state approximation methods for atmospheric chemistry integration in the last fifteen years the quasi steady state approximation qssa method has been a commonly used method for integrating stiff ordinary differential equations arising from atmospheric chemistry problems in this paper a theoretical analysis of the qssa method is developed stressing its strengths and its weaknesses this theory leads to practical improvements to the qssa method new algorithms including symmetric and extrapolated qssa are presented

design of software for odes we discuss the design of software that is easy to use for simple problems but still capable of solving complicated problems nowadays a design should accommodate related but fundamentally different tasks such as solving daes and ddes in addition to ivps and bvps for odes major issues are illustrated with the matlab ode suite and selected solvers written in fortran 90 and maple

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems

output in extrapolation codes abstract codes for the solution of the initial value problem for a system of ordinary differential equations which are based on extrapolation of the explicit midpoint rule are adversely affected by frequent output at specified points a good algorithm for the selection of order reduces the sensitivity to output significantly a complementary algorithm is developed here which comes into play when variation of the order is no longer helpful when it is applicable the number of specific output points does not matter

extrapolation discontinuous galerkin method for ultraparabolic equations ultraparabolic equations arise from the characterization of the performance index of stochastic optimal control relative to ultradiffusion processes they evidence multiple temporal variables and may be regarded as parabolic along characteristic directions we consider theoretical and approximation aspects of a temporally order and step size adaptive extrapolation discontinuous galerkin method coupled with a spatial lagrange second order finite element approximation for a prototype ultraparabolic problem as an application we value a so called asian option from mathematical finance

extrapolated implicit explicit time stepping this paper constructs extrapolated implicit explicit time stepping methods that allow one to efficiently solve problems with both stiff and nonstiff components the proposed methods are based on euler steps and can provide very high order discretizations of odes index 1 daes and pdes in the method of lines framework implicit explicit schemes based on extrapolation are simple to construct easy to implement and straightforward to parallelize this work establishes the existence of perturbed asymptotic expansions of global errors explains the convergence orders of these methods and studies their linear stability properties numerical results with stiff ode dae and pde test problems confirm the theoretical findings and illustrate the potential of these methods to solve multiphysics multiscale problems

a new support integration scheme for the weakform in mesh free methods a new support integration technique is proposed which is similar to those used in truly mesh free methods the contribution of this paper is to exploit the divergence free condition for the support integrals to construct quadrature formulas that only require three integration points per particle in two dimensions numerical examples show that the proposed integration method can achieve results that agree with manufactured closed form solutions copyright 2009 john wiley sons ltd

a semi implicit mid point rule for stiff systems of ordinary differential equations the paper introduces a new semi implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations the existence of a quadratic asymptotic expansion in terms of the stepsize is shown moreover the new discretization is analyzed in the light of well known stability models the efficiency of the new integrator is clearly demonstrated by solving a series of challenging test problems including real life examples