towards a general integration algorithm for time dependent one dimensional systems of parabolic partial differential equations using the method of lines abstract an algorithm for the solution of nonlinear systems of parabolic partial differential equations is presented in this algorithm the second order spatial derivatives are approximated by a three point five point or seven point central differencing scheme the first order spatial derivatives are approximated by hybrid schemes that combine central differencing with upwinding schemes that are based on two three four and five points interpolating polynomial formulas the system of nonlinear odes that result from the spatial discretization is solved using a variable time step variable order algorithm based on the backward differentiation formulas due to the inclusion of higher order spatial differencing and upwinding the resultant algorithm is capable of solving a wide class of nonlinear systems of one dimensional parabolic pdes efficiently realistic numerical examples are given with emphasis on the convection diffusion equation

operator splitting methods for pricing american options under stochastic volatility we consider the numerical pricing of american options under heston s stochastic volatility model the price is given by a linear complementarity problem with a two dimensional parabolic partial differential operator we propose operator splitting methods for performing time stepping after a finite difference space discretization the idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps with this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint our analysis suggests that the crank nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy the numerical experiments show that the operator splitting methods have comparable discretization errors they also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations

the lem exponential integrator for advection diffusion reaction equations we implement a second order exponential integrator for semidiscretized advection diffusion reaction equations obtained by coupling exponential like euler and midpoint integrators and computing the relevant matrix exponentials by polynomial interpolation at leja points numerical tests on 2d models discretized in space by finite differences or finite elements show that the leja euler midpoint lem exponential integrator can be up to 5 times faster than a classical second order implicit solver

uniform convergence of compact and bdf methods for the space fractional semilinear delay reaction diffusion equations abstract in this article two classes of finite difference methods are constructed to solve the space fractional semilinear delay reaction diffusion equations firstly fractional centered finite difference method in space and backward differential formula bdf2 in time are employed to discrete the original equations secondly the existence uniqueness and convergence of the numerical method are analyzed at length by g norm technique and the convergence order is o 2 h 2 then the compact technique of finite difference method is used to further increase the accuracy in spatial direction and a method with fourth accuracy in spatial dimension is obtained the convergence orders are proved to be o 2 h 4 in the sense of three classes of norms by ga norm technique especially we obtain the uniform convergence for the error estimation which demonstrates competitive performance compared with the preceding work in the related references finally two numerical examples are provided to verify our theoretical results and demonstrate the effectiveness of both methods when applied to simulate space fractional delay diffusive nicholson s blowflies equation

a new operator splitting method for american options under fractional black scholes models abstract a new operator splitting method is proposed for american options under time fractional black scholes models the fractional linear complementarity problem is split into two easy sub problems with the leading coefficients separated from the convolution sum and matched through a general correction step the method is implementation friendly in the sense that one can easily modify a fractional european solver to obtain the proposed method since the correction step is decoupled and is trivial to solve the method is validated through numerical experiments and demonstrated to be superior to the traditional approach the paper also provides numerical studies including the effect of fractional orders and the comparison of fractional models

a stable and accurate scheme for nonlinear diffusion equations application to atmospheric boundary layer stability concerns are always a factor in the numerical solution of nonlinear diffusion equations which are a class of equations widely applicable in different fields of science and engineering in this study a modified extended backward differentiation formulae me bdf scheme is adapted for the solution of nonlinear diffusion equations with a special focus on the atmospheric boundary layer diffusion process the scheme is first implemented and examined for a widely used nonlinear ordinary differential equation and then extended to a system of two nonlinear diffusion equations a new temporal filter which leads to significant improvement of numerical results is proposed and the impact of the filter on the stability and accuracy of the results is investigated noteworthy improvements are obtained as compared to other commonly used numerical schemes linear stability analysis of the proposed scheme is performed for both systems and analytical stability limits are presented

dynamical behavior of reaction diffusion neural networks and their synchronization arising in modeling epileptic seizure a numerical simulation study abstract excessive synchronizations of neurons in the brain networks can be a reason for some episodic disorders such as epilepsy in this paper we simulate neural dynamic models and their synchronizations numerically by a new numerical algorithm in order to overcome the complexity of the problem a suitable combination of the legendre spectral element method and operator splitting technique is proposed to convert a complex system of partial differential equations into sparse linear algebraic systems the simulations are explored in different aspects such as accuracy computing the wavefront and angular speeds computing time the condition number of obtained algebraic systems to demonstrate the proposed scheme effectiveness the numerical simulations and comparisons confirm the method is suitable in terms of accuracy and speed

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

on simple moving grid methods for one dimensional evolutionary partial differential equations abstract two moving grid algorithms for the numerical integration of evolutionary one dimensional partial differential equations are considered both algorithms discretize the equation on trapezoidal space time elements and combine a prediction step with a remeshing technique in order to determine the orientation of the sides of the trapezoids joining nodes at consecutive time levels the behavior of the discretizations employed is forecast by means of the modified equation technique and then studied in a series of numerical experiments

a composite integration scheme for the numerical solution of systems of parabolic pdes in one space dimension abstract carroll j a composite integration scheme for the numerical solution of systems of parabolic pdes in one space dimension journal of computational and applied mathematics 46 1993 327 343 we apply the numerical method of lines to approximate the solution of a system of parabolic partial differential equations in one space dimension by approximating the spatial derivatives by discrete values of the solution at a set of mesh points in space we use these approximations to represent the partial differential equations at each mesh point giving a semidiscrete system of ordinary differential equations in the time direction the resulting ode system is integrated numerically using a second order l stable composite integration scheme the author this journal 1989 using variable stepsize integration an implementation of the composite scheme is applied to approximate the solution of some test problems selected from the literature using a uniform spatial mesh we present the results of a number of numerical experiments which allow an appraisal of the proposed approach with published works

l0 stable split linear multistep formulas for parabolic pdes abstract new types of split linear multistep methods slmms are applied to the numerical solution of parabolic differential equations the formulas derived achieve up to fourth order in time and unlike the crank nicolson method are l0 stable

a polynomial based iterative method for linear parabolic equations a new polynomial based method pbm is developed to integrate multi dimensional linear parabolic initial boundary value problems it is based on l sb2 approximations to f z 1 exp z z f 0 1 over ellipses in the complex plane using expansions of f in chebychev polynomials the calculation of the fourier coefficients requires numerical integration over only a single line segment in the complex plane whose length and orientation depend on the step size and the parabolic operator itself the simplicity with which these coefficients are obtained rests on special properties of the chebychev polynomials most of the work in pbm consists of matrix vector multiplications involving a matrix l which arises from the spatial discretization of the differential operator to be specific pbm integrates the semi discrete problem u sb t l t u b t u b in r sp n and l in r sp n times n and requires only a modest amount of storage a few vectors of order n due to the analyticity of f it has good convergence properties and compares favorably to other standard methods from the classes of hopscotch alternating direction implicit adi and locally one dimensional lod schemes as measured by the cpu times required on a single cpu of a cray x mp 24 it is also competitive with crank nicolson which we couple with two proven iterative solvers i recommend pbm on problems which require fourth order spatial accuracy problems whose solutions contain significant high frequency components and problems whose operators cannot be split conveniently in an adi or lod fashion for example problems with mixed derivatives

unimodality convexity and applications properties of univariate unimodal distributions concepts of multivariate unimodality some more notions of unimodality unimodality for discrete distributions unimodality of infinitely divisible distributions unimodality and notions of dependence ordering of distributions by peakedness applications of unimodality in statistical inference convexity in reliability theory appendix

a composite integration scheme for the numerical solution of systems of ordinary differential equations abstract a generalization of a composite linear multistep method 2 is developed and applied to the approximate integration of systems of ordinary differential equations the proposed scheme is second order accurate and l stable an algorithm based on the integration formula derived in this paper is applied to approximate the solutions of a number of standard test problems the numerical results indicate that the method is competitive with other fixed order methods particularly in terms of computational overhead and could provide the basis for efficient temporal integration in the semidiscretization of time dependent partial differential equations

two new finite difference schemes for parabolic equations two new classes of finite difference schemes are applied to the numerical solution of parabolic partial differential equations the formulae derived are self starting are at least second order in time are unconditionally stable and unlike the crank nicolson method are l 0 stable in the sense of gourlay and morris the stability of the new schemes is examined using a linear stability analysis and some numerical results are presented

the integration of stiff initial value problems in odes using modified extended backward differentiation formulae abstract a class of modified extended backward differentiation formulae mebdf suitable for the approximate numerical integration of first order systems of stiff ordinary differential equations is introduced the computational aspects of this new class of formulae are examined in detail in particular algorithms for varying both order and stepsize are given and this leads to a variable step variable order process using highly stable formulae of order 2 8 extensive numerical results for the well known detest set are given and on this basis a comparison is made between a code incorporating mebdf a code due to hindmarsh based on conventional backward differentiation formulae and a code due to skeel and kong based on blended linear multistep methods it is shown that on this test set the mebdf code is very reliable and is competitive with the other two codes on a significant class of problems a computer listing of the mebdf code used to obtain the results presented in this paper is available from the author

interpolating discrete advection diffusion propagators at leja sequences we propose and analyze the relpm real leja points method for evaluating the propagator tb v via matrix interpolation polynomials at spectral leja sequences here b is the large sparse nonsymmetric matrix arising from stable 2d or 3d finite difference discretization of linear advection diffusion equations and z is the entire function z ez 1 z the corresponding stiff differential system y t by t g y 0 y 0 is solved by the exact time marching scheme y i 1 y i t i t i b by i g i 0 1 where the time step is controlled simply via the variation percentage of the solution and can be large numerical tests show substantial speed ups up to one order of magnitude with respect to a classical variable step size crank nicolson solver

high order computational methods for option valuation under multifactor models abstract many of the different numerical techniques in the partial differential equations framework for solving option pricing problems have employed only standard second order discretization schemes a higher order discretization has the advantage of producing low size matrix systems for computing sufficiently accurate option prices and this paper proposes new computational schemes yielding high order convergence rates for the solution of multi factor option problems these new schemes employ galerkin finite element discretizations with quadratic basis functions for the approximation of the spatial derivatives in the pricing equations for stochastic volatility and two asset option problems and time integration of the resulting semi discrete systems requires the computation of a single matrix exponential the computations indicate that this combination of high order finite elements and exponential time integration leads to efficient algorithms for multi factor problems highly accurate european prices are obtained with relatively coarse meshes and high order convergence rates are also observed for options with the american early exercise feature various numerical examples are provided for illustrating the accuracy of the option prices for heston s and bates stochastic volatility models and for two asset problems under merton s jump diffusion model

efficient variable stiffness methods for cooling of hot rolled steel sections abstract an industrial finite element package for the simulation of alternative cooling strategies for hot rolled steel sections has been enhanced by the incorporation of a variable stiffness second order time integration scheme based on a specially developed family of extended stability explicit runge kutta methods and an l stable semi implicit formula the integration scheme uses local error estimation to vary step size and runge kutta method selection is achieved by monitoring the numerical stiffness through the solution period using a computationally inexpensive estimate for the spectral radius of the system jacobian numerical tests on a range of section cooling problems indicate that the variable stiffness rk code developed is considerably more efficient than standard first order integration methods for the levels of accuracy required for industrial simulations the efficiency compares favorably with other high efficiency codes such as the variable order backward differentiation formulae vode and the nearly stiff rkc explicit solver

stability analysis of six point finite difference schemes for the constant coefficient convective diffusion equation abstract a comprehensive and systematic study is presented to derive stability properties of various two level six point finite difference schemes in particular difference schemes of pade type for the approximation to the constant coefficient convective diffusion equation first the modified equivalent partial differential equation mepde for a general six point difference scheme is derive the mepde provides direct information on the order of accuracy of a difference scheme the von neumann and matrix methods are then employed to deduce the necessary and sufficient conditions for the numerical stability for the six point difference schemes an unified technique is developed to find the stability regions for the difference schemes some new second and third order six point difference schemes for the approximation of the constant coefficient convective diffusion equation are presented

an iterative splitting method for pricing european options under the heston model abstract in this paper we propose an iterative splitting method to solve the partial differential equation pde in option pricing problems we focus on the heston stochastic volatility model and the derived two dimensional pde we take the european option as an example and conduct numerical experiments using different boundary conditions the iterative splitting method transforms the two dimensional equation into two quasi one dimensional equations with the variable on the other dimension fixed which helps to lower the computational cost numerical results show that the iterative splitting method together with an artificial boundary condition abc based on the method by li and huang 2019 gives the most accurate option price and greeks compared to the classic finite difference method with the commonly used boundary conditions in heston 1993

solving partial integro differential option pricing problems for a wide class of infinite activity levy processes in this paper numerical analysis of finite difference schemes for partial integro differential models related to european and american option pricing problems under a wide class of levy models is studied apart from computational and accuracy issues qualitative properties such as positivity are treated consistency of the proposed numerical scheme and stability in the von neumann sense are included gauss laguerre quadrature formula is used for the discretization of the integral part numerical examples illustrating the potential advantages of the presented results are included

solving time dependent pdes via an improved modified extended bdf scheme in a previous article the known mebdf scheme had been extended to solve odes that arise in the method of lines solution of time dependent pdes in that article it was shown that mebdf is superior to the widely used bdf approach in certain important classes of problems in this paper by following a different logic we were able to improve the stability characteristics of mebdf appropriate modifications to the code have also been made consequently we investigate whether the current approach is better to the one previously presented it is indeed shown that the instigated improvements have had a positive effect on both accuracy and efficiency numerical results are given to support this claim

preconditioning strategies for fully implicit radiation diffusion with material energy transfer in this paper we present a comparison of four preconditioning strategies for jacobian systems arising in the fully implicit solution of radiation diffusion coupled with material energy transfer the four preconditioning methods are block jacobi schur complement and operator splitting approaches that split the preconditioner solve into two steps one splitting method includes the coupling of the radiation and material fields that appears in the matrix diagonal in the first solve and the other method puts this coupling into the second solve all preconditioning approaches use multigrid methods to invert blocks of the matrix formed from the diffusion operator the schur complement approach is clearly seen to be the most effective for a large range of weightings between the diffusion and energy coupling terms in addition tabulated opacity studies were conducted where again the schur preconditioner performed well last a parallel scaling study was done showing algorithmic scalability of the schur preconditioner

structured dispersion and validity in linear inference abstract structured matrices i n e e ggee arise in nonstandard linear models where e 1 1 1 n and gg 1 n n their properties are studied including expressions for eigenvalues conditions for positive definiteness and conditioning of be as varies it is shown that if majorizes 0 then the condition numbers are ordered as c c 0 for every condition number c generated by the unitarily invariant matrix norms applications are noted in linear inference and in outlier detection

numerical analysis of two level finite difference schemes for unsteady diffusion convection problems for the linear diffusion convection model problem this paper develops the numerical analysis of two classes of three time level second order finite difference schemes weighed schemes similar to two step schemes for differential equations and lf schemes based on the classical conservative approximation of the wave equation as in our paper devoted to two level schemes fourier analysis is an essential tool which yields several key properties stability dissipation numerical diffusion and dispersion it also permits the analysis of parasitic solutions characteristic of three level schemes and their control in terms of the starting scheme the analysis is completed by giving results on the accuracy positivity and parabolicity of the numerical schemes the main conclusions are the expected superiority of these schemes over two level schemes when the diffusion phenomenon is not negligible cell reynolds number less than 1 and the very limited possibilities of obtaining satisfactory numerical results even with upstream differencing when convection is strongly predominating

split linear multistep methods for the numerical integration of stiff differential systems a new approach to the problem of numerically integrating stiff differential systems is described in this approach a linear multistep method the basic method is split into a kind of predictor corrector scheme where the predictor is also implicit if this splitting is done in an appropriate manner the modified method has considerably better stability properties than the basic method as a result splitting methods are particularly useful for problems where conventional integration methods experience stability difficulties in particular some highly stable split linear multistep methods based on backward differentiation formulae are derived and a highly stable variable step implementation is proposed

numerical methods for the solution of the third and fifth order dispersive korteweg de vries equations abstract two numerical methods are proposed for the solution of the third and fifth order korteweg de vries equations the first method is derived using central differences to replace the space derivatives with a predictor corrector time stepping and the second method by linearizing the implicit corrector scheme in which the solution is then found by solving a linear algebraic system at each time step rather than a nonlinear algebraic system which is more usual an important advantage to be gained from the use of the linearized implicit method over the predictor corrector method which is optimally stable is the ability to vary the mesh length the methods are analysed with respect to stability criteria and numerical dispersion numerical results portraying a single soliton solution and the interaction of more than one soliton are reported for the third order korteweg de vries equation numerical results for the fifth order korteweg de vries equation using the linearized implicit method are also reported

an imex scheme for pricing options under stochastic volatility models with jumps partial integro differential equation pide formulations are often preferable for pricing options under models with stochastic volatility and jumps especially for american style option contracts we consider the pricing of options under such models namely the bates model and the so called stochastic volatility with contemporaneous jumps svcj model the nonlocality of the jump terms in these models leads to matrices with full matrix blocks standard discretization methods are not viable directly since they would require the inversion of such a matrix instead we adopt a two step implicit explicit imex time discretization scheme the imex cnab scheme where the jump term is treated explicitly using the second order adams bashforth ab method while the rest is treated implicitly using the crank nicolson cn method the resulting linear systems can then be solved directly by employing lu decomposition alternatively the systems can be iterated under a scalable algebraic multigrid amg method f

linear unbiased approximators of the disturbances in the standard linear model abstract the least squares residuals from the standard linear model have a variance matrix which is a function of the n q matrix of observations on the regressors we examine two classes of residuals which do not suffer from this defect our first class of residuals luz residuals has a variance matrix which is a scalar multiple of an n n idempotent matrix of rank n q specified by the user and our second class of residuals lus residuals has a variance matrix which is a scalar multiple of the n q n q identity matrix

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

how many k step linear block methods exist and which of them is the most efficient and simplest one abstract there have appeared in the literature a lot of k step block methods for solving initial value problems the methods consist in a set of k simultaneous multistep formulas over k non overlapping intervals a feature of block methods is that there is no need of other procedures to provide starting approximations and thus the methods are self starting sharing this advantage of runge kutta methods all the formulas are usually obtained from a continuous approximation derived via interpolation and collocation at k 1 points nevertheless all the k step block methods thus obtained may be considered as different formulations of one of them which results to be the most efficient and simple formulation of all of them the theoretical analysis and the numerical experiments presented support this claim

compact finite difference schemes for mixed initial boundary value problems this paper discusses a class of compact second order accurate finite difference equations for mixed initial boundary value problems for hyperbolic and convective diffusion equations convergence is proved by means of energy arguments and both types of equations are solved by similar algorithms for hyperbolic equations an extension of the lax wendroff method is described which incorporates dissipative boundary conditions upwind downwind differencing techniques arise as the formal hyperbolic limit of the convective diffusion equation finally a finite difference chain rule transforms the schemes from rectangular to quadrilateral subdomains

on nonlinear instabilities in leap frog finite difference schemes abstract briggs newell and sarie j comput phys 51 83 1983 have discussed a mechanism for the destabilisation of finite difference approximations to nonlinear partial differential equations their ideas were developed using the leap frog approximation to the advection equation here the same situation is examined in a manner which compares the basic solution to a periodic wavetrain an investigation is made into the stability of the basic solution to small disturbances which take the form of side band fourier modes the relation between side band growth and envelope modulation is discussed

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

time step size selection for radiation diffusion calculations the purpose of this note is to describe a time step control technique as applied to radiation diffusion standard practice only provides a heuristic criteria related to the relative change in the dependent variables the authors propose an alternative based on relatively simple physical principles this time step control applies to methods of solution that are unconditionally stable and converges nonlinearities within a time step in the governing equations commonly nonlinearities in the governing equations are evaluated using existing old time data the authors refer to this as the semi implicit si method when a method converges nonlinearities within a time step the entire governing equation including all nonlinearities is self consistently evaluated using advance time data with appropriate time centering for accuracy

on the dynamic behaviour of polymers under finite strains constitutive modelling and identification of parameters abstract in this paper we apply a model of finite viscoelasticity and propose an identification technique to represent the dynamic properties of polymers the model is based on a multiplicative split of the deformation gradient into a thermal and a mechanical part the latter being decomposed further into elastic and viscous parts in order to formulate the constitutive equations we transfer the concept of discrete relaxation spectra to finite strains specify the free energy as a function of elastic strain tensors and evaluate the dissipation principle of thermodynamics in the form of the clausius duhem inequality then we investigate the dynamic moduli of a polyethylene melt under harmonic shear deformations and determine the material parameters to this end we linearise the constitutive model and calculate the analytical solution of the evolution equations in addition we formulate a second model which represents the experimental data on the basis of a fairly small number of fitting parameters this so called substitute model is based on the fractional calculus and corresponds to a continuous relaxation spectrum in order to identify the material constants of the finite strain model we are looking for we proceed as follows we determine the parameters of the substitute model calculate the so called cumulative relaxation spectrum and approximate it by means of a series of step functions the height of the steps corresponds to the stiffness parameters of the finite strain model and their locations to the inverse relaxation times

linearized adi schemes for two dimensional space fractional nonlinear ginzburg landau equation abstract space and time approximations for two dimensional space fractional complex ginzburg landau equation are examined the schemes under consideration are discreted by the second order backward differential formula bdf2 in time and two classes of the fractional centered finite difference methods in space a linearized technique is employed by the extrapolation we prove the unique solvability and stability for both numerical methods the convergence of both numerical methods is analyzed at length utilizing the energy argument and the convergence orders under the optimal step size ratio are o 2 h 2 and o 2 h 4 in the sense of the discrete l 2 norm where is the time step size h max h x h y and h x h y are spatial grid sizes in the x direction and y direction respectively in addition we construct a multistep alternating direction implicit adi scheme and a multistep compact adi scheme based on bdf2 for the efficiently numerical implementation finally numerical examples are carried out to verify our theoretical results

high order difference methods for linear variable coefficient parabolic equation abstract in this paper we derive o k 2 h 4 two level three point finite difference methods for the solution of the general linear variable coefficient parabolic equation in one dimension u t a x u xx b x u x c x u 0 x 1 t 0 under suitable initial and boundary conditions the stability of the new schemes is examined using a linear stability analysis in particular we derive unconditionally stable o k 2 h 4 methods for the solution of the convection diffusion type equation in cylindrical and spherical coordinates viz u t u xx x u x cu 0

numerical simulation of reaction diffusion neural dynamics models and their synchronization desynchronization application to epileptic seizures abstract cognitive disorders especially epilepsy are closely linked with synchronization desynchronization of neurons in the brain in this paper the dynamical modeling and behavior analysis of a fitzhugh nagumo neuron and also synchronization control of a network of fitzhugh nagumo neurons which promise the understanding of cognitive processing are studied to numerically simulate these models and in order to overcome their difficulties such as computational complexity multi dimensionality non linearity having large spatial and temporal domains and also having a discrete initial data we propose the use of a strongly meshless technique based on wendland compactly supported radial basis function in conjunction with an operator splitting algorithm the main advantage of the proposed algorithm is its computational complexity for example o n x 1 3 n x 2 3 of the inversion of all matrices which is much lower than the complexity of pure radial basis functions method o n x 1 3 n x 2 3 of the inversion of matrices numerical experiments are presented showing that the proposed approaches are extremely accurate and fast

on the convergence of projected triangular decomposition methods for pricing american options with stochastic volatility numerical pricing of american options with heston stochastic volatility model is considered the complementarity problem with a two dimensional parabolic partial differential operator is discretized by the craig sneyd alternative direction implicit scheme and the resulted linear complementarity problems at each time step are solved by the projected triangular decomposition methods which are constructed as an extension of the classical brennan schwartz algorithm the convergence theorems are established when the system matrix is an m matrix numerical experiments show that the proposed methods with alternative direction implicit schemes are efficient and outperform the classical psor method and operator splitting method

numerical solution of linear and nonlinear black scholes option pricing equations this paper deals with the numerical solution of black scholes option pricing partial differential equations by means of semidiscretization technique for the linear case a fourth order discretization with respect to the underlying asset variable allows a highly accurate approximation of the solution for the nonlinear case of interest modeling option pricing with transaction costs semidiscretization technique provides a competitive numerical solution with respect to others recently given in b during m fournier a jungel convergence of a high order compact finite difference scheme for a nonlinear black scholes equation esaim math modelling numer anal modelisation mathematique et analyse numerique 38 2004 359 369 b during black scholes type equations mathematical analysis parameter identification numerical solution dissertation university mainz july 2005

on the computational efficiency of certain upwinding schemes abstract the computational efficiency of certain upwinding schemes for the discretization of the one dimensional advection dominated transport equation are examined in particular the first order upwind the second order upwind the weighted scheme and the quick scheme are considered the total computational cost for a prescribed accuracy of these schemes was compared using a general purpose algorithm for the integration of time dependent partial differential equations using the method of lines the method of lines approach enables us to utilize the well developed methodology for the assessment of computational efficiency of ode integrators moreover in the time integration the resulting system of odes is integrated by an efficient variable order stepsize numerical integration scheme the results indicate that for small cell peclet number central differences are superior for moderate values of the cell peclet number the quick scheme is the most attractive under high advective flow conditions however the second order upwinding is the most reliable

parallel lod methods for second order time dependent pdes abstract time dependent problems in partial differential equations pdes are often solved by the method of lines mol in many important instances the exact solution of the resulting system of ordinary differential equations odes satisfies recurrence relations involving the matrix exponential function this function is approximated by a new type of rational function possessing real and distinct poles which consequently readily admits a partial fraction expansion thereby allowing the distribution of the work in solving the corresponding linear algebraic systems in essentially backward euler like solves on concurrent processors locally one dimensional lod splitting methods with enhanced parallelization are developed for multispace problems utilizing strang like splitting techniques the use of rational approximants with distinct real poles in the temporal direction in collusion with splitting techniques in the spatial directions creates the potential for efficient coarse grain time stepping parallel algorithms on mimd machines the resulting parallel algorithms possess appropriate stability properties and are implemented on various parabolic and hyperbolic pdes from the literature in higher space dimensional problems

radial basis function generated finite differences for option pricing problems abstract in this paper we present a numerical method to price options based on radial basis function generated finite differences rbf fd in space and the backward differentiation formula of order 2 bdf 2 in time we use gaussian rbfs that depend on a shape parameter e the choice of this parameter is crucial for the performance of the method we chose e as const h 1 and we derive suitable values of the constant for different stencil sizes in 1d and 2d this constant is independent of the problem parameters such as the volatilities of the underlying assets and the interest rate in the market in the literature on option pricing with rbf fd a constant value of the shape parameter is used we show that this always leads to ill conditioning for decreasing h whereas our proposed method avoids such ill conditioning we present numerical results for problems in 1d 2d and 3d demonstrating the useful features of our method such as discretization sparsity flexibility in node placement and easy dimensional extendability which provide high computational efficiency and accuracy

stability analysis of finite difference schemes for the advection diffusion equation we present a collection of stability results for finite difference approximations to the advection diffusion equation u t a u x b u xx the results are for centered difference schemes in space and include explicit and implicit schemes in time up to fourth order and schemes that use different space and time discretizations for the advective and diffusive terms the results are derived from a uniform framework based on the schur cohn theory of simple von neumann polynomials and are necessary and sufficient for the stability of the cauchy problem some of the results are believed to be new

computing option pricing models under transaction costs this paper deals with the barles soner model arising in the hedging of portfolios for option pricing with transaction costs this model is based on a correction volatility function j solution of a nonlinear ordinary differential equation in this paper we obtain relevant properties of the function j which are crucial in the numerical analysis and computing of the underlying nonlinear black scholes equation consistency and stability of the proposed numerical method are detailed and illustrative examples are given

stability concepts in the numerical solution of difference and differential equations abstract several algorithms have been proposed for the stable numerical computation of non dominant solutions of linear difference equations the unifying approach of these algorithms is that they replace the task of solving the given initial value problem by that of solving a boundary value problem which is in some sense equivalent a similar task is involved in the numerical solution of initial value problems for ordinary differential equations using linear multistep methods however in this case it is always assumed that the numerical solution will be found by forward recurrence and this assumption imposes severe restrictions on the numerical methods that can be used by adapting some of the ideas that have been proposed for linear recurrence relations new algorithms are derived for the numerical solution of initial value problems and it is shown that these new methods can be competitive with the best methods currently available

a multigrid preconditioner and automatic differentiation for non equilibrium radiation diffusion problems we study the efficient solution of non equilibrium radiation diffusion problems an implicit time discretization leads to the solution of systems of non linear equations which couple radiation energy and material temperature we consider the implicit euler method the mid point scheme the two step backward differentiation formula and a two stage implicit runge kutta method for time discretization we employ a newton krylov method in the solution of arising non linear problems we describe the computation of the jacobian matrix for newton s method using automatic differentiation based on the operator overloading in fortran 90 for gmres iterations we propose a simple multigrid preconditioner applied directly to the coupled linearized problems we demonstrate the efficiency and scalability of the proposed solution procedure by solving one dimensional and two dimensional model problems

meshfree exponential integrators for the numerical solution of time dependent partial differential equations a class of meshfree exponential integrators is proposed these methods are of particular interest in situations where the solution of the differential equation concentrates on a small part of the computational domain which may vary in time for the space discretization radial basis functions with compact support are suggested the reasons for this choice are the stability and robustness of the resulting interpolation procedure the time integration is performed with an exponential rosenbrock method the required matrix functions are computed by newton interpolation based on leja points the proposed integrators are fully adaptive in space and time numerical examples that illustrate the robustness and the good stability properties of the method are included

matrix equivalence classes with applications abstract a definite matrix is represented in terms of a semidefinite matrix and a vector such that may be recovered explicitly from equivalence classes are generated over the cone of positive definite matrices first on fixing then on fixing these classes in turn associate naturally as scale parameters in ensembles of elliptical measures each ensemble is seen to admit a distinct linear solution to the problem of minimal risk equivariant estimation for a location scalar as do all mixtures over a particular ensemble to include star contoured measures

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

numerical analysis of the stress concentration near holes originating in previously loaded viscoelastic bodies at finite strains abstract the state of stress and strain of previously loaded viscoelastic bodies with holes originating in them successively or simultaneously is analyzed under finite plane deformations the problem statement and solution are based on the theory of repeatedly superimposed large deformations the material mechanical properties are described using integral relations of the convolution type over time with a weakly singular kernel the problem solving is based on the finite element method to calculate the integral of the convolution type a recurrence formula is used that can be obtained by approximating the initial kernel with a linear combination of exponential functions the truncated prony s series the nonlinear effects and the effect of the interaction between holes on the stress concentration are analyzed for the dynamic problems the results for incompressible and weakly compressible materials are compared

downside risk measurement in regime switching stochastic volatility abstract risk measurement is important to firms to enable management of risks and ensure profitability during different firm and market events in particular downside risk is an important risk measure as it is a coherent risk measure and it is also compatible with industry risk management approaches such as stop losses whilst regime switching models have been used for downside risk measurement the regime switching models for stochastic volatility dynamics have been limited and so restrict risk measurement in this paper we propose a new regime switching model that incorporates non trivial stochastic volatility dynamics hence we are able to measure risk more realistically we derive the downside risk measure associated with our regime switching model for risk measurement including and excluding jump risk we prove that the regime switching model converges to the underlying continuous time asset pricing model hence our risk measurement is consistent we provide a discretisation for the variance risk process which is locally consistent and enables computational implementation we also provide numerical experiments to illustrate our method