stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations abstract this paper deals with the stability analysis of one step methods for the numerical solution of initial value problems both stiff ordinary and partial differential equations are included the problem is considered how to restrict the stepsize in the methods in order that they behave stable we review and extend some recent results on this problem that are based on the use of stability regions in the complex plane we focus on differential equations that are essentially more general than the classical test equation u t u t with a complex constant further the emphasis is on stability of the methods with respect to the maximum norm

the sharpness of stability estimates corresponding to a general resolvent condition abstract in the stability analysis of numerical processes the problem arises of establishing moderate upper bounds for the norm of the n th power of given s s matrices recently new upper bounds were derived under generalized versions of the resolvent condition occurring in the kreiss matrix theorem in the present paper the question is answered of whether these new bounds are sharp with respect to their behaviour as n and s

the importance of eigenvectors for local preconditioners of the euler equations the design of local preconditioners to accelerate the convergence to a steady state for the compressible euler equations has so far been solely based on eigenvalue analysis however numerical evidence exists that the eigenvector structure also has an influence on the performance of preconditioners and should therefore be included in the design process in this paper we present the mathematical framework for the eigenvector analysis of local preconditioners for the multi dimensional euler equations the non normality of the preconditioned system is crucial in determining the potential for transient amplification of perturbations several existing local preconditioners are shown to possess a highly non normal structure for low mach numbers this non normality leads to significant robustness problems at stagnation points a modification to these preconditioners which eliminates the non normality is suggested and numerical results are presented showing the marked improvement in robustness

runge kutta solutions of a hyperbolic conservation law with source term spurious long term solutions of a finite difference method for a hyperbolic conservation law with a general nonlinear source term are studied results are contrasted with those that have been established for nonlinear ordinary differential equations various types of spurious behavior are examined including spatially uniform equilibria that exist for arbitrarily small time steps nonsmooth steady states with profiles that jump between fixed levels and solutions with oscillations that arise from nonnormality and exist only in finite precision arithmetic it appears that spurious behavior is associated in general with insufficient spatial resolution the potential for curbing spuriosity by using adaptivity in space or time is also considered

a boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2d we present a new method for the evolution of inextensible vesicles immersed in a stokesian fluid we use a boundary integral formulation for the fluid that results in a set of nonlinear integro differential equations for the vesicle dynamics the motion of the vesicles is determined by balancing the non local hydrodynamic forces with the elastic forces due to bending and tension numerical simulations of such vesicle motions are quite challenging on one hand explicit time stepping schemes suffer from a severe stability constraint due to the stiffness related to high order spatial derivatives and a milder constraint due to a transport like stability condition on the other hand an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step we present two semi implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme we discretize the equations by using a spectral method in space and a multistep third order accurate scheme in time we use the fast multipole method fmm to efficiently compute vesicle vesicle interaction forces in a suspension with a large number of vesicles we report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme

weak ill posedness of spatial discretizations of absorbing boundary conditions for schrodinger type equations when we wish to solve numerically a differential problem defined on an infinite domain it is necessary to consider a finite subdomain and to use artificial boundary conditions in such a way that the solutions in the finite subdomain approximate the original solution these boundary conditions are called absorbing when small reflections to the interior domain are allowed in this paper we develop a general class of absorbing boundary conditions for schrodinger type equations by using rational approximations to the transparent boundary conditions with this approach previous absorbing boundary conditions in the literature are included in this class we use the method of lines for the discretization of the initial boundary value problems obtained this way we show that the ordinary differential systems that arise after the spatial discretization are weakly ill posed explaining a previous conjecture of fevens and jiang the time discretization is carried out with a stable runge kutta methods where the high order ones may be used to compensate for the possible troubles present in the problems semidiscretized in space

a generalization of the numerical range of a matrix abstract a new generalization of the classical numerical range of a matrix is introduced various properties and basic characterizations of this generalized numerical range are established one of these characterizations is used in proving a resolvent inequality for a discretized version of the differential operator d 2 dx 2

a pseudospectral based method of lines for solving integro differential boundary layer equations application to the mixed convection over a heated horizontal plate abstract the method of lines is well suited for solving numerically parabolic boundary layer equations because it avoids the numerical difficulties associated to the integration of the continuity equation which is subsumed into the momentum equations as an integral of the main velocity component to deal with these integrals as well as with any other integral operator entering the boundary layer equations in some particular problems it is very efficient to discretize the transversal coordinate using pseudospectral methods the resulting ordinary differential equations odes can be then written in a very compact form suitable for general purpose methods and software developed for the numerical integration of odes we present here such a numerical method applied to the boundary layer equations governing the mixed convection over a heated horizontal plate these parabolic equation can be written in such a way that the natural convection appears as an integro differential term in the usual horizontal momentum equation so that the discretization by pseudospectral methods of the vertical coordinate derivative is very appropriate several matlab based solvers are compared to integrate the resulting odes to validate the numerical results they are compared with analytic solutions valid near the leading edge of the boundary layer

on the 2d bioheat equation with convective boundary conditions and its numerical realization via a highly accurate approach abstract a series solution for a 2d bioheat pennes conduction model with convective boundary conditions is established by using the classical fourier method to validate and compare results a numerical method for constructing highly accurate numerical solutions is also derived and numerically illustrated both the series solution and the proposed numerical method open interesting possibilities of application in inverse perfusion coefficient identification problems

an instability phenomenon in spectral methods the eigenvalues of chebyshev and legendre spectral differentiation matrices which determine the allowable time step in an explicit time integration are extraordinarily sensitive to rounding errors and other perturbations on a grid of n points per space dimension machine rounding leads to errors in the eigenvalues of size o n 2 this phenomenon may lead to inconsistency between predicted and observed time step restrictions one consequence of it is that spectral differentiation by interpolation in legendre points which has a favorable o n 1 time step restriction for the model problem u t u x in theory is subject to an o n 2 restriction in practice the same effect occurs with chebyshev points for the model problem u t xu x another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision we know of no other examples in numerical computation of this kind of precision dependent stability

several concepts to investigate strongly nonnormal eigenvalue problems eigenvalue analysis plays an important role in understanding physical phenomena however if one deals with strongly nonnormal matrices or operators the eigenvalues alone may not tell the full story a well known tool which can be useful to get more insight in the reliability or sensitivity of eigenvalues is varepsilon pseudospectra apart from varepsilon pseudospectra we consider other tools which might help to learn more about the eigenvalue problem viz condition numbers of the eigenvalues condition numbers of sets of eigenvectors and angles between invariant subspaces all these concepts will be studied and compared for both standard and generalized eigenvalue problems the tools can be used to analyze large eigenvalue problems we apply the different concepts to a generalized eigenvalue problem obtained from magnetohydrodynamics in this problem one is interested in an interior part of the spectrum called the alfven spectrum

optimal runge kutta methods for first order pseudospectral operators new runge kutta methods for method of lines solution of systems of ordinary differential equations arising from discretizations of spatial derivatives in hyperbolic equations by chebyshev or modified chebyshev methods are introduced these runge kutta methods optimize the time step necessary for stable solutions while holding dispersion and dissipation fixed it is found that maximizing dispersion minimizes dissipation and vice versa optimal methods with respect to large stability intervals on the imaginary axis and with respect to the eigenvalue spectra of the underlying pseudospectral discretizations are developed in the latter case stability regions are optimized to include the outliers of the spatial operators performance on a model problem in computational aeroacoustics is evaluated the optimized schemes have two more function evaluations per timestep than the standard fourth order runge kutta method but allow timesteps up to 1 7 times larger moreover dissipation and dispersion are reduced

characterizations of the range of neumann problem for semilinear elliptic equations on etudie le probleme de neumann au gu i j 1 n a ij x u x i x j g u f ce bu x j 1 n a ij x u x i cos x j 0 xe ou est un domaine de r n a frontiere lisse est la normale exterieure unite de t a ij x e 1 a est un operateur uniformement fortement elliptique dans et g t r r est une fonction continue et bornee on donne un theoreme d existences des solutions faibles

stable spectral collocation solutions to a class of benjamin bona mahony initial value problems we are concerned with stable spectral collocation solutions to non periodic benjamin bona mahony bbm modified bbm and benjamin bona mahony burgers bbm b initial value problems on the real axis the spectral collocation is based alternatively on the scaled hermite and sinc functions in order to march in time we use several one step and linear multistep finite difference schemes such that the method of lines mol involved is stable in sense of lax the method based on hermite functions ensures the correct behavior of the solutions at large spatial distances and in long time periods in order to prove the stability we use the pseudospectra of the linearized spatial discretization operators the extent at which the energy integral of bbm model is conserved over time is analyzed for hermite collocation along with various finite difference schemes this analysis has been fairly useful in optimizing the scaling parameter the effectiveness of our approach has been confirmed by some numerical experiments

spectra pseudospectra and localization for random bidiagonal matrices there has been much recent interest initiated by work of the physicists hatano and nelson in the eigenvalues of certain random non hermitian periodic tridiagonal matrices and their bidiagonal limits these eigenvalues cluster along a bubble with wings in the complex plane and the corresponding eigenvectors are localized in the wings delocalized in the bubble here in addition to eigenvalues pseudospectra are analyzed making it possible to treat the nonperiodic analogues of these random matrix problems inside the bubble the resolvent norm grows exponentially with the dimension outside it grows subexponentially in a bounded region that is the spectrum of the infinite dimensional operator localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase respectively with distance from the diagonal this article presents theorems that characterize the spectra pseudospectra and numerical range for the four cases of finite bidiagonal matrices infinite bidiagonal matrices stochastic toeplitz operators finite periodic matrices and doubly infinite bidiagonal matrices stochastic laurent operators c 2001 john wiley sons inc

amf runge kutta formulas and error estimates for the time integration of advection diffusion reaction pdes the convergence of a family of amf runge kutta methods in short amf rk for the time integration of evolutionary partial differential equations pdes of advection diffusion reaction type semi discretized in space is considered the methods are based on very few inexact newton iterations applied to implicit runge kutta formulas by combining the use of a natural splitting for the underlying jacobians and the approximate matrix factorization amf technique this approach allows a very cheap implementation of the runge kutta formula under consideration particular amf rk methods based on radau iia formulas are considered these methods have given very competitive results when compared with important formulas in the literature for multidimensional systems of non linear parabolic pde problems uniform bounds for the global time space errors on semi linear pdes when simultaneously the time step size and the spatial grid resolution tend to zero are derived numerical illustrations supporting the theory are presented

on the use of stability regions in the numerical analysis of initial value problems this paper deals with the stability analysis of one step methods in the numerical solution of initial boundary value problems for linear ordinary and partial differential equations restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size these restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration the errors in the one step methods are measured in arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of the heat equation and some other differential equations where the error growth is measured in the maximum norm

the equivalence of l2 stability the resolvent condition and strict h stability abstract the kreiss matrix theorem asserts that a family of n n matrices is l 2 stable if and only if either a resolvent condition r or a hermitian norm condition h is satisfied we give a direct considerably shorter proof of the power boundedness of an n n matrix satisfying r sharpening former results by showing that power boundedness depends at most linearly on the dimension n we also show that l 2 stability is characterized by an h condition employing a general h numerical radius instead of the usual h norm thus generalizing a sufficient stability criterion due to lax and wendroff

long time behavior of x led reactions single chemical reactions with a finite but arbitrary number of chemicals are considered and the properties of the dynamical systems which arise in modeling the time evolution of the corresponding concentrations are studied no space variables are included in the models as it is assumed that the reactions take place in well stirred reactor tanks the time variable is assumed to take discrete values which are multiples of a finite interval t the links between such dynamical systems and the nonlinear stochastic processes derived from statistical dynamics are presented also the concept of x led reactions is introduced as a generalization of the dynamics associated with density and activity led reactions in all cases maps of the interval whose convergence properties are investigated are obtained the fact that taking t 0 reproduces better and better the solution of the associated differential equations is briefly discussed

a noncommutative central limit theorem for ccr algebras abstract a noncommutative generalization of the central limit theorem for even completely positive mappings between two ccr algebras is proved quasi free completely positive mappings are found to be the generalizations of the gaussian distributed random variables

stepsize restrictions for stability of one step methods in the numerical solution of initial value problems this paper deals with the analysis of general one step methods for the numerical solution of initial boundary value problems for stiff ordinary and partial differential equations restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size these restrictions are related to disks contained in the stability region of the method and the errors are measured with arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of a diffusion convection problem where the error growth is measured with the maximum norm

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

on the uniform power boundedness of a family of matrices and the applications to one leg and linear multistep methods we study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy dt y and constant stepsizeh lets be the region of the absolute stability of the method and letd be a closed subset ofs on the riemann sphere mathbb c it is shown that the solutions of these difference equations are bounded forn 0 uniformly for h d s is itself closed in mathbb c iff s is free of cusps the question is studed by means of contractivity analysis and a matrix theorem derived from the matrix theorem of kreiss

on a generalization of the resolvent condition in the kreiss matrix theorem this paper deals with a condition on the resolvent of s x s matrices a in one of the equivalent assertions of the kreiss matrix theorem the spectral norm of the resolvent of a at 4 must satisfy an inequality for all 4 lying outside the unit disk in c we consider a generalization in which domains different from the unit disk and more general norms are allowed under this generalized resolvent condition an upper bound is derived for the norms of the nth powers of s x s matrices b here b depends on a via a relation b 0 a where p is an arbitrary rational function the upper bound grows linearly with s 1 and is independent of n 1 this generalizes an upper bound occurring in the kreiss theorem where b a like the classical kreiss theorem the upper bound derived in this paper can be used in the stability analysis of numerical methods for solving differential equations

stability of the method of lines it is well known that a necessary condition for the lax stability of the method of lines is that the eigenvalues of the spatial discretization operator scaled by the time stepk lie within a distanceo k of the stability region of the time integration formula ask 0 in this paper we show that a necessary and sufficient condition for stability except for an algebraic factor is that the pseudo eigenvalues of the same operator lie within a distanceo o k of the stability region ask 0 our results generalize those of an earlier paper by considering a runge kutta and other one step formulas b implicit as well as explicit linear multistep formulas c weighted norms d algebraic stability e finite and infinite time intervals and f stability regions with cusps in summary the theory presented in this paper amounts to a transplantation of the kreiss matrix theorem from the unit disk for simple power iterations to an arbitrary stability region for method of lines calculations

rapid prolate pseudospectral differentiation and interpolation with the fast multipole method pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials however the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix vector products which if evaluated directly entail o n2 memory requirement and computational complexity where n is the number of unknowns utilized for discretization and interpolation in this work we show that the fast multipole method fmm can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to o n example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver

about the sharpness of the stability estimates in the kreiss matrix theorem one of the conditions in the kreiss matrix theorem involves the resolvent of the matrices a under consideration this so called resolvent condition is known to imply for all n 1 the upper bounds an ek n 1 and an ek n 1 here is the spectral norm k is the constant occurring in the resolvent condition and the order of a is equal to n 1 1 it is a long standing problem whether these upper bounds can be sharpened for all fixed k 1 to bounds in which the right hand members grow much slower than linearly with n 1 and with n 1 respectively in this paper it is shown that such a sharpening is impossible the following result is proved for each e 0 there are fixed values c 0 k 1 and a sequence of n 1 n 1 matrices an satisfyirig the resolvent condition such that an n c n 1 1 e c n 1 1 e for n n 1 2 3 the result proved in this paper is also relevant to matrices a whose e pseudospectra lie at a distance not exceeding ke from the unit disk for all e 0

time compact high order difference methods for wave propagation we construct one step explicit difference methods for solution of wave propagation problems with fourth order accuracy in both space and time by using a principle that can be generalized to arbitrary order we use the first order system form and a staggered grid the fourth order accuracy in time is obtained by transferring time derivatives in the truncation error to space derivatives discontinuous coefficients corresponding to interfaces between different materials are considered as a special case of variable coefficients and the method is applied across the discontinuities the accuracy is much improved compared to second order methods even for this type of problem a certain norm is shown to be conserved ensuring good accuracy even for long time integration

resolvent conditions and bounds on the powers of matrices with relevance to numerical stability of initial value problems we deal with the problem of establishing upper bounds for the norm of the nth power of square matrices this problem is of central importance in the stability analysis of numerical methods for solving linear initial value problems for ordinary partial or delay differential equations a review is presented of upper bounds which were obtained in the literature under the resolvent condition occurring in the kreiss matrix theorem as well as under variants of that condition moreover we prove new bounds under resolvent conditions which generalize some of the reviewed ones the paper concludes by applying one of the new upper bounds in a stability analysis of the trapezoidal rule for delay differential equations

convergence of the quantum boltzmann map we consider a non linear map on the space of density matrices which we call the boltzmann map it is the composition of a doubly stochastic mapt on the space ofn body states and the conditional expectation onto the one body space whent is ergodic then the iterates of take any initial state to the uniform distribution if the energy levels are equally spaced andt conserves energy and is ergodic on each energy shell then iterates of take any initial state of finite energy to a canonical distribution

multiple solutions of the periodic boundary value problem for some forced pendulum type equations if f x c 0 and e is a constant t then integrating 0 1 over 0 272 after multiplication of both members of the equation by x shows that the only possible solutions are constant and hence satisfy a sin x c consequently if we exclude the trivial translation by a multiple of 275 0 1 will have no solution if dj a one solution if lp a and two solutions if lc l a in 0th er t erms the intersection of the range of the operator d2 dt2 c d dt f u sin acting on zn periodic functions of class g with the subspace of constant functions in the space c o 2711 of real continuous functions on 0 27r is the closed interval a a whose interior points are images of two distinct solutions and boundary points of one we try to obtain similar results for the case where e is no more constant and

eigenvalue stability of radial basis function discretizations for time dependent problems differentiation matrices obtained with infinitely smooth radial basis function rbf collocation methods have under many conditions eigenvalues with positive real part preventing the use of such methods for time dependent problems we explore this difficulty at theoretical and practical levels theoretically we prove that differentiation matrices for conditionally positive definite rbfs are stable for periodic domains we also show that for gaussian rbfs special node distributions can achieve stability in 1 d and tensor product nonperiodic domains as a more practical approach for bounded domains we consider differentiation matrices based on least squares rbf approximations and show that such schemes can lead to stable methods on less regular nodes by separating centers and nodes least squares techniques open the possibility of the separation of accuracy and stability characteristics

a jacobi waveform relaxation method for odes a jacobi waveform relaxation wr method for solving initial value problems for ordinary differential equations odes is presented in each window the method uses a technique called dynamic fitting and a pair of continuous runge kutta rk formulas to produce the initial waveform after which a fixed number of waveform iterates are computed the reliability and efficacy of the method are demonstrated numerically by applying it to qualitatively different problems for linear tridiagonal odes

chebyshev pseudospectral method for computing numerical solution of convection diffusion equation abstract a method for computing highly accurate numerical solutions of 1d convection diffusion equations is proposed in this method the equation is first discretized with respect to the spatial variable transforming the original problem into a set of ordinary differential equations and then the resulting system is integrated in time by the fourth order runge kutta method spatial discretization is done by using the chebyshev pseudospectral collocation method before describing the method we review a finite difference based method by salkuyeh d khojasteh salkuyeh on the finite difference approximation to the convection diffusion equation appl math comput 179 2006 79 86 and contrary to the proposal of the author we show that this method is not suitable for problems involving time dependent boundary conditions which calls for revision stability analysis based on pseudoeigenvalues to determine the maximum time step for the proposed method is also carried out superiority of the proposed method over a revised version of salkuyeh s method is verified by numerical examples

stability of a chebychev pseudospectral solution of the wave equation with absorbing boundaries abstract stability of the pseudospectral chebychev collocation solution of the two dimensional acoustic wave problem with absorbing boundary conditions is investigated the continuous one dimensional problem with one absorbing boundary and one dirichlet boundary has previously been shown to be far from normal consequently the spectrum of that problem says little about the stability behavior of the solution our analysis proves that the discrete formulation with dirichlet boundaries at all boundaries is near normal and hence the formulation with absorbing boundaries at all boundaries either for one dimensional or two dimensional wave propagation is not far from normal the near normality follows from the near normality of the second order derivative pseudospectral differential operator further the nearness to normality is independent of the boundary discretization stability limits on the timestep are however dependent on the boundary operator with an explicit euler method having the most restrictive condition the crank nicolson implementation has a stability limit the same as the dirichlet formulation furthermore in this case the restriction scales by 1 2 in moving from one dimension to two dimensions exactly as in the central finite difference approximation numerical results confirm the predicted values on allowable timesteps obtained from a spectral analysis for both chebychev and modified chebychev implementations we conclude that the spectrum of the evolution operator is informative for predicting the behavior of the numerical solution

a quantum central limit theorem for sums of independent identically distributed random variables we formulate and prove a general central limit theorem for sums of independent identically distributed noncommutative random variables

stability estimates for families of matrices of nonuniformly bounded order this paper addresses the problem of establishing upper bounds for the norm of the nth power of square matrices the focus is on upper bounds that are valid for families of matrices of finite but nonuniformly bounded order such upper bounds are relevant to the stability analysis of numerical methods for solving differential equations in the famous kreiss matrix theorem a condition on the resolvent of matrices occurs which implies that the norm of the nth power does not grow faster than linearly with n in this paper a slightly stronger version of this resolvent condition is studied which is often satisfied in cases of practical interest we prove that this version implies growth at a rate that is essentially lower than in the case of the classical kreiss resolvent condition we consider also a condition on the iterated resolvent of matrices which is related to one of the assumptions occurring in the hille yosida theorem this condition is known to imply growth at a rate b n we prove that when a slightly stronger version of the iterated resolvent condition is in force there is growth at a rate that is at most n 1 1 with 0 q p i

pseudospectra of waveform relaxation operators abstract the performance of the waveform relaxation method for solving systems of odes depends largely on the choices that are made for splitting size of time window and preconditioning although it is known that superlinear convergence is obtained on finite time windows the convergence may be slow in the first few iterations we propose the use of pseudospectra to analyze the convergence ratio of the first few iterations when waveform relaxation is applied to linear systems of odes through pseudospectral radii we can examine the effect of preconditioning and overlapping on the rate of convergence we may also use this to estimate a suitable size of the time window numerical experiments performed on a system of odes arising from the discretization of an advection diffusion equation confirm the validity of the obtained estimates

the errors in calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points abstract we discuss here the errors incurred using the standard formula for calculating the pseudospectral differentiation matrices for cebysev gauss lobatto points we propose explanations for these errors and suggest more precise methods for calculating the derivatives and their matrices

higher order time stepping methods for time dependent reaction diffusion equations arising in biology abstract this paper demonstrates the use of higher order methods to solve some time dependent stiff pdes in the past the most popular numerical methods for solving system of reaction diffusion equations was based on the combination of low order finite difference method with low order time stepping method we extend in this report the compatibility of fourth order finite difference scheme in space coupled with fourth order time stepping methods such as imexlm4 imexpc4 imexrk4 and etdrk4 b in time for direct integration of reaction diffusion equations in one space dimension some interesting numerical anomaly phenomenons associated with steady state solutions of the examples chosen from the literature are well presented to address the naturally arising points and queries our findings have led to the understanding of pattern formation such as spiral waves and patchy structures as well as some spatio temporal dynamical structures

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws

entropy and the central limit theorem in quantum mechanics gives a short proof that any infinitely divisible even representation of a clifford algebra or a car algebra is quasi free and of hudson s central limit theorem 1973 for fermions the author shows that the entropy of any even state of the car clifford or ccr algebra is less than or equal to the entropy of the quasi free state with the same two point function