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

pseudospectra for the wave equation with an absorbing boundary for systems which can be described by ut au with a highly nonnormal matrix or operator a the spectrum of a may describe the behavior of the system poorly one such operator arises from the one dimensional wave equation on a finite interval with a homogeneous dirichlet condition at one end and a linear damping condition at the other in this paper the pseudospectra norm of the resolvent of this operator are computed in an energy norm using analytical techniques and computations with discrete approximations when the damping condition is perfectly absorbing the pseudospectra are half planes parallel to the imaginary axis and in other cases they are periodic in the imaginary direction and approximate strips of finite thickness the nonnormality of the operator is related to the behavior of the system and the limitations of spectral analysis

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

hopf bifurcation of reaction diffusion and navier stokes equations under discretization the long time behaviour of numerical approximations to the solutions of a semilinear parabolic equation undergoing a hopf bifurcation is studied in this paper the framework includes reaction diffusion and incompressible navier stokes equations it is shown that the phase portrait of a supercritical hopf bifurcation is correctly represented by runge kutta time discretization in particular the bifurcation point and the hopf orbits are approximated with higher order a basic tool in the analysis is the reduction of the dynamics to a two dimensional center manifold a large portion of the paper is therefore concerned with studying center manifolds of the discretization

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

stability and non normality of the k e equations abstract the analytical and numerical solutions of the equations of the k e turbulence model are analyzed under certain conditions on the boundary values and the interior values of k and e the analytical and numerical solutions are bounded if the steady state solution is obtained numerically by a runge kutta time stepping method then severe constraints on the time step and the non normality of the jacobian matrix make the convergence very slow the simplifications and conclusions are supported by data from a numerical solution of flow over a flat plate

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

nonnormality effects in a discretised nonlinear reaction convection diffusion equation what is the long time effect of adding convention to a discretised reaction diffusion equation for linear problems it is well known that convection may denormalise the process and in particular eigenvalue based stability predictions may be overoptimistic this work deals with a related issue with a nonlinear reaction term the nonnormality can greatly influence the long time dynamics for a nonlinear model problem with dirichlet boundary conditions it is shown that the basin of attraction of the correct steady state can be shrunk in a directionally biased manner a normwise analysis provides lower bounds on the basin of attraction and a more revealing picture is provided by pseudo eigenvalues in extreme cases the computed solution can converge to a spurious bounded steady state that exists only in finite precision arithmetic the impact of convection on the existence and stability of spurious periodic solutions is also quantified

ordering ambiguity versus representation in this work we show that the ordering ambiguity on quantization depends on the representation choice this property is then used to solve unambiguously some particular systems finally we speculate on the consequences for more involved cases

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

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

pseudospectra of the orr sommerfeld operator this paper investigates the pseudospectra and the numerical range of the orr sommerfeld operator for plane poiseuille flow a number z in bf c is in the epsilon pseudospectrum of a matrix or operator a if zi a 1 geq epsilon 1 or equivalently if z is in the spectrum of a e for some perturbation e satisfying e leq epsilon the numerical range of a is the set of numbers of the form au u where cdot cdot is the inner product and u is a vector or function with huh u 1 the spectrum of the orr sommerfeld operator consists of three branches it is shown that the eigenvalues at the intersection of the branches are highly sensitive to perturbations and that the sensitivity increases dramatically with the reynolds number the associated eigenfunctions are nearly linearly dependent even though they form a complete set to understand the high sensitivity of the eigenvalues a model operator is considered related to the airy equation tha

stability and boundedness in the numerical solution of initial value problems abstract this paper concerns the theoretical analysis of step by step meth ods for solving initial value problems in ordinary and partial differential equa tions the main theorem of the paper answers a natural question arising in thelinear stability analysis of such methods it guarantees a strong version ofnumerical stability under a stepsize restriction related to the stability regionof the numerical method and to a circle condition on the differential equation the theorem also settles an open question related to the properties total variation diminishing strong stability preserving monotonic and total variation bounded under a monotonicity condition on the forward eulermethod the theorem specifies a stepsize condition guaranteeing boundednessfor linear problems the main theorem is illustrated by applying it to linear multistep methods for important classes of these methods conclusions are thus obtained whichsupplement earlier results in the literature

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

a spectral fc solver for the compressible navier stokes equations in general domains i explicit time stepping we present a fourier continuation fc algorithm for the solution of the fully nonlinear compressible navier stokes equations in general spatial domains the new scheme is based on the recently introduced accelerated fc method which enables use of highly accurate fourier expansions as the main building block of general domain pde solvers previous fc based pde solvers are restricted to linear scalar equations with constant coefficients the fc methodology presented in this text thus constitutes a significant generalization of the previous fc schemes as it yields general domain fc solvers for nonlinear systems of pdes while not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent fourier expansions including high order convergence essentially spectrally accurate dispersion relations and much milder cfl constraints than those imposed by polynomial based spectral methods since for example the spectral radius of the fc first derivative grows linearly with the number of spatial discretization points we demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid dynamics and nonlinear acoustics

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

method of lines for hyperbolic differential equations runge kutta methods are applied to the numerical solution of hyperbolic partial differential equations conditions that they are locally stable are derived

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

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

stability analysis of preconditioned approximations of the euler equations on unstructured meshes this paper analyses the stability of a discretisation of the euler equations on 3d unstructured grids using an edge based data structure first order characteristic smoothing a block jacobi preconditioner and runge kutta timemarching this is motivated by multigrid navier stokes calculations in which this inviscid discretisation is the dominant component on coarse grids the analysis uses algebraic stability theory which allows at worst a bounded linear growth in a suitably defined perturbation energy provided the range of values of the preconditioned spatial operator lies within the stability region of the runge kutta algorithm the analysis also includes consideration of the effect of solid wall boundary conditions and the addition of a low mach number preconditioner to accelerate compressible flows in which the mach number is very low in a significant portion of the flow numerical results for both inviscid and viscous applications confirm the effectiveness of the numerical algorithm and show that the analysis provides accurate stability bounds

block newton method and block rayleigh quotient iteration for computing invariant subspaces of general complex matrices abstract we consider the block newton method and a modification of it the block rayleigh quotient iteration for approximating a simple p dimensional invariant subspace x im x and the corresponding eigenvalues collected in the projection l x h x 1 x h a x of an arbitrary complex matrix a c n n both methods generate a sequence of bases u k such that the subspaces im u k approximate the target subspace x the block newton method is newton s method applied to the extended system a x x l 0 w h x i with respect to x l where w is a normalization matrix we give a direct convergence analysis which in contrast to the approach by beyn kless thummler 2001 explicitly exploits the product form of the second derivative and therefore leads to new and essentially sharper error recursions for each of the sequences u k and k where u k and k are the k th x and l iterates resp then following losche schwetlick timmermann 1998 where the case of real symmetric a is considered we modify the block newton method as follows in step k we set w u k and use only the x update from the newton method to improve u k and then with the improved basis perform a rayleigh schur process and take the orthonormal basis and the generalized rayleigh quotient matrix obtained from it as new iterates so the method can be considered as a block rayleigh quotient iteration we show that for a sufficiently good initial approximation u 0 the method converges in that the sequence sin k with k im u k x converges q quadratically to zero provided that the invariant subspace x is simple i e that the eigenvalues corresponding to x which are the eigenvalues of the projection l x h x 1 x h a x of a onto x are separated from those corresponding to x which are given by a l the convergence results are directly proven without exploiting standard newton convergence results by using the angles between the corresponding subspaces instead of working with the norm differences of the bases as done by beyn kless thummler 2001 as a byproduct the method is shown to converge cubically if a a h as in the case p 1

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

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