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

stability properties of implicit runge kutta methods new stability concepts bs stability and bsi stability internal bs stability are introduced bs stability is a modification and extension of b stability and enables the derivation of order results for runge kutta methods applied to general nonlinear stiff initial value problems these order results b consistency b convergence are not affected by stiffness several classes of implicit runge kutta methods are shown to be bs stable gauss radau ia and radau iia schemes

on the convergence of multistep methods for nonlinear stiff differential equations convergence estimates are given fora stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems the approach taken here combines perturbation arguments with frequency domain techniques

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

stability of linear multistep methods for nonlinear neutral delay differential equations in banach space this paper is devoted to investigating the nonlinear stability properties of linear multistep methods for the solution to neutral delay differential equations in banach space two approaches to numerically treating the neutral term are considered which allow us to prove several results on numerical stability of linear multistep methods these results provide some criteria for choosing the step size such that the numerical method is stable some examples of application and a numerical experiment which further confirms the main results are given

highly efficient strong stability preserving runge kutta methods with low storage implementations strong stability preserving ssp runge kutta methods were developed for time integration of semidiscretizations of partial differential equations ssp methods preserve stability properties satisfied by forward euler time integration under a modified time step restriction we consider the problem of finding explicit runge kutta methods with optimal ssp time step restrictions first for the case of linear autonomous ordinary differential equations and then for nonlinear or nonautonomous equations by using alternate formulations of the associated optimization problems and introducing a new more general class of low storage implementations of runge kutta methods new optimal low storage methods and new low storage implementations of known optimal methods are found the results include families of low storage second and third order methods that achieve the maximum theoretically achievable effective ssp coefficient independent of stage number as well as low storage fourth order methods that are more efficient than current full storage methods the theoretical properties of these methods are confirmed by numerical experiment

positive and conservative schemes for mass action kinetics abstract some implicit schemes for the discretization of mass action kinetics are presented and discussed although all the schemes are classical the solution procedure takes great advantage of the peculiar structure of the ode all the schemes maintain linear first integrals such as atomic mass conservation the main features of the first order scheme is positivity preservation for arbitrarily large time steps while for the higher order schemes there is numerical evidence that this property is maintained with fairly large time steps for the first order scheme the existence of a nonnegative solution of the algebraic system arising in the advancing step is proved for arbitrarily large time steps while uniqueness is proved for limited time steps finally an efficient procedure is presented for solving the nonlinear systems involved in the time step these systems are solved without the use of jacobian matrices or their approximations but by the repeated inversion of m matrices a procedure which is both easy and fast considerably simplifying and accelerating the computer implementation of the schemes

a vertex based finite volume method applied to non linear material problems in computational solid mechanics a vertex based finite volume fv method is presented for the computational solution of quasi static solid mechanics problems involving material non linearity and infinitesimal strains the problems are analysed numerically with fully unstructured meshes that consist of a variety of two and threedimensional element types a detailed comparison between the vertex based fv and the standard galerkin fe methods is provided with regard to discretization solution accuracy and computational efficiency for some problem classes a direct equivalence of the two methods is demonstrated both theoretically and numerically however for other problems some interesting advantages and disadvantages of the fv formulation over the galerkin fe method are highlighted

absolute monotonicity of rational functions occurring in the numerical solution of initial value problems this paper deals with rational functions o z approximating the exponential function exp z related to numerical procedures for solving initial value problems motivated by positivity and contractivity requirements imposed on these numerical procedures we study the greatest nonnegative numberr denoted byr o such that o is absolutely monotonic on r 0 an algorithm for the computation ofr o is presented application of this algorithm yields the valuer o for the well known pade approximations to exp z for some specific values ofm n andp we determine the maximum ofr o when o varies over the class of all rational functions o with degree of the numerator m degree of the denominator n and o z exp z z p 1 forz 0

contractivity preserving implicit linear multistep methods we investigate contractivity properties of implicit linear multistep methods in the numerical solution of ordinary differential equations the emphasis is on nonlinear and linear systems d u t f t u t where f satisfies a so called circle condition in an arbitrary norm the results for the two types of systems turn out to be closely related we construct optimal multistep methods of given order and stepnumber which allow the use of a maximal stepsize

numerical simulation of coupled fluid solid problems abstract a multi grid finite volume method is presented for the prediction of coupled fluid solid problems in complex geometries whereby a unified finite volume approach based on the concept of block structured grids is employed for the handling of the complexity of the geometry and the fluid solid coupling a high numerical efficiency is obtained with a non linear multi grid method using a pressure correction based scheme as a smoother a number of numerical experiments were used to verify the accuracy and numerical efficiency of the method additionally to illustrate the capabilities of the approach results for two practical applications are presented

order results for implicit runge kutta methods applied to stiff systems an extensive order analysis is given for the local and global errors of certain classes of runge kutta methods applied to general nonlinear stiff initial value problems it turns out that there are three order levels for linear problems y ay the classical orders apply for general nonlinear problems however two types of order reductions occur there a distinction has to be made between stiff problems where only f y is large and stiff problems where also some of the other derivatives f t f tt f yy cdots are large for these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h the order levels for both classes of problems are considerably lower than the respective conventional orders these order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems

extending convergence of bdf methods for a class of nonlinear strongly stiff problems abstract in 1995 a new approach was presented to overcome the drawbacks of existing convergence theory for stiff initial value problems by auzinger et al bit 36 4 1996 635 652 and illustrated by means of a convergence argument for the implicit euler scheme in the present paper our intention is to obtain extending quantitive convergence results of bdf methods for a class of nonlinear strongly stiff problems and to illustrate further this new approach some numerical examples confirm our results

stability and b convergence properties of multistep runge kutta methods this paper continues earlier work by the same author concerning the stability and b convergence properties of multistep runge kutta methods for the numerical solution of nonlinear stiff initial value problems in a hilbert space a series of sufficient conditions and necessary conditions for a multistep runge kutta method to be algebraically stable diagonally stable b or optimally b convergent are established by means of which six classes of high order algebraically stable and b convergent multistep runge kutta methods are constructed in a unified pattern these methods include the class constructed by burrage in 1987 as special case and most of them can be regarded as extension of the gauss radauia radauiia and lobattoiiic runge kutta methods we find that the classes of multistep runge kutta methods constructed in the present paper are superior in many respects to the corresponding existing one step runge kutta schemes

a finite volume method for analysing the bending deformation of thick and thin plates the formulation of a novel finite volume based procedure for determining deformations and moment distributions within transversely loaded plates is described in detail the formulation employs the mindlin theory of plate deformation transverse displacement and bending moment distributions are determined for a number of test problems the effect of mesh refinement upon the accuracy of the displacement and moment fields is investigated the finite volume procedure exhibits a number of interesting features it is conservative in the sense that equilibrium is satisfied locally within each cell and across cell boundaries it has the capacity to represent the behaviour of thick and thin plates and it is geometrically versatile because the cell shapes can take the form of multifaceted polygons

b convergence of lobatto iiic formulas a completion ofb convergence results of lobatto iiic schemes is presented in particular it is shown that lobatto iiic schemes with more than two stages areb convergent when applied to ivps with a negative one sided lipschitz constantm they are notb convergent however for ivps with a non negativem

stepsize restrictions for the total variation diminishing property in general runge kutta methods much attention has been paid in the literature to total variation diminishing tvd numerical processes in the solution of nonlinear hyperbolic differential equations for special runge kutta methods conditions on the stepsize were derived that are sufficient for the tvd property see e g shu and osher j comput phys 77 1988 pp 439 471 and gottlieb and shu math comp 67 1998 pp 73 85 various basic questions are still open regarding the following issues 1 the extension of the above conditions to more general runge kutta methods 2 simple restrictions on the stepsize which are not only sufficient but at the same time necessary for the tvd property and 3 the determination of optimal runge kutta methods with the tvd property in this paper we propose a theory by means of which we are able to clarify the above questions moreover by applying our theory we settle analogous questions regarding the related strong stability preserving ssp property see e g gottlieb shu and

galerkin runge kutta discretizations of nonlinear parabolic equations global error bounds are derived for full galerkin runge kutta discretizations of nonlinear parabolic problems including the evolution governed by the p laplacian with p 2 the analysis presented here is not based on linearization procedures but on the fully nonlinear framework of logarithmic lipschitz constants and an extended b convergence theory the global error is bounded in l 2 by dx r email protected q where r is the convergence order of the galerkin method applied to the underlying stationary problem and q is the stiff order of the algebraically stable runge kutta method

on the existence of solution of stage equations in implicit runge kutta methods abstract this paper is concerned with the unique solvability of stage equations which arise when implicit runge kutta methods apply to nonlinear stiff systems of differential equations y f t y denoting by a the matrix of coefficients of the runge kutta method and by 2 j the logarithmic norm of the matrix j associated with the l2 norm several authors crouzeix et al bit 23 1983 84 91 hundsdorfer and spijker siam j numer anal 24 1987 583 594 kraaijevanger and schneid numer math 59 1991 129 157 liu and kraaijevanger bit 28 4 1988 825 838 have obtained conditions on a that ensure for a given the unique solvability of stage equations for all stepsize h and stiff system with h 2 f t y

two classes of implicit explicit multistep methods for nonlinear stiff initial value problems the initial value problems of nonlinear ordinary differential equations which contain stiff and nonstiff terms often arise from many applications in order to reduce the computation cost implicit explicit imex methods are often applied to these problems i e the stiff and non stiff terms are discretized by using implicit and explicit methods respectively in this paper we mainly consider the nonlinear stiff initial value problems satisfying the one sided lipschitz condition and a class of singularly perturbed initial value problems and present two classes of the imex multistep methods by combining implicit one leg methods with explicit linear multistep methods and explicit one leg methods respectively the order conditions and the convergence results of these methods are obtained some efficient methods are constructed some numerical examples are given to verify the validity of the obtained theoretical results

modern convergence theory for stiff initial value problems abstract in this paper we give a brief review of available theoretical results about convergence and error structures for discretizations of stiff initial value problems we point out limitations of the various approaches and discuss some recent developments

optimized strong stability preserving imex runge kutta methods we construct and analyze robust strong stability preserving implicit explicit runge kutta imex rk methods for models of flow with diffusion as they appear in astrophysics and in many other fields where equations with similar structure arise it turns out that besides the optimization of the region of absolute monotonicity some other properties of the methods are crucial for the success of such simulations in particular the models in our focus dictate to also take into account the step size limits associated with dissipativity positivity of the stiff parabolic terms which represent transport by diffusion the uniform convergence with respect to different stiffness properties of those same terms etc furthermore in the literature some other properties like the inclusion of a part of the imaginary axis in the stability region have been argued to be relevant in this paper we construct several new imex rk methods which differ from each other by taking various or even all of these constraints simultaneously into account it is demonstrated for some simple examples as well as for the problem of double diffusive convection that the newly constructed schemes provide a significant computational advantage over other methods from the literature due to their accumulation of different stability properties the optimized imex rk methods obtained in this paper are robust schemes that may also be useful for general models which involve the solution of advection diffusion equations or other transport equations with similar stability requirements

finite volume analysis of stress and deformation in hygro thermo elastic orthotropic body abstract a numerical method originally developed for the stress analysis in isotropic bodies of arbitrary shape subjected to thermo mechanical loads is extended to handle general anisotropic bodies subjected to hygro thermo mechanical loads starting from an integral form of the equations governing heat mass and momentum balance and using the constitutive equations which account for mechanical as well as thermal and hygral stresses an unconditionally stable second order in both time and space finite volume discretisation is performed resulting in a set of algebraic equations which are then solved by an efficient segregated iterative procedure in order to demonstrate the methods capabilities a transient analysis of stresses and deformation in an orthotropic body subjected to hygro thermal loads is performed and shown that results compare favourably with experimental data

contractivity preserving explicit linear multistep methods we investigate contractivity properties of explicit linear multistep methods in the numerical solution of ordinary differential equations the emphasis is on the general test equation frac d dt u t au t wherea is a square matrix of arbitrary orders 1 the contractivity is analysed with respect to arbitrary norms in thes dimensional space which are not necessarily generated by an inner product for given order and stepnumber we construct optimal multistep methods allowing the use of a maximal stepsize

a matrix free implicit unstructured multigrid finite volume method for simulating structural dynamics and fluid structure interaction a new three dimensional 3d matrix free implicit unstructured multigrid finite volume fv solver for structural dynamics is presented in this paper the solver is first validated using classical 2d and 3d cantilever problems it is shown that very accurate predictions of the fundamental natural frequencies of the problems can be obtained by the solver with fast convergence rates this method has been integrated into our existing fv compressible solver x lv y zhao et al an efficient parallel unstructured multigrid preconditioned implicit method for simulating 3d unsteady compressible flows with moving objects journal of computational physics 215 2 2006 661 690 based on the immersed membrane method imm x lv y zhao et al as mentioned above results for the interaction between the fluid and an immersed fixed free cantilever are also presented to demonstrate the potential of this integrated fluid structure interaction approach

on positivity shape and norm bound preservation of time stepping methods for semigroups abstract we use functional calculus methods to investigate qualitative properties of c 0 semigroups that are preserved by time discretization methods preservation of positivity concavity and other qualitative shape properties which can be described via positivity are treated in a banach lattice framework preservation of contractivity or norm bound of the semigroup is investigated in the banach space setting the use of the hille phillips h p functional calculus instead of the dunford taylor functional calculus allows us to extend fundamental qualitative results concerning time discretization methods and simplify their proofs including results on multi step schemes and variable step sizes since the h p functional calculus is used throughout the paper we present an elementary introduction to it based on the riemann stieltjes integral

an algebraic characterization ofb convergent runge kutta methods in the analysis of discretization methods for stiff intial value problems stability questions have received most part of the attention in the past b stability and the equivalent criterion algebraic stability are well known concepts for runge kutta methods applied to dissipative problems however for the derivation ofb convergence results error bounds which are not affected by stiffness it is not sufficient in many cases to requireb stability alone in this paper necessary and sufficient conditions forb convergence are determined

contractivity of waveform relaxation runge kutta iterations and related limit methods for dissipative systems in the maximum norm contractivity properties of runge kutta methods are analyzed with suitable interpolation implemented using waveform relaxation strategy for systems of ordinary differential equations that are dissipative in the maximum norm in general this type of implementation which is quite appropriate in a parallel computing environment improves the stability properties of runge kutta methods as a result of this analysis a new class of methods is determined which is different from runge kutta methods but closely related to them and which combines its high order of accuracy and unconditional contractivity in the maximum norm this is not possible for classical runge kutta methods

an extension and analysis of the shu osher representation of runge kutta methods in the context of solving nonlinear partial differential equations shu and osher introduced representations of explicit runge kutta methods which lead to stepsize conditions under which the numerical process is total variation diminishing tvd much attention has been paid to these representations in the literature in general a shu osher representation of a given runge kutta method is not unique therefore of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them several basic questions are still open notably regarding the following issues 1 the formulation of a simple and general strategy for finding a best possible shu osher representation for any given runge kutta method 2 the question of whether the tvd property of a given runge kutta method can still be guaranteed when the stepsize condition corresponding to a best possible shu osher representation of the method is violated 3 the generalization of the shu osher approach to general possibly implicit runge kutta methods in this paper we give an extension and analysis of the original shu osher representation by means of which the above questions can be settled moreover we clarify analogous questions regarding properties which are referred to in the literature by the terms monotonicity and strong stability preserving ssp

some simple characterizations of contractivity regions abstract we present necessary and sufficient conditions for a multiderivative multistep method to be contractive in a sector and in disk of the complex plane

implicit explicit one leg methods for nonlinear stiff neutral equations abstract in this paper by adapting the underlying implicit explicit imex one leg methods cf 1 2 a class of extended imex one leg eieol methods are suggested for solving nonlinear stiff neutral equations snes it is proven under some suitable conditions that eieol methods are d convergent of order 2 and stable for nonlinear snes several numerical examples are given to testify the obtained theoretical results and the computational effectiveness of eieol methods moreover a comparison with the fully implicit one leg methods is presented which shows that eieol methods have the higher computational efficiency

cell centered finite volume method for elastic deformation of heterogeneous media with full tensor properties abstract we propose a cell centered finite volume method for the heterogeneous anisotropic linear elasticity problem the internal traction vector is represented as a discrete flux on the interface this elastic flux is decomposed into a discrete two point flux approximation and a semi discrete transversal part we introduce an interpolation method in the presence of discontinuity in the full tensor material properties the scheme yields an accurate reconstruction of the gradient of the displacement in each cell the formulation is based on the assumption of linearity of the displacement and we enforce continuity of the internal traction vector and the displacement at the interface the treatment of the boundary conditions can be complicated in finite volume methods here we describe a general treatment of boundary conditions that does not entail the introduction of additional degrees of freedom the finite volume method is tested for a series of challenging elasticity problems

partially implicit bdf2 blends for convection dominated flows in this paper we consider various blends of implicit and explicit time integration schemes based on the well known bdf2 method applied to convection diffusion problems with dominating convection a fully implicit treatment of convection terms is often not very efficient we shall deal with second order schemes that are implicit in the convection terms only locally in space without introducing the internal inconsistencies that are common with many time splitting methods along with implementation aspects of the implicit relations we shall discuss accuracy of the schemes positivity and monotonicity properties

a stabilized dg cut cell method for discretizing the linear transport equation we present new stabilization terms for solving the linear transport equation on a cut cell mesh using the discontinuous galerkin dg method in two dimensions with piecewise linear polynomials the

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

d convergence of general linear methods for stiff delay differential equations abstract this paper is concerned with the numerical solution of delay differential equations ddes we focus on the error behaviour of general linear methods for stiff ddes a restricted type of interpolation procedure is considered for general linear methods d convergence properties of general linear methods with this interpolation procedure are investigated some extant results are unified

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

a low order finite element method for three dimensional linear elasticity problems with general meshes abstract the paper is concerned with a low order finite element method namely the staggered cell centered finite element method which has been proposed and analyzed in ong et al 2015 for two dimensional compressible and nearly incompressible linear elasticity problems in this work we extend the results to the three dimensional case and focus on the creating of the meshes in particular from a general primal mesh m we construct a polygonal dual mesh m and its submesh m in a way such that each dual control volume of m corresponds to a primal vertex and is a union macro element of some fixed number of adjacent tetrahedral elements of m the displacement is approximated by piecewise trilinear functions on the subdual mesh m and the pressure by piecewise constant functions on the dual mesh m as for two dimensional case such construction of the meshes and approximation spaces satisfies the macroelement condition which implies stability and convergence of the scheme numerical experiments are carried out to investigate the performance of the proposed method on various mesh types

jordan structures in harmonic functions and fourier algebras on homogeneous spaces we study the jordan structures and geometry of bounded matrix valued harmonic functions on a homogeneous space and their analogue the harmonic functionals in the setting of fourier algebras of homogeneous spaces

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

global optimization of explicit strong stability preserving runge kutta methods strong stability preserving runge kutta ssprk methods are a type of time discretization method that are widely used especially for the time evolution of hyperbolic partial differential equations pdes under a suitable stepsize restriction these methods share a desirable nonlinear stability property with the underlying pde e g positivity or stability with respect to total variation this is of particular interest when the solution exhibits shock like or other nonsmooth behaviour a variety of optimality results have been proven for simple ssprk methods however the scope of these results has been limited to low order methods due to the detailed nature of the proofs in this article global optimization software baron is applied to an appropriate mathematical formulation to obtain optimality results for general explicit ssprk methods up to fifth order and explicit low storage ssprk methods up to fourth order throughout our studies allow for the possibility of negative coefficients which correspond to downwind biased spatial discretizations guarantees of optimality are obtained for a variety of third and fourth order schemes where optimality is impractical to guarantee specifically for fifth order methods and certain low storage methods extensive numerical optimizations are carried out to derive numerically optimal schemes as a part of these studies several new schemes arise which have theoretically improved time stepping restrictions over schemes appearing in the recent literature

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

dual banach algebras associated to the coset space of a locally compact group abstract for h a closed group of a locally compact group g b forrest 4 has defined the fourier and fourier stieltjes algebras associated to the coset space g h a g h and b g h respectively he proved that when h is compact it is possible to extend many classical results to this new setting we continue this investigation with the study of the dual space of a g h showing that it can be identified with a particular type of w closed ideal of v n g determined by h we obtain a characterization of all w closed left ideals of v n g that are of this particular form we introduce and study the analogs of some of the classical subspaces of v n g u b c g w g and a p g in the new setting we obtain results similar to those of d e ramirez 2 e granirer 6 and a t lau 9 obtained for these spaces in the group setting

stepsize conditions for boundedness in numerical initial value problems for runge kutta methods rkms linear multistep methods lmms and classes of general linear methods glms much attention has been paid in the literature to special nonlinear stability requirements indicated by the terms total variation diminishing strong stability preserving and monotonicity stepsize conditions guaranteeing these properties were derived by shu osher j comput phys 77 1988 pp 439 471 and in numerous subsequent papers these special stability requirements imply essential boundedness properties for the numerical methods among which the property of being total variation bounded unfortunately for many well known methods the above special requirements are violated so that one cannot conclude in this way that the methods are total variation bounded in this paper we focus on stepsize conditions for boundedness directly rather than via the detour of the above special stability properties we present a generic framework for deriving best possible stepsize conditions which guarantee boundedness of actual rkms lmms and glms thereby generalizing results on the special stability properties mentioned above

a method for constructing generalized runge kutta methods abstract in the implementation of an implicit runge kutta formula we need to solve systems of nonlinear equations in this paper we analyze the newton iteration process and a modified newton iteration process for solving these equations then we propose the methods in which we take only a fixed finite number of iterations and adopt the last iterate as the approximate solution for the nonlinear equations our methods are a kind of generalized runge kutta methods with the same order as the original runge kutta formula and inherit its linear stability properties of the original implicit runge kutta formula for example an stability l stability and s stability based on this fact we construct three methods of order five imbedding a fourth order formula for error estimation finally test results for 25 stiff problems are discussed

unconditional contractivity in the maximum norm of diagonally split runge kutta methods we consider the new class of diagonally split runge kutta dsrk methods 1 5 for the standard initial value problem 1 1 by using the theory of p series we derive the order conditions on the method s coefficients stability analysis is also carried out by showing that the method is unconditionally contractive in the maximum norm if and only if the method is based on particular pairs of discrete and continuous runge kutta methods for an effective implementation of the method an iteration scheme is proposed which preserves the contractivity of any iterate

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

contractivity in the numerical solution of initial value problems consider a linear autonomous system of ordinary differential equations with the property that the norm u t of each solutionu t satisfies u t u 0 t 0 we call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations a givenk step method is said to be unconditionally contractive if for each stepsizeh 0 the numerical process is contractive in this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity it turns out that unconditionally contractive methods are subject to an order barrierp 1 further the concept of a contractivity threshold is studied which makes it possible to compare the contractivity behaviour of methods with an orderp 1 as well most theoretical results in this paper are formulated for differential equations in arbitrary banach spaces applications are given to numerical methods for solving ordinary as well as partial differential equations

asymptotic error expansions for stiff equations an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case the structure of the global discretization error is studied for the implicit midpoint and trapezoidal rules applied to nonlinearstiff initial value problems the point is that in general the global error contains nonsmooth oscillating terms at the dominanth 2 level however it is shown in the present paper that for special classes of stiff problems these nonsmooth terms contain an additional factor where 1 is the magnitude of the stiff eigenvalues in these cases a full asymptotic error expansion exists in thestrongly stiff case sufficiently small compared to the stepsizeh the general case where the oscillating error components areo h 2 and noto h 2 and applications of our results extrapolation and defect correction algorithims will be studied in separate papers

the high fidelity generalized method of cells with arbitrary cell geometry and its relationship to the parametric finite volume micromechanics abstract this contribution is motivated by the recent ijss article titled formulation of the high fidelity generalized method of cells with arbitrary cell geometry for refined micromechanics and damage in composites by haj ali and aboudi in the present discussion we critically compare these authors approach with the parametric finite volume direct averaging micromechanics fvdam theory developed in a sequence of papers dating to 2004 we show that the extension proposed by haj ali and aboudi follows the framework of the parametric fvdam theory and not the original hfgmc model as claimed just as importantly we demonstrate that the proposed extension is fundamentally flawed as it violates accepted principles which every mechanics theory must satisfy namely correct reducibility and coordinate frame indifference direct comparison of predictions generated by the parametric fvdam theory the proposed extension called hfgmc with arbitrary cell geometry and finite element method demonstrates no need for the extension carried out in the specific manner proposed by haj ali and aboudi which in fact produces inferior results

numerical method for coupled fluid flow heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology abstract a numerical method is presented that can be used for both solid body stress analysis and fluid flow predictions independently as well as in a coupled manner the method uses an integral form of equations governing mass energy and momentum balance for an arbitrary control volume a detailed description is provided of a novel second order accurate spatial discretisation technique which can accommodate unstructured moving meshes with cells of arbitrary topology this is accompanied by a fully implicit temporal discretisation which makes the method stable for any time step size the resulting set of coupled non linear algebraic equations is solved by employing a segregated approach leading to a decoupled set of linear algebraic equations for each dependent variable with a sparse diagonally dominant coefficient matrix these equations are solved by an iterative conjugate gradient solver which retains the sparsity of the coefficient matrix thus achieving a very efficient use of computer resources the method has been tested on a number of fluid flow and stress analysis problems and numerical calculations show favourable agreement with analytical and or experimental results in order to demonstrate the full capabilities of the present method an example of numerically complex coupled fluid flow stress analysis and heat transfer calculations is presented

stability with large step sizes for multistep discretizations of stiff ordinary differential equations in this paper we consider a large set of variable coefficient linear systems of ordinary differential equations which possess two different time scales a slow one and a fast one a small parameter e characterizes the stiffness of these systems we approximate a system of ode s in this set by a general class of multistep discretizations which includes both one leg and linear multistep methods we determine sufficient conditions under which each solution of a multistep method is uniformly bounded with a bound which is independent of the stiffness of the system of ode s when the step size resolves the slow time scale but not the fast one we call this property stability with large step sizes the theory presented in this paper lets us compare properties of one leg methods and linear multistep methods when they approximate variable coefficient systems of stiff ode s in particular we show that one leg methods have better stability properties with large step sizes than their linear multistep counterparts this observation is consistent with results obtained by dahlquist and lindberg 11 nevanlinna and liniger 33 and van veldhuizen 42 our theory also allows us to relate the concept of d stability van veldhuizen 42 to the usual notions of stability and stability domains and to the propagation of errors for multistep methods which use large step sizes

asymptotic error expansions for stiff equations the implicit euler scheme this paper discusses the existence of an asymptotic expansion for the global error of the implicit euler scheme applied to stiff nonlinear systems of ordinary differential equations it is shown that in strongly stiff situations a full asymptotic expansion exists at all gridpoints for the mildly stiff case it is shown that the full order of the remainder term which inevitably breaks down at the first gridpoints after a stepsize change reappears at the subsequent gridpoints our analysis is based on singular perturbation techniques

the dichotomy of stiffness pragmatism versus theory the problem of stiffness occuring in ordinary differential equations is now well understood and various concepts such as nonlinear stability b convergence and stiff order have been introduced as being relevant to an understanding of the behaviour of numerical methods when applied to stiff problems however it is often the case that some of these properties are not always attainable when methods are implemented efficiently and in this paper we will investigate this paradox in some detail in addition we will give a resume of some of the recent theoretical developments in the applicability of numerical methods to stiff problems and make some comparisons between existing codes based on runge kutta and linear multistep methods we will conclude this discussion with the introduction of a very general family of methods called multivalue methods and show how the important concepts of cheap implementation stage order and stability apply to these methods

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

the c algebra generated by operators with compact support on a locally compact group abstract let g be a locally compact group and vn g be the von neumann algebra generated by the left regular representation of g let ucb denote the c subalgebra generated by operators in vn g with compact support when g is abelian ucb corresponds to the space of bounded uniformly continuous functions on the dual group of g in this paper we prove among other things that for a large class of locally compact groups which include the heisenberg group the ax b group and the motion group the centre of the banach algebra ucb is the fourier stieltjes algebra b g

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

a finite volume method for incompressible linear elasticity abstract this paper describes development of a finite volume based method for modelling of incompressible linear elastic body problems the method is based on the solution of the integral form of conservation equations governing momentum balance and the introduction of pressure as an additional variable the incompressibility is enforced by employing the volumetric constraint equation a collocated variable arrangement is used and the spatial domain is discretised using finite volumes of arbitrary polyhedral shapes a segregated approach is employed to solve resulting set of coupled linear algebraic equations embedding a simple based algorithm for displacement pressure coupling numerical experiments showed that the developed method appears to be locking free

a unified formulation for continuum mechanics applied to fluid structure interaction in flexible tubes this paper outlines the development of a new procedure for analysing continuum mechanics problems with a particular focus on fluid structure interaction in flexible tubes a review of current methods of fluid structure coupling highlights common limitations of high computational cost and solution instability it is proposed that these limitations can be overcome by an alternative approach in which both fluid and solid components are solved within a single discretized continuum domain a single system of momentum and continuity equations is therefore derived that governs both fluids and solids and which are solved with a single mesh using finite volume discretization schemes the method is validated first by simulating dynamic oscillation of a clamped elastic beam it is then applied to study the case of interest wave propagation in highly flexible tubes in which a predicted wave speed of 8 58m s falls within 2 of an approximate analytical solution the method shows further good agreement with analytical solutions for tubes of increasing rigidity covering a range of wave speeds from those found in arteries to that in the undisturbed fluid

stability and b convergence of general linear methods abstract this paper continues earlier work by the same author concerning the stability and b convergence of general linear methods for the numerical solution of nonlinear stiff initial value problems in a hilbert space in a previous paper we have proved that bh consistency resp bh consistency together with bh stability implies optimal b convergence resp b convergence in this paper by means of bs and bsi stability properties the sufficient conditions for a method to be b bh bh or bh consistent bh or weakly bh stable b or optimally b convergent respectively are further established

on symmetrizers for gauss methods in this paper the maximum attainable order of a special class of symmetrizers for gauss methods is studied in particular it is shown that a symmetrizer of this type for thes stage gauss method can attain order 2s 1 only for 1 s 3 and that these symmetrizers arel stable a classification of the maximum attainable order of symmetrizers for some higher stages is presented anl stable symmetrizer is also shown to exist for each of the methods studied

finite volume method for thermo elasto plastic stress analysis abstract in this paper a new numerical method for prediction of stresses and displacements in thermo elasto plastic material is presented the method is based on the direct solution of the governing equations for thermal energy and momentum balance expressed in terms of temperature and displacements as dependent variables via corresponding constitutive relations and discretized by the finite volume technique as a result the unsteady fields of temperature displacements and stresses as well as residual stresses in a solid body subjected to a non uniform temperature and or external loads are obtained the method has been applied to a number of test cases and the results are compared with the analytical solution calculations obtained by another numerical method or experimental data results show that the method is accurate stable and efficient and that it can be used as a useful tool for predicting thermo mechanical deformation processes in a solid body

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

on monotonicity and boundedness properties of linear multistep methods in this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with runge kutta starting procedures this allows many multistep methods of practical interest to be included in the theory in a related manner we also consider contractivity and stability in arbitrary norms

the concept of b convergence a useful concept of convergence for nonlinear stiff ode s will be developed which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem this concept will be discussed for three simple methods for stiff systems the implicit euler scheme the implicit midpoint rule and the implicit trapezoidal rule

boundedness and strong stability of runge kutta methods in the literature much attention has been paid to runge kutta methods rkms satisfying special nonlinear stability requirements indicated by the terms total variation diminishing tvd strong stability preserving ssp and monotonicity stepsize conditions guaranteeing these properties were derived by shu osher j comput phys 77 1988 pp 439 471 and in numerous subsequent papers these special stability requirements imply essential boundedness properties for the numerical methods among which the property of being total variation bounded unfortunately for many rkms the above special requirements are violated so that one cannot conclude in this way that the methods are total variation bounded in this paper we study stepsize conditions for boundedness directly rather than via the detour of the above special stability properties we focus on stepsize conditions which are optimal in that they are not unnecessarily restrictive we nd that in situations where the special stability properties mentioned above are violated boundedness can be present only within a class of very special rkms as a byproduct our analysis sheds new light on the known theory of monotonicity for rkms we obtain separate results for internal and external monotonicity as well as a new proof of the fundamental relation between monotonicity and kraaijevanger s coecient this proof distinguishes itself from older ones in that it is shorter and more transparent while it requires more simple assumptions on the rkms under consideration

absolute monotonicity of polynomials occurring in the numerical solution of initial value problems this paper deals with polynomial approximations o x to the exponential function exp x related to numerical procedures for solving initial value problems motivated by positivity and contractivity requirements imposed on these numerical procedures we study the smallest negative argument denoted by r o at which o is absolutely monotonic for given integersp 1 m 1 we determine the maximum ofr o when o varies over the class of all polynomials of a degree m with phi left x right exp left x right mathcal o left x p 1 right left for x to 0 right forx 0

preserving stability implicit euler method for nonlinear volterra and neutral functional differential equations in banach space this paper is concerned with the contractivity and asymptotic stability properties of the implicit euler method iem for nonlinear functional differential equations fdes these properties are first analyzed for volterra fdes and then the analysis is extended to the case of neutral fdes nfdes such an extension is particularly important since nfdes are more general and have received little attention in the literature the main result we establish is that the iem with linear interpolation can completely preserve these stability properties of the analytical solution to such fdes

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

computation of optimal monotonicity preserving general linear methods monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution we formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations and propose an efficient algorithm for its solution this algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and or stages the optimality of some recently proposed methods is verified and many more efficient methods are found we use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type including methods for hyperbolic pdes that use downwind biased operators

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