the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

some remarks on operator norm convergence of the trotter product formula on banach spaces abstract we extend the trotter kato chernoff theory of strong approximation of c 0 semigroups on banach spaces to operator norm approximation of analytic semigroups with error estimate as application we obtain a criterion for the operator norm convergence of the trotter product formula on banach spaces with error estimate n 1 log n provided one of the generators has a bounded h functional calculus for both results we present versions for approximation in operator ideal norms such as the trace norm or the hilbert schmidt norm finally we give some remarks on the operator norm convergence of the trotter product formula for semigroups acting on a scale of l p spaces

a p stable eighth order method for the numerical integration of periodic initial value problems an eighth order p stable two step method for the numerical integration of second order periodic initial value problems is developed in this paper this method has seven stages per iteration and an interval of periodicity equal to 0 i e it is p stable numerical and theoretical results obtained for several well known problems show the efficiency of the new method

characteristic based rotated upwind scheme for the euler equations a rotated upwind scheme is developed that calculates shock waves oblique to the grid with nearly the same resolution as shocks that are aligned with the grid the scheme applies flux difference splitting along two orthogonal directions for each cell face with the directions computed on the basis of pressure gradient data rotated extrapolations and flux limiting are used to generate second order spatial accuracy a filtering of the pressure gradient data allows the convergence of the rotated scheme with continuously updated upwinding angles at a rate comparable to classical schemes flux evaluations based on characteristi c variables and characteristic boundary conditions are discussed and implemented the method has been tested on two simple shock reflection problems and formally second order results capture the oblique shocks with only two to three cells in the shock transition region

two step runge kutta methods and hyperbolic partial differential equations the purpose of this study is the design of efficient methods for the solution of an ordinary differential system of equations arising from the semidiscretization of a hyperbolic partial differential equation jameson recently introduced the use of one step runge kutta methods for the numerical solution of the euler equations improvements in efficiency up to 80 may be achieved by using two step runge kutta methods instead of the classical onestep methods these two step runge kutta methods were first introduced by byrne and lambert in 1966 they are designed to have the same number of function evaluations as the equivalent one step schemes and thus they are potentially more efficient by solving a nonlinear programming problem which is specified by stability requirements optimal two step schemes are designed the optimization technique is applicable for stability regions of any shape

high algebraic high phase lag order embedded numerov type methods for oscillatory problems an embedded numerov type method of algebraic order eight with phase lag orders 10 20 for the numerical integration of second order periodic initial value problems is developed in this paper the method has 10 stages an error estimation based on varying the phase lag order is introduced and an embedded technique is described each method of the embedded pair has a large interval of periodicity theoretical evidence and numerical and theoretical results obtained for several well known problems show the efficiency of the new methods

construction of highly stable two step w methods for ordinary differential equations we describe the construction of linearly implicit two step w methods of high stage order and desirable stability properties for the numerical solution of stiff differential systems the presented methods are demonstrated to be a and l stable when the stepsize is held constant they also preserve good stability properties in a variable stepsize environment under quite demanding conditions imposed on the stepsize pattern

explicit high order methods for the numerical integration of periodic initial value problems two explicit two step hybrid methods of order 7 and 8 for the numerical integration of second order periodic initial value problems are developed in this paper the first of them has a large interval of periodicity and the other a minimal phase lag each of them has seven stages per iteration numerical and theoretical results obtained for several well known problems show the efficiency of the new methods

derivation of continuous explicit two step runge kutta methods of order three we describe a construction of continuous extensions to a new representation of two step runge kutta methods for ordinary differential equations this representation makes possible the accurate and reliable estimation of local discretization error facilitates the efficient implementation of these methods in variable stepsize environment and adapts readily to the numerical solution of a class of delay differential equations a number of numerical tests carried out on the obtained methods of order 3 with quadratic interpolants show their efficiency and robust performance which allow them to compete with the state of the art dde23 code from matlab

numerical treatment of o d es the theory ofa methods almost all commonly used methods for o d es and their most miscellaneous compositions area methods i e they can be reduced toz o z j az j 1 h x j 1 z j 1 z j h z j s a s s j 1 1 m this paper presents a general theory fora methods and discusses its practical consequences an analysis of local discretization error l d e accumulation results in a general order criterium and reveals which part of the l d e effectively influences the global error this facilitates the comparison of methods and generalizes considerably the concept of error constants it is shown as a consequence that the global error cannot be safely controlled by the size of the l d e and that the conventional error control may fail in important cases furthermore butcher s effective order methods the concept of nordsieck forms and gear s interpretation of lineark step schemes as relaxation methods are generalized the stability of step changing is shortly discussed

implementation of two step runge kutta methods for ordinary differential equations we investigate the potential for efficient implementation of two step runge kutta methods tsrk a new class of methods introduced recently by jackiewicz and tracogna for numerical integration of ordinary differential equations the implementation issues addressed are the local error estimation changing stepsize using nordsieck technique and construction of interpolants the numerical experiments indicate that the constructed error estimates are very reliable in a fixed and variable stepsize environment

stepsize conditions for general monotonicity in numerical initial value problems for runge kutta methods and linear multistep methods much attention has been paid in the literature to special nonlinear stability properties indicated by the terms total variation diminishing tvd strong stability preserving ssp and monotonicity stepsize conditions guaranteeing these properties were studied e g by shu and osher j comput phys 77 1988 pp 439 471 gottlieb shu and tadmor siam rev 43 2001 pp 89 112 hundsdorfer and ruuth monotonicity for time discretizations dundee conference report na 217 2003 university of dundee dundee uk 2003 pp 85 94 higueras j sci comput 21 2004 pp 193 223 and siam j numer anal 43 2005 pp 924 948 spiteri and ruuth siam j numer anal 40 2002 pp 469 491 gottlieb j sci comput 25 2005 pp 105 128 and ferracina and spijker siam j numer anal 42 2004 pp 1073 1093 and math comp 74 2005 pp 201 219 in the present paper we obtain a special stepsize condition guaranteeing the above properties for a generic numerical process this condition is best possible in a well defined and natural sense it is applicable to the important class of general linear methods and it can also be used to answer some open questions for methods of which the above stability properties were studied earlier

order conditions for integrators and mappings with applications abstract we regard the elementary weight function as a mapping from input to output so that the mappings allow us to derive the order conditions for a given method more easily and concisely this paper presents the order conditions of one step runge kutta methods runge kutta type methods with derivatives a class of two step runge kutta methods and three step runge kutta methods based on the elementary weight mappings the composition formula of two mappings as well as the mappings i d r e defined on the set t of all rooted trees from an input to an output it is pointed out that the order conditions can be easily achieved for linear multistep methods and second derivative linear multistep methods of order p and it suffices to consider only bushy trees of order up to p together with the empty tree 0 for a linear multistep method or a second derivative linear multistep method of order p

two step runge kutta methods implicit two step runge kutta methods are studied it will be shown that these methods require fewer stages to achieve the same order as one step runge kutta methods which means the two step methods are potentially more efficient than one step methods order conditions are derived and examples of two step one stage methods of order 2 and two step two stage methods of order 4 are presented stability properties of these methods with respect to y ay are studied and a stable two step methods of order 2 are characterized two step two stage methods of order 4 which are a stable are found by an extensive computer search semi implicit two stage methods of order 4 were also constructed this is in contrast to the situation encountered in the runge kutta theory where the unique two stage method of order 4 is not semi implicit

a new theoretical approach to runge kutta methods runge kutta rk methods are treated here as linear multistage methods within the author s concept of a methods as a consequence the traditional difference in treatment of rk methods and linear multistep methods vanishes in particular it is not necessary to consider elementary differentials it is shown how the order of a rk method depends on the error constants of its stages and the conditions nonlinear equations for order p are generated from a very simple recursion

stability analysis of two step runge kutta methods for delay differential equations abstract we investigate stability properties of two step runge kutta methods with respect to the linear test equation y t ay t by t t 0 y t g t t 0 where a and b are complex parameters it is known that the solution y t to this equation tends to zero as t if b a we will show that under some conditions this property is inherited by any a stable two step runge kutta method applied on a constrained mesh to delay differential equations i e that the corresponding method is p stable

approximation of the semigroup generated by the robin laplacian in terms of the gaussian semigroup we consider the laplacian r subject to robin boundary conditions u u 0 on the space c where is a smooth bounded open subset of rn it is known that r generates an analytic contraction semigroup we show how this semigroup can be obtained from the gaussian semigroup on c0 rn via a trotter formula as the main ingredient we construct a positive contractive linear extension operator e from c to c0 rn which maps an operator core for r into the domain of the generator of the gaussian semigroup

order conditions for general two step runge kutta methods the use of b series for the derivation of order conditions for general two step runge kutta methods is illustrated

strong stability preserving transformed dimsims abstract in this paper we investigate the strong stability preserving ssp property of transformed diagonally implicit multistage integration methods dimsims within this class examples of ssp methods of order p 1 2 3 and 4 and stage order q p have been determined and also suitable starting and finishing procedure have been constructed the numerical experiments performed on a set of test problems have shown that ssp transformed dimsims can be more accurate and competitive with ssp runge kutta methods of the same order

numerical search for algebraically stable two step almost collocation methods we investigate algebraic stability of the new class of two step almost collocation methods for ordinary differential equations these continuous methods are obtained by relaxing some of the interpolation and collocation conditions to achieve strong stability properties together with uniform order of convergence on the whole interval of integration we describe the search for algebraically stable methods using the criterion based on the nyquist stability function proposed recently by hill this criterion leads to a minimization problem in one variable which is solved using the subroutine fminsearch from matlab examples of algebraically stable methods in this class are also presented

on hille type approximation of degenerate semigroups of operators the result that goes essentially back to euler 15 says that for any element a of a unital banach algebra a with unit u the limit lime 0 u ea e 1t where denotes the integral part exists for all t r and equals eta as developed by e hille 22 thm 12 2 1 in the case where a is replaced by the generator a of a strongly continuous semigroup eta t 0 in a banach space x a proper counterpart of this formula is eta lime 0 ix ea e 1t strongly in x motivated by an example from mathematical biology related to rotenberg s model of cell growth 40 we study convergence of a similar approximation in which u resp ix is replaced by j a resp j l x such that for some l 2 jl u resp jl ix

construction of highly stable parallel two step runge kutta methods for delay differential equations it is shown that any a stable two step runge kutta method of order p and stage order q p for ordinary differential equations can be extended to the p stable method of uniform order p p for delay differential equations

construction of two step runge kutta methods with large regions of absolute stability we describe the construction of explicit two step runge kutta methods of order p and stage order q p 1 or q p with large regions of absolute stability this process is illustrated for the methods of order p 2 and 3 and leads to new methods which are competitive with explicit runge kutta methods of the same order

rotating dissipation for accurate shock capture we introduce some concepts related to the upwind multidimensional schemes into a standard central method we present a new approach to rotated differencing that is simple and efficient in terms of computational efforts the basic idea is first to compute the standard grid aligned flux advection artificial dissipation and finally to use a correction that is rotated with respect to the grid and aligned with the direction normal to the shock wave in this way we can use any grid aligned scheme and obtain a better description of oblique discontinuities

implicit flux split schemes for the euler equations progress in the development of implicit algorithms for the euler equations using the flux vector splitting method is described comparisons of the relative efficiency of relaxation and spatially split approximately factored methods on a vector processor for transonic and supersonic twodimensional channel flows are made a hybrid threedimensional algorithm is developed that uses relaxation in one coordinate direction and approximate factorization in the crossflow plane the scheme is completely vectorizable and recovers conventional space marching schemes for fully supersonic flows the method is applied to a forebody shape in supersonic flow with an embedded pocket of subsonic flow

explicit two step peer methods we present a new class of explicit two step peer methods for the solution of nonstiff differential systems a construction principle for methods of order p s s the number of stages with optimal zero stability is given two methods of order p 6 found by numerical search are tested in matlab on several representative nonstiff problems the comparison with ode45 confirms the high potential of the new class of methods

numerical solutions of fuzzy differential equations by an efficient runge kutta method with generalized differentiability in this paper an extended fourth order runge kutta method is studied to approximate the solutions of first order fuzzy differential equations using a generalized characterization theorem in this method new parameters are utilized in order to enhance the order of accuracy of the solutions using evaluations of both f and f instead of using the evaluations of f only the proposed extended runge kutta method and its error analysis which guarantees pointwise convergence are given in detail furthermore the accuracy and efficiency of the proposed method are demonstrated in a series of numerical experiments

elements of a general theory of composite integration methods this contribution is a survey of results on general linear methods after an introduction into the concept of a methods which is a slight generalization of general linear methods and its main results rk methods are presented in the a method notation which offers the following advantages a the order conditions are obtained from a simple recursion without relying on rooted trees or elementary differentials b they appear in a transparent form as orthogonality relations in dependence of the error constants of the stages of the rk method c their orthogonal structure permits elegant existence proofs and yields new auxiliary conditions d the error accumulation within the stages of the method is completely understood the proof of the main result on rk methods theorem 5 is simplified

starting methods for two step runge kutta methods of stage order 3 and order 6 jackiewicz and tracogna siam j numer anal 32 1995 1390 1427 proposed a general formulation of two step runge kutta tsrk methods using formulas for two step pairs of tsrk methods constructed in japan jiam 19 2002 227 248 jackiewicz and verner obtain results for order 8 pairs that fail to show this designated order hairer and wanner siam j numer anal 34 1997 2087 2089 identify the problem by using b series to formulate a complete set of order conditions for tsrk methods and emphasize that special starting methods are necessary for the first step of implementation they observe that for methods with stage order at least p 1 and design order p starting methods of order at least p are sufficient in this paper the more general challenge to provide correct starting values for methods of low stage order is met by showing how perturbed starting values should be selected for methods of order 6 and stage order 3 the approach is sufficiently general that it may and later will be provided for such methods of higher orders evidence of the accompanying improvement in the implementation of tsrk methods illustrates that carefully designed starting methods are essential for efficient production codes based on methods of low stage order

directional diffusion regulator ddr for some numerical solvers of hyperbolic conservation laws a computational tool called directional diffusion regulator ddr is proposed to bring forth real multidimensional physics into the upwind discretization in some numerical schemes of hyperbolic conservation laws the direction based regulator when used with dimension splitting solvers is set to moderate the excess multidimensional diffusion and hence cause genuine multidimensional upwinding like effect the basic idea of this regulator driven method is to retain a full upwind scheme across local discontinuities with the upwind bias decreasing smoothly to a minimum in the farthest direction the discontinuous solutions are quantified as gradients and the regulator parameter across a typical finite volume interface or a finite difference interpolation point is formulated based on fractional local maximum gradient in any of the weak solution flow variables say density pressure temperature mach number or even wave velocity etc ddr is applied to both the non convective as well as whole unsplit dissipative flux terms of some numerical schemes mainly of local lax friedrichs to solve some benchmark problems describing inviscid compressible flow shallow water dynamics and magneto hydrodynamics the first order solutions consistently improved depending on the extent of grid non alignment to discontinuities with the major influence due to regulation of non convective diffusion the application is also experimented on schemes such as roe jameson schmidt turkel and some second order accurate methods the consistent improvement in accuracy either at moderate or marked levels for a variety of problems and with increasing grid size reasonably indicate a scope for ddr as a regular tool to impart genuine multidimensional upwinding effect in a simpler framework

order conditions for general linear methods we describe the derivation of order conditions without restrictions on stage order for general linear methods for ordinary differential equations this derivation is based on the extension of the albrecht approach proposed in the context of runge kutta and composite and linear cyclic methods this approach was generalized by jackiewicz and tracogna to two step runge kutta methods by jackiewicz and vermiglio to general linear methods with external stages of different orders and by garrappa to some classes of runge kutta methods for volterra integral equations with weakly singular kernels this leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods nordsieck methods and general linear methods with inherent runge kutta stability exact coefficients for several low order methods with some desirable stability properties are presented for illustration

exponentially fitted two step runge kutta methods construction and parameter selection abstract we derive exponentially fitted two step runge kutta methods for the numerical solution of y f x y specially tuned to the behaviour of the solution such methods have nonconstant coefficients which depend on a parameter to be suitably estimated the construction of the methods is shown and a strategy of parameter selection is presented some numerical experiments are provided to confirm the theoretical expectations

a class of implicit explicit two step runge kutta methods this work develops new implicit explicit time integrators based on two step runge kutta methods the class of schemes of interest is characterized by linear invariant preservation and high stage orders theoretical consistency stability and stiff convergence analyses are performed to reveal the excellent properties of these methods the new framework offers extreme flexibility in the construction of partitioned integrators since no coupling conditions are necessary practical schemes of orders three four and six are constructed and are used to solve several test problems numerical results confirm the theoretical findings

a general class of two step runge kutta methods for ordinary differential equations a general class of two step runge kutta methods that depend on stage values at two consecutive steps is studied these methods are special cases of general linear methods introduced by butcher and are quite efficient with respect to the number of function evaluations required for a given order general order conditions are derived using the approach proposed recently by albrecht and examples of methods are given up to the order 5 these methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments

efficiency comparisons of methods for integrating odes abstract considerable work has been done comparing the relative efficiencies of linear multi step runge kutta and extrapolation methods within their respective classes however efficiency comparisons between these classes have been relegated to numerical case studies which can be misleading because they depend on a small set of test problems and particular implementations of the methods in this paper a unified way of comparing efficiency is presented that makes it possible to assess theoretically the relative efficiencies of methods from the different classes

search for efficient general linear methods for ordinary differential equations we analyze implicit general linear methods with s internal stages and r s 1 external stages of order p s 1 and stage order q s or q s 1 these methods might eventually lead to more efficient formulas than the class of dimsims and the class of general linear methods with inherent runge kutta stability we analyze also error propagation and estimation of local discretization errors examples of such methods which are a and l stable are derived up to the stage order q 3 or q 4 and order p 4

a new recurrence for computing runge kutta truncation error coefficients researchers investigating runge kutta methods use truncation error coefficients to derive formulas and assess their quality in this paper albrecht s recurrence for generating order conditions is refined to produce truncation error coefficients the refined recurrence provides a fast and relatively simple way of calculating truncation error coefficients of the albrecht expansion for general runge kutta methods

on the relation of closed forms and trotter s product formula abstract the aim of this paper is to give a characterization in hilbert spaces of the generators of c 0 semigroups associated with closed sectorial forms in terms of the convergence of a generalized trotter s product formula in the course of the proof of the main result we also present a similarity result which can be of independent interest for any unbounded generator a of a c 0 semigroup e ta it is possible to introduce an equivalent scalar product on the space such that e ta becomes non quasi contractive with respect to the new scalar product

two step almost collocation methods for volterra integral equations in this paper we construct a new class of continuous methods for volterra integral equations these methods are obtained by using a collocation technique and by relaxing some of the collocation conditions in order to obtain good stability properties