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

zero stability properties of the three ordinate variable stepsize variable formula methods consider the systemy f x y y a x a b y r n wheref is continuous and lipschitzian with respect to the second argument very often linear multistep variable stepsize variable formula methods lm vsvfm s are used to computey k y xk on the points of the grid a x 0 x1 x2 xn b the general lm vsvfm is based on formulae of the following type y k sum limits i 1 s k alpha i bar h k sk y k i sum limits i 0 s k h k i beta i bar h k s k f x k i y k i whereh k xk xk 1 bar h k h k h k 1 ldots h k s k s k k k 1 1 n the coefficients i and i depend on the lasts k 1 stepsizes and on the formula used at stepk only the zero stability properties of some special classes of lm vsvfm s as for example those based on adams formulae were investigated in the literature a class of three ordinate lm vsvfm s is defined in this paper some results concerning the zero stability properties of these methods are proved it is shown that some well known results are simple corollaries of the results found for the three ordinate lm vsvfm s it is easily seen that similar results hold for the corresponding one leg vsvfm s finally the use of the theoretical results in the practical implementations is briefly discussed

staggered time integrators for wave equations we consider variations of the adams bashforth backward differentiation and runge kutta families of time integrators to solve systems of linear wave equations on uniform time staggered grids these methods are found to have smaller local truncation errors and to allow larger stable time steps than traditional nonstaggered versions of equivalent orders we investigate the accuracy and stability of these methods analytically experimentally and through the use of a novel root portrait technique

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

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

spatial finite difference approximations for wave type equations the simplest finite difference approximations for spatial derivatives are centered explicit and applied to regular equispaced grids well established generalizations include the use of implicit compact approximations and staggered grids we find here that the combination of these two concepts together with high formal order of accuracy is very effective for approximating the first derivatives in space that occur in many wave type pdes

a numerical model for calculating the distributions of velocity and boundary shear stress across irregular straight open channels a model is presented for calculating the distributions of velocity and boundary shear stress across irregular straight open channels a turbulent kinetic energy transport equation and a two dimensional ramp function for the turbulent length scale provide turbulence closure a modified sigma transformation maps the channel cross section onto a rectangular domain the transformed equations are solved by a simple finite difference algorithm velocity profiles generated by the model are logarithmic close to the bed but in the core of the flow predicted velocities follow a wake profile similar to that observed experimentally the boundary shear stresses calculated by the model for smooth one dimensional flows are accurate to within 0 02 the model also reproduces flow and boundary shear stress data for smooth rectangular and trapezoidal laboratory channels the average boundary shear stress is calculated to within about 1 and the distributions of velocity and shear stress are similar to those obtained from laboratory measurements local errors are caused by ignoring weak secondary currents and by using a simplistic symmetry assumption for the boundary condition at the water s surface the model overestimates the discharge by an average of 8 this error probably occurs because the model ignores the resistance due to secondary currents

numerical integrators for the hybrid monte carlo method we construct numerical integrators for hamiltonian problems that may advantageously replace the standard verlet time stepper within hybrid monte carlo and related simulations past attempts have often aimed at boosting the order of accuracy of the integrator and or reducing the size of its error constants order and error constant are relevant concepts in the limit of vanishing step length we propose an alternative methodology based on the performance of the integrator when sampling from gaussian distributions with not necessarily small step lengths we construct new splitting formulae that require two three or four force evaluations per time step limited proof of concept numerical experiments suggest that the new integrators may provide an improvement on the efficiency of the standard verlet method especially in problems with high dimensionality

high order difference schemes with reduced dispersion for hyperbolic differential equations abstract we investigate difference schemes for systems of first order hyperbolic differential equations in two space dimensions possessing the following characteristics 1 the spatial discretizations are fourth order accurate 2 the time discretization is of explicit runge kutta type and is also fourth order accurate 3 the scaled stability boundary is approximately 1 2 2 4 the weights in the space discretizations and the runge kutta parameters can be adapted in order to reduce the dispersion of dominant fourier components this method is illustrated by applying it to the shallow water equations simulating the motion of water in a shallow sea due to tidal forces since in such problems the dominant frequencies in the solution are known in advance the method can take fully advantage of the possibility to tune the various parameters to these dominant frequencies

explicit nordsieck methods with extended stability regions abstract we describe the construction of explicit nordsieck methods of order p and stage order q p with large regions of absolute stability we also discuss error propagation and estimation of local discretization errors the error estimators are derived for examples of general linear methods constructed in this paper some numerical experiments are presented which illustrate the effectiveness of proposed methods

triple product p adic l functions for balanced weights we construct p adic triple product l functions that interpolate square roots of central critical l values in the balanced region thus our construction complements that of harris and tilouine there are four central critical regions for the triple product l functions and two opposite settings according to the sign of the functional equation in the first case three of these regions are of interpolation having positive sign they are called the unbalanced regions and one gets three p adic l functions one for each region of interpolation this is the harris tilouine setting in the other setting there is only one region of interpolation called the balanced region we produce the corresponding p adic l function our triple product p adic l function arises as p adic period integrals interpolating normalizations of the local archimedean period integrals the latter encode information about classical representation theoretic branching laws the main step in our construction of p adic period integrals is showing that these branching laws vary in a p adic analytic fashion this relies crucially on the ash stevens theory of highest weight representations over affinoid algebras

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

imex extensions of linear multistep methods with general monotonicity and boundedness properties for solving hyperbolic systems with stiff sources or relaxation terms time stepping methods should combine favorable monotonicity properties for shocks and steep solution gradients with good stability properties for stiff terms in this paper we consider implicit explicit imex multistep methods suitable methods will be constructed based on explicit methods with general monotonicity and boundedness properties for hyperbolic equations numerical comparisons are made with several implicit explicit runge kutta methods

from semidiscrete to fully discrete stability of runge kutta schemes by the energy method the integration of semidiscrete approximations for time dependent problems is encountered in a variety of applications the runge kutta rk methods are widely used to integrate the ode systems which arise in this context resulting in large ode systems called methods of lines these methods of lines are governed by possibly ill conditioned systems with a growing dimension consequently the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading instead we present here a stability analysis of rk methods for well posed semidiscrete approximations based on a general energy method we review the stability question for such rk approximations and highlight its intricate dependence on the growing dimension of the problem in particular we prove the strong stability of general fully discrete rk methods governed by coercive approximations we conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection diffusion equations with variable coefficients a straightforward implementation of our results verify the strong stability of rk methods for local finite difference schemes as well as global spectral approximations since our approach is based on the energy method which is carried in the physical space and since it avoids the von neumann analysis which is carried in the dual fourier space we are able to easily adapt additional extensions due to nonperiodic boundary conditions general geometries etc

a numerical model for flow in meandering channels with natural bed topography a finite volume numerical model is presented for the calculation of three dimensional turbulent flow in meandering channels with natural bed configuration the governing equations are written in a general curvilinear coordinate system so that all singly connected geometrical configurations can be accommodated for simplicity the cartesian velocity components are retained as dependent variables the numerical procedure is very general so that it can deal with complex cross sections and separated flow turbulence closure is through the standard k model computed results show good agreement with experimental data in meandering channels with rectangular and natural bed configurations this work provides a hydrodynamic basis for the study of the mechanisms for the formation of river meanders

dual control monte carlo method for tight bounds of value function in regime switching utility maximization abstract in this paper we study the dual control approach for the optimal asset allocation problem in a continuous time regime switching market we find the lower and upper bounds of the value function that is a solution to a system of fully coupled nonlinear partial differential equations these bounds can be tightened with additional controls to the dual process we suggest a monte carlo algorithm for computing the tight lower and upper bounds and show the method is effective with a variety of utility functions including power non hara and yaari utilities the latter two utilities are beyond the scope of any current methods available in finding the value function

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

high order finite differences and the pseudospectral method on staggered grids finite difference approximations for the first derivative valid halfway between equidistant gridpoints are in general much more accurate than the corresponding approximations which are valid at gridpoints the pseudospectral fourier method can be viewed as the limit of finite difference approximations when the order of accuracy tends to infinity a fundamentally different and more accurate pseudospectral method is obtained from the halfway approximations this study derives these and related methods and discusses their accuracies

stability and accuracy of time discretizations for initial value problems this paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems here the treatment is extended to cover also implicit schemes and by placing the accuracy of the schemes into a more central position in the discussion general method free statements are again obtained more specialized results are additionally given for linear multistep methods and for the taylor series method

finite volume methods for two dimensional incompressible flows with complex boundaries abstract the paper discusses possible ways of developing accurate and efficient finite volume methods for complex incompressible flows the issues of boundary fitted grid generation choice of coupled or uncoupled procedures cartesian or grid aligned velocity components staggered or non staggered variable arrangement discretization methods solution algorithms and convergence acceleration by multigrid methods are addressed an example of a finite volume method for general two dimensional flows is described in some detail and applications of this method to a variety of flows are presented

mathematical model for studying the sulphur pollution over europe the long range transport of sulphur pollutants in the atmosphere is an important environmental problem a mathematical model consisting of a system of partial differential equations has been developed at the danish agency of environmental protection for studying this phenomenon the model is handled numerically a pseudospectral fourier algorithm is used in the space discretization phase a variable stepsize variable formula method vsvfm based on predictor corrector schemes may be specified during the time discretization part the time integration formulae are especially constructed so that a the predictor corrector schemes have good absolute stability properties on the imaginary axis and b the fundamental properties consistency zero stability and convergence of the global vsvfm are not affected by the changes of the stepsize and or the predictor corrector scheme both the accuracy requirements and the stability requirements are taken into account in the stepsize selection strategy as well as in the formula selection strategy this double check leads to a great efficiency in the numerical treatment of problems arising in atmospheric environments this is demonstrated by many numerical results the question of the reliability of the results obtained in runs of real physical problems is discussed some results obtained in the comparison of calculated results with measurements are reported an illustration where the model is applied in studying the air pollution in the area around denmark due to danish sources is presented

high order linear multistep methods with general monotonicity and boundedness properties we consider linear multistep methods that possess general monotonicity and boundedness properties strict monotonicity in terms of arbitrary starting values for the multistep schemes is only valid for a small class of methods under very stringent step size restrictions this makes them uncompetitive with the strong stability preserving ssp runge kutta methods by relaxing these strict monotonicity requirements a larger class of methods can be considered including many methods of practical interest in this paper we construct linear multistep methods of high order up to six that possess relaxed monotonicity or boundedness properties with optimal step size conditions numerical experiments show that the new schemes perform much better than the classical monotonicity preserving multistep schemes moreover there is a substantial gain in efficiency compared to recently constructed ssp runge kutta ssprk methods

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

essentially optimal explicit runge kutta methods with application to hyperbolic parabolic equations optimal explicit runge kutta methods consider more stages in order to include a particular spectrum in their stability domain and thus reduce time step restrictions this idea so far used mostly for real line spectra is generalized to more general spectra in the form of a thin region in thin regions the eigenvalues may extend away from the real axis into the imaginary plane we give a direct characterization of optimal stability polynomials containing a maximal thin region and calculate these polynomials for various cases semi discretizations of hyperbolic parabolic equations are a relevant application which exhibit a thin region spectrum as a model linear scalar advection diffusion is investigated the second order stabilized explicit runge kutta methods derived from the stability polynomials are applied to advection diffusion and compressible viscous fluid dynamics in numerical experiments due to the stabilization the time step can be controlled solely from the hyperbolic cfl condition even in the presence of viscous fluxes

regions of stability equivalence theorems and courant friedichs lewy condition the celebrated cfl condition for discretizations of hyperbolic pdes is shown to be equivalent to some results of jeltsch and nevanlinna concerning regions of stability ofk step m stage linear methods for the integration of odes we characterize the methods for the numerical integration of the model equation ut ux which are weakly stable when the mesh ratio takes the maximum value allowed by the cfl condition we provide new equivalence theorems between stability and convergence which improve on the classical results

increasing the real stability boundary of explicit methods abstract based on the simplest well known integration rules such as the forward euler scheme and the classical runge kutta method an extension is proposed to enlarge the real stability boundary the main characteristic of the resulting schemes is that the computational complexity is hardly increased

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

a step size selection strategy for explicit runge kutta methods based on lyapunov exponent theory using a stability result for variable time step runge kutta methods applied to nonautonomous real linear scalar test problem that decays exponentially fast a step size selection algorithm is devised the step size selection algorithm is based partly on stability information obtained by estimating the discrete lyapunov exponent of a runge kutta method applied to a nonautonomous linear scalar problem the utility of the approach is illustrated in numerical experiments that demonstrate how the new algorithm performs against a standard accuracy based step size selection strategy

stability of semidiscretizations of hyperbolic problems semidiscretization of hyperbolic systems yields systems of ordinary differential equations of first or second order in both cases we investigate the stability of a general class of methods namely the m stage k step methods when applied to linear test equations a map sigma is introduced which allows us at least for constant linear systems to investigate whether a second order system should be integrated directly or should first be reduced to a first order system

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

an efficient algorithm based on splitting for the time integration of the schrodinger equation we present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the schrodinger equation when the hamiltonian operator is either time independent or changes slowly with time in the later case the evolution operator can be effectively approximated in a step by step manner first divide the time integration interval in sufficiently short subintervals and then successively solve a schrodinger equation with a different time independent hamiltonian operator in each of these subintervals when discretized in space the schrodinger equation with the time independent hamiltonian operator obtained for each time subinterval can be recast as a classical linear autonomous hamiltonian system corresponding to a system of coupled harmonic oscillators the particular structure of this linear system allows us to construct a set of highly efficient schemes optimized for different precision requirements and time intervals sharp local error bounds are obtained for the solution of the linear autonomous hamiltonian system considered in each time subinterval our schemes can be considered in this setting as polynomial approximations to the matrix exponential in a similar way as methods based on chebyshev and taylor polynomials the theoretical analysis supported by numerical experiments performed for different time independent hamiltonians indicates that the new methods are more efficient than schemes based on chebyshev polynomials for all tolerances and time interval lengths the algorithm we present automatically selects for each time subinterval the most efficient splitting scheme among several new optimized splitting methods for a prescribed error tolerance and given estimates of the upper and lower bounds of the eigenvalues of the discretized version of the hamiltonian operator

on the adoption of velocity variable and grid system for fluid flow computation in curvilinear coordinates abstract the issues of adopting the velocity components as dependent velocity variables including the cartesian and culvilinear ones for the navier stokes flow computations are investigated the viewpoint advocated is that a numerical algorithm should preferably honor both the physical conservation law in differential form and the geometric conservation law in discrete form it is demonstrated that with the curvilinear velocity vectors the curvatures of the grid lines introduce extra source terms into the governing equations with the cartesian velocity vector on the other hand the governing equations in curvilinear coordinates can retain the full conservation law form and honor the physical conservation laws the nonlinear combinations of the metric terms also cause the algorithms based on curvilinear velocity components to be more difficult to satisfy the geometric conservation law and hence more sensitive to grid skewness effect for the combined utilization of the cartesian velocity vector and the staggered grid arrangement the implications of spurious pressure oscillation arising from the 90 turning are discussed it is demonstrated that these spurious oscillations can possibly appear only under a very specific circumstance namely the meshes in the region with 90 turning must be parallel to the cartesian coordinates and of uniform spacing along coordinates otherwise no spurious oscillations can appear several flow solutions for domain with 90 and 360 turnings are presented to demonstrate that satisfactory results can be obtained by using the cartesian velocity components and the staggered grid arrangement

accuracy bounds for semidiscretizations of hyperbolic problems bounds are given for the error constant of stable finite difference methods for first order hyperbolic equations in one space dimension which use r downwind and s upwind points in the discretization of the space derivatives and which are of optimal order p min r s 2 r 2 2 s it is known that this order can be obtained by interpolatory methods examples show however that their error constants can be improved

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

constructive characterization ofa stable approximations to exp z and its connection with algebraically stable runge kutta methods all rational approximations to exp z of order 2m m denotes the maximal degree of nominator and denominator are given by a closed formula involving real parameters using the theory of order stars 9 necessary and sufficient conditions fora stability respectivelyi stability are given on the basis of this characterization relations between the concepts ofa stability and algebraic stability for implicit runge kutta methods are investigated in particular we can partly prove the conjecture that to any irreduciblea stabler z of oderp 0 there exist algebraically stable runge kutta methods of the same order withr z as stability function

stability of parallel explicit ode methods general purpose numerical methods for ordinary initial value problems are usually found in the class of linear multistage multivalue methods first formulated by j c butcher among these the explicit methods are easiest to implement for this reason there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in par allel each such group of concurrent function evaluations can be regarded as a stage of the method it is desirable that methods have a large region of stability so that stability dictated restrictions on the stepsize will occur less frequently in order to do a fair comparison of stability regions one should consider the scaled stability region this is defined using not the nominal stepsize but rather the stepsize per unit of work each stage counts as one work unit in the context of parallel computing each stage may involve any number of independent function evaluations very important results on scaled stability regions were obtained by jeltsch and nevanlinna in the context of serial computation they identify a type of method actually a type of alge braic function whose scaled stability region is optimal this means that the region is not a proper subset of some other scaled stability region the distinguished type of algebraic functions is one that satisfies property c which requires that the principal branch of the algebraic function be analytic outside the stability region except for a pole at infinity whose multiplicity equals the number of work units of the method it is shown that by modifying property c the result on optimal stability regions holds also in the context of parallel computa tion an important consequence is that although the use of parallelism enlarges the class of scaled stability regions it does not result in scaled stability regions that properly contain those obtained without parallelism roughly stated parallelism cannot improve the stability of an explicit linear method but it can even for the standard test problem improve the accuracy

order stars and stiff integrators order stars introduced in g wanner e hairer s p norsett order stars and stability theorems bit 18 1978 475 489 have become a fundamental tool for the understanding of order and stability properties of numerical methods for stiff differential equations this survey retraces their discovery and their principal achievements we also sketch some later extensions and describe some recent developments

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

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

two results concerning the stability of staggered multistep methods staggered spatial node layouts are widely used in the context of solving wave equations when this idea of staggering is extended to time integrators for wave equations one finds methods that offer both better accuracy and less restrictive stability conditions for an equivalent computational cost here we prove two results concerning staggered multistep methods we first prove that staggered backwards differentiation methods are unstable for all orders p ge 5 we then extend dahlquist s first stability barrier to explicit staggered multistep methods

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

the teitelbaum conjecture in the indefinite setting let f be a new cusp form on gamma 0 n of even weight k 2 geq 2 suppose that there is a prime p parallel n and that we may write n pn n where n is the squarefree product of an even number of primes there is a darmon style cal l invariant cal l n f attached to this factorization which is the orton cal l invariant when n 1 we prove that cal l n f does not depend on the chosen factorization of n and it is equal to the other known cal l invariants we also give a formula for the computation of the logarithmic p adic abel jacobi image of the darmon cycles this formula is crucial for the computations of the derivatives of the p adic l functions of the weight variable attached to a real quadratic field k bbb q such that the primes dividing n are split and the primes dividing pn are inert