systems of numeration and fractal functions relating to substitutions french abstract let a be a finite alphabet a substitution over a un n n a fixed point for and for each a a a a real number we establish under some assumptions an asymptotic formula concerning the sum s n i n u i n n this result generalizes some previous results from coquet or brillhart erdos and morton moreover relations with self affine functions in a sense which generalizes a definition from kamae are proved the calculi leave over systems of representation of integers and real numbers

solving second order initial value problems by a hybrid multistep method without predictors abstract a three step seventh order hybrid linear multistep method hlmm with three non step points is proposed for the direct solution of the special second order initial value problems ivps of the form y f x y with an extension to y f x y y the main method and additional methods are obtained from the same continuous scheme derived via interpolation and collocation procedures the stability properties of the methods are discussed by expressing them as a one step method in higher dimension the methods are then applied in block form as simultaneous numerical integrators over non overlapping intervals numerical results obtained using the proposed block form reveal that it is highly competitive with existing methods in the literature

fully implicit mixed hybrid finite element discretization for general purpose subsurface reservoir simulation abstract we present a new fully implicit mixed hybrid finite element mhfe discretization scheme for general purpose compositional reservoir simulation the locally conservative scheme solves the coupled momentum and mass balance equations simultaneously and the fluid system is modeled using a cubic equation of state we introduce a new conservative flux approach for the mass balance equations for this fully implicit approach we discuss the nonlinear solution procedure for the proposed approach and we present extensive numerical tests to demonstrate the convergence and accuracy of the mhfe method using tetrahedral elements we also compare the method to other advanced discretization schemes for unstructured meshes and tensor permeability finally we illustrate the applicability and robustness of the method for highly heterogeneous reservoirs with unstructured grids

trust region based solver for nonlinear transport in heterogeneous porous media we describe a new nonlinear solver for immiscible two phase transport in porous media where viscous buoyancy and capillary forces are significant the flux fractional flow function f is a nonlinear function of saturation and typically has inflection points and can be non monotonic the non convexity and non monotonicity of f are major sources of difficulty for nonlinear solvers of coupled multiphase flow and transport in natural porous media we describe a modified newton algorithm that employs trust regions of the flux function to guide the newton iterations the flux function is divided into saturation trust regions delineated by the inflection unit flux and end points the updates are performed such that two successive iterations cannot cross any trust region boundary if a crossing is detected the saturation value is chopped back to the appropriate trust region boundary the proposed trust region newton solver which is demonstrated across the parameter space of viscous buoyancy and capillary effects is a significant extension of the inflection point strategy of jenny et al jcp 2009 5 for viscous dominated flows we analyze the discrete nonlinear transport equation obtained using finite volume discretization with phase based upstream weighting then we prove convergence of the trust region newton method irrespective of the timestep size for single cell problems numerical results across the full range of the parameter space of viscous gravity and capillary forces indicate that our trust region scheme is unconditionally convergent for 1d transport that is for a given choice of timestep size the unique discrete solution is found independently of the initial guess for problems dominated by buoyancy and capillarity the trust region newton solver overcomes the often severe limits on timestep size associated with existing methods to validate the effectiveness of the new nonlinear solver for large reservoir models with strong heterogeneity we compare it with state of the art nonlinear solvers for two phase flow and transport using the spe 10 model compared with existing nonlinear solvers our trust region solver results in superior convergence performance and achieves reduction in the total newton iterations by more than an order of magnitude together with a corresponding reduction in the overall computational cost

adaptive implicit finite element methods for multicomponent compressible flow in heterogeneous and fractured porous media this work presents adaptive implicit first order and second order discontinuous galerkin dg methods for the transport of multicomponent compressible fluids in heterogeneous and fractured porous media discretized by triangular quadrilateral and hexahedral grids the adaptive implicit method aim combines the advantages of purely explicit or implicit methods in time in grid cells with high fluxes or low pore volumes the transport update is done implicitly to alleviate the courant friedrichs lewy cfl time step constraints of the conditionally stable explicit approach grid cells with a large cfl condition are updated explicitly combined this allows higher efficiency than explicit methods but it reduces the penalty of implicit methods which exhibit high numerical dispersion and are more computationally and storage expensive per time step the advantages of aim are modest for uniform grids and rock properties however in heterogeneous or fractured reservoirs explicit methods may become impractical while a fully implicit approach introduces unnecessary numerical dispersion and is overkill for low permeability layers and matrix blocks in such applications aim is shown to be significantly more efficient and accurate the division between explicit and implicit grid cells is made adaptively in space and time this allows for a high level of explicitness and can also adapt to high fluxes caused by e g viscous and gravitational flow instabilities numerical examples demonstrate the powerful features of aim to model e g solute transport carbon sequestration in saline aquifers and miscible gas injection in fractured oil and gas reservoirs

perturbed collocation and runge kutta methods it is well known thatsome implicit runge kutta methods are equivalent to collocation methods this fact permits very short and natural proofs of order anda b an bn stability properties for this subclass of methods see 9 and 10 the present paper answers the natural question ifall rk methods can be considered as a somewhat perturbed collocation after having introduced this notion we give a proof on the order of the method and discuss their stability properties much of known theory becomes simple and beautiful

implicit hybrid upwinding for two phase flow in heterogeneous porous media with buoyancy and capillarity abstract we consider the numerical solution of the partial differential equations governing multiphase flow in porous media for highly nonlinear problems the temporal discretization of choice is often the unconditionally stable fully implicit method however the nonlinear systems often solved with newton s method are difficult to solve thus the computational cost is strongly dependent on the nonlinear convergence rate and enhancing this convergence property is key to speed up subsurface flow simulation we focus on the case of spatially discontinuous capillary pressure between rock regions to efficiently and accurately simulate the flow dynamics in heterogeneous porous media the flux computation combines implicit hybrid upwinding with transmission conditions between different rock regions this leads to a scheme that correctly represents the trapping mechanisms while improving the nonlinear convergence we extend our previous results hamon et al 2016 18 by generalizing the scheme to fully implicit coupled flow and transport to address realistic problems in multiple dimensions the generalized scheme is supported by an analysis of its mathematical properties our multidimensional numerical examples which range from buoyancy driven flow with capillary barriers to viscous dominated flow demonstrate that the implicit hybrid upwinding scheme improves the accuracy compared to the standard phase based upwinding scheme while leading to significant reductions in the number of nonlinear iterations in multiple dimensions

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

petrov galerkin methods for linear volterra integro differential equations in this paper we study a class of petrov galerkin solutions that have global optimal convergence rates for linear volterra integro differential equations these solutions also possess certain local and global superconvergence asymptotic expansions of the errors in these solutions are established which can be used to form higher order approximations by richardson extrapolation and defect corrections several postprocessing techniques are introduced to enhance these solutions as by products these higher order numerical approximations can be used to generate a posteriori error estimators representative numerical results are also provided

an iterative multistep kernel based method for nonlinear volterra integral and integro differential equations of fractional order abstract an iterative multistep kernel based method is proposed for the nonlinear volterra integral equations and nonlinear volterra integro differential equations of fractional order which can produce reliable globally smooth numerical solutions an error estimate of the positive definite kernel interpolation and the convergence and error analysis of the proposed iterative scheme are investigated here we focused on positive definite radial basis kernels and further a new and applicable shape parameter selection strategy is proposed the proposed multi step method set up and solve several small local problems instead of a single large problem which makes it suitable for problems with long time simulations in order to show the efficiency and versatility of the proposed method some numerical experiments are reported the comparison of the numerical results with the analytical solutions and the best reported results in the literature confirm the good accuracy and applicability of the proposed method

high performance parallel numerical methods for volterra equations with weakly singular kernels non stationary discrete time waveform relaxation methods for abel systems of volterra integral equations using fractional linear multistep formulae are introduced fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed a significant expression of the error is proved which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance the numerical experiments confirm the theoretical expectations

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

fast numerical solution of weakly singular volterra integral equations abstract we give a fortran program for the numerical solution of abel volterra convolution equations of the second and first kind we use a fractional linear multistep method bdf4 1 2 and use fast fourier transform techniques to exploit the convolution structure a detailed description of the computation of the quadrature weights using fft is given numerical examples are included

b convergence properties of multistep runge kutta methods by using the theory of b convergence for general linear methods to the special case of multistep runge kutta methods a series of b convergence results for multistep runge kutta methods is obtained and it is proved that the family of algebraically stable r step s stage multistep runge kutta methods with parameters al a2 ar presented by burrage in 1987 is optimally bconvergent of order at least s and b convergent of order s 1 provided that r s and aj o j 1 2 r furthermore this family of methods is optimally b convergent of order s 1 if some other additional conditions are satisfied

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

on one leg multistep methods the properties and possibilities of one leg methods are presented in a manner that admits the generalization to smoothly varying step size a new definition is given of the local truncation error and of the order of consistency some data are given for the most accurate one leg methods i e for p k and p k 1 where p is the order of consistency and k is the step number of the method

one step collocation uniform superconvergence predictor corrector method local error estimate in this paper we consider the one step collocation method at gaussian points for the numerical solution of ivp s for ode s it is well known that this method has a superconvergence rate o h 2n at the nodes while it uniformly converges with the lower order o h n 1 we give a rule to reach the uniform superconvergence by some kind of idec method an interesting by product of this procedure is an accurate estimation of the local discretization error moreover we give a modified version of the method in a predictor corrector mode for which all the previous results keep on holding in particular we propose a new method of order four with step size control

a parallel algorithm for large systems of volterra integral equations of abel type a significative number of recent applications require numerical solution of large systems of abel volterra integral equations here we propose a parallel algorithm to numerically solve a class of these systems designed for a distributed memory mimd architecture in order to achieve a good efficiency we employ a fully parallel and fast convergent waveform relaxation wr method and evaluate the lag term by using fft techniques to accelerate the convergence of the wr method and to best exploit the parallel architecture we develop special strategies the performances of the resulting code nswr4 are illustrated on some examples

fast and accurate coarsening simulation with an unconditionally stable time step we present cahn hilliard and allen cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy controlled simulation our stability analysis is based on eyre s theorem and unconditional von neumann stability analysis both of which we present numerical tests confirm the accuracy of the von neumann approach which is straightforward and should be widely applicable in phase field modeling for the cahn hilliard case we show that accuracy can be controlled with an unbounded time step dt that grows with time t as dt t a we develop a classification scheme for the step exponenta and demonstrate that a class of simple linear algorithms gives a51 3 for this class the speedup relative to a fixed time step grows with n the linear size of the system as n lnn with conservative choices for the parameters controlling accuracy and finite size effects we find that an 8192 2 lattice can be integrated 300 times faster than with the euler method

potential based reduced newton algorithm for nonlinear multiphase flow in porous media we present a phase based potential ordering that is an extension of the cascade ordering introduced by appleyard and cheshire john r appleyard ian m cheshire the cascade method for accelerated convergence in implicit simulators in european petroleum conference 1982 pp 113 122 the proposed ordering is valid for both two phase and three phase flow and it can handle countercurrent flow due to gravity and or capillarity we show how this ordering can be used to reduce the nonlinear algebraic system that arises from the fully implicit method fim into one with only pressure dependence the potential based reduced newton algorithm is then obtained by applying newton s method to this reduced order system numerical evidence shows that our potential based reduced newton solver is able to converge for time steps that are much larger than what the standard newton s method can handle in addition whenever standard newton converges so does the reduced newton algorithm and the number of global nonlinear iterations required for convergence is significantly reduced compared with the standard newton s method

algebraic multiscale solver for flow in heterogeneous porous media an algebraic multiscale solver ams for the pressure equations arising from incompressible flow in heterogeneous porous media is described in addition to the fine scale system of equations ams requires information about the superimposed multiscale dual and primal coarse grids ams employs a global solver only at the coarse scale and allows for several types of local preconditioners at the fine scale the convergence properties of ams are studied for various combinations of global and local stages these include multiscale finite element msfe and multiscale finite volume msfv methods as the global stage and correction functions cf block incomplete lower upper factorization bilu and ilu as local stages the performance of the different preconditioning options is analyzed for a wide range of challenging test cases the best overall performance is obtained by combining msfe and ilu as the global and local preconditioners respectively followed by msfv to ensure local mass conservation comparison between ams and a widely used algebraic multigrid amg solver 1 indicates that ams is quite efficient a very important advantage of ams is that a conservative fine scale velocity can be constructed after any msfv stage

an adaptive method for volterra fredholm integral equations on the half line in this paper we develop a direct quadrature method for solving volterra fredholm integral equations on an unbounded spatial domain these problems when related to some important physical and biological phenomena are characterized by kernels that present variable peaks along space the method we propose is adaptive in the sense that the number of spatial nodes of the quadrature formula varies with the position of the peaks the convergence of the method is studied and its performances are illustrated by means of a few significative examples the parallel algorithm which implements the method and its performances are described

paperfolding morphisms planefilling curves and fractal tiles an interesting class of automatic sequences emerges from iterated paperfolding the sequences generate curves in the plane with an almost periodic structure we generalize the results obtained by davis and knuth on the self avoiding and planefilling properties of these curves giving simple geometric criteria for a complete classification finally we show how the automatic structure of the sequences leads to self similarity of the curves which turns the planefilling curves in a scaling limit into fractal tiles for some of these tiles we give a particularly simple formula for the hausdorff dimension of their boundary

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

high order algebraically stable multistep runge kutta methods in this paper we examine some of the stability properties of a special class of multivalue methods called in k burrage order properties of multivalue methods ima j numer anal submitted multistep runge kutta methods and construct families of algebraically stable methods of arbitrarily high order

simulation of incompressible two phase flow in porous media with large timesteps abstract multiphase flow in porous media occurs in several disciplines including petroleum reservoir engineering petroleum systems analysis and co 2 sequestration while simulations often use a fully implicit discretization to increase the time step size restrictions on the time step often exist due to non convergence of the nonlinear solver e g newton s method here this problem is addressed for the buckley leverett equations which model incompressible immiscible two phase flow with no capillary potential the equations are recast as a gradient flow using the phase field method and a convex energy splitting scheme is applied to enable large timesteps even for high degrees of heterogeneity in permeability and viscosity by using the phase field formulation as a homotopy map the underlying hyperbolic flow equations can be solved with large timesteps for a heterogeneous test problem the new homotopy method allows the timestep to be increased by more than six orders of magnitude relative to the unmodified equations while maintaining convergence

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

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

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

unconditionally convergent nonlinear solver for hyperbolic conservation laws with s shaped flux functions this paper addresses the convergence properties of implicit numerical solution algorithms for nonlinear hyperbolic transport problems it is shown that the newton raphson nr method converges for any time step size if the flux function is convex concave or linear which is in general the case for cfd problems in some problems e g multiphase flow in porous media the nonlinear flux function is s shaped not uniformly convex or concave as a result a standard nr iteration can diverge for large time steps even if an implicit discretization scheme is used to solve the nonlinear system of equations in practice when such convergence difficulties are encountered the current time step is cut previous iterations are discarded a smaller time step size is tried and the nr process is repeated the criteria for time step cutting and selection are usually based on heuristics that limit the allowable change in the solution over a time step and or nr iteration here we propose a simple modification to the nr iteration scheme for conservation laws with s shaped flux functions that converges for any time step size the new scheme allows one to choose the time step size based on accuracy consideration only without worrying about the convergence behavior of the nonlinear solver the proposed method can be implemented in an existing simulator e g for co 2 sequestration or reservoir flow modeling quite easily the numerical analysis is confirmed with simulation studies using various test cases of nonlinear multiphase transport in porous media the analysis and numerical experiments demonstrate that the modified scheme allows for the use of arbitrarily large time steps for this class of problems

nonlinear analysis of multiphase transport in porous media in the presence of viscous buoyancy and capillary forces nonlinear convergence problems in numerical reservoir simulation can lead to unacceptably large computational time and are often the main impediment to performing simulation studies of large scale problems we analyze the nonlinearity of the discrete transport mass conservation equation for immiscible incompressible two phase flow in porous media in the presence of viscous buoyancy and capillary forces although simulation problems are multi dimensional with large numbers of cells and variables we find that the essence of the nonlinear behavior can be understood by studying the discretized numerical flux function for the interface between two cells the numerical flux is expressed in terms of the saturations of the two cells discontinuities in the first order derivative of the flux function referred to as kinks and inflection lines are identified as the cause of convergence difficulty these critical features kinks and inflections change the curvature of the numerical flux function abruptly and can lead to overshoots oscillations or divergence in newton iterations based on our understanding of the nonlinearity a nonlinear solver is developed referred to as the numerical trust region ntr solver the solver is able to guide the newton iterations safely and efficiently through the different saturation trust regions delineated by the kinks and inflections specifically overshoots and oscillations that often lead to convergence failure are avoided numerical examples demonstrate that our ntr solver has superior convergence performance compared with existing methods in particular convergence is achieved for a wide range of timestep sizes and courant friedrichs lewy cfl numbers spanning several orders of magnitude in addition a discretization scheme is proposed for handling heterogeneities in capillary pressure saturation relationship the scheme has less degree of nonlinearity compared with the standard single point phase based upstream weighting scheme leading to an improved nonlinear convergence performance especially when used together with our ntr solver our proposed numerical solution strategy that is based on the numerical flux and handles capillarity extends the previous work by jenny et al 2009 6 and wang and tchelepi 2013 7 significantly the nonlinearity of multiphase transport in porous media is analyzed the analysis considers viscous buoyancy and capillary forces the critical features that cause nonlinear convergence failures are identified a nonlinear solver that converges without oscillations and divergence is developed the solver converges faster than existing methods and allows large timesteps

analysis of a darcy cahn hilliard diffuse interface model for the hele shaw flow and its fully discrete finite element approximation in this paper we present pde and finite element analyses for a system of pdes consisting of the darcy equation and the cahn hilliard equation which arises as a diffuse interface model for the two phase hele shaw flow in the model the two sets of equations are coupled through an extra phase induced force term in the darcy equations and a fluid induced transport term in the cahn hilliard equation we propose a fully discrete implicit finite element method for approximating the pde system which consists of the implicit euler method combined with a convex splitting energy strategy for the temporal discretization the standard finite element discretization for the pressure and a split or mixed finite element discretization for the fourth order cahn hilliard equation it is shown that the proposed numerical method satisfies a mass conservation law in addition to a discrete energy law that mimics the basic energy law for the darcy cahn hilliard phase field model and holds uniformly in the phase field

multistep collocation methods for volterra integro differential equations multistep collocation methods for volterra integro differential equations are derived and analyzed they increase the order of convergence of classical one step collocation methods at the same computational cost the numerical stability analysis is carried out and classes of a 0 stable methods are provided numerical experiments confirm theoretical expectations and make comparisons with the one step collocation methods

operator based linearization approach for modeling of multiphase multi component flow in porous media a new approach for the linearization of governing equations that describe flow and transport in porous media is proposed in this work it is based on an approximate representation of the exact physics of the problem which is similar to an approximate representation of space and time discretization performed in conventional simulation the governing equations are introduced as a combination of operators dependent on spatially altered properties and operators fully controlled by nonlinear properties of fluid and rock next a parametrization in the physics space of the problem is introduced the property based operators are approximated using direct interpolation in the space of nonlinear unknowns the discrete version of the governing equations is constructed as a combination of operators that approximate both nonlinear physics and discretization in time and space this approach is applied to the reservoir simulation of miscible and immiscible displacement processes the performance of the method demonstrates a convergence of simulation results by resolution in physical space with the improved performance

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

the stability function for multistep collocation methods c polynomials for rational approximation to the exponential function was introduced by norsett 7 to study stability properties of one step methods for one step collocation methods thec polynomial has a very simple form in this paper we studyc polynomials for multistep collocation methods and obtain results that generalize those in the one step case and provide a way to analyze linear stability of such methods

mellin transforms and asymptotics digital sums abstract arithmetic functions related to number representation systems exhibit various periodicity phenomena for instance a well known theorem of delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function we show that such periodicity phenomena can be analyzed rather systematically using classical tools from analytic number theory namely the mellin perron formulae this approach yields naturally the fourier series involved in the expansions of a variety of digital sums related to number representation systems

upstream differencing for multiphase flow in reservoir simulation upstream weighting for multiphase flow in reservoir simulation is analyzed the associated numerical flux is shown to be well defined monotone lipschitz continuous and consistent in the case of a two phase flow the corresponding numerical flux is compared to that of godunov and engquist and osher finally simple way to obtain a higher order scheme is outlined

an efficient and fast parallel method for volterra integral equations of abel type in this paper we present an efficient and fast parallel waveform relaxation method for volterra integral equations of abel type obtained by reformulating a nonstationary waveform relaxation method for systems of equations with linear coefficient constant kernel to this aim we consider the laplace transform of the equation and here we apply the recurrence relation given by the chebyshev polynomial acceleration for algebraic linear systems back in the time domain we obtain a three term recursion which requires at each iteration the evaluation of convolution integrals where only the laplace transform of the kernel is known for this calculation we can use a fast convolution algorithm numerical experiments have been done also on problems where it is not possible to use the original nonstationary method obtaining good results in terms of improvement of the rate of convergence with respect the stationary method

power and exponential sums for generalized coding systems by a measure theoretic approach a measure theoretic approach to study the power and the exponential sums for the usual coding system has been developed since the 1990s in this paper we first introduce a new coding system and then give explicit formulas for the power and the exponential sums for the coding system by the measure theoretic approach an expression for the power sum using the generalized takagi function will also be given

analysis of hybrid upwinding for fully implicit simulation of three phase flow with gravity the systems of algebraic equations arising from implicit backward euler finite volume discretization of the conservation laws governing multiphase flow in porous media are quite challenging for nonlinear solvers in the presence of countercurrent flow due to buoyancy the numerical flux obtained with single point phase potential upwinding ppu is not differentiable which causes convergence difficulties for nonlinear solvers recently lee efendiev and tchelepi adv water resour 82 2015 pp 27 38 proposed a hybrid upwinding strategy for two phase flow that gives rise to a differentiable numerical flux across the entire viscous gravity parameter space here we first present an implicit hybrid upwinding ihu scheme for hyperbolic conservation laws extending the work of lee efendiev and tchelepi to an arbitrary number of fluid phases we show that the numerical flux obtained with the ihu is consistent and a monotone function of its own saturation it is also a differentiable function of t

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

fully discrete finite element approximations of the navier stokes cahn hilliard diffuse interface model for two phase fluid flows this paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the navier stokes equations and the cahn hilliard equation which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids in the model the two sets of equations are coupled through an extra phase induced stress term in the navier stokes equations and a fluid induced transport term in the cahn hilliard equation fully discrete mixed finite element methods are proposed for approximating the coupled system it is shown that the proposed numerical methods satisfy a mass conservation law and a discrete energy law which is analogous to the basic energy law for the phase field model the convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law as a by product the convergence result also provides a constructive proof of the existence of weak solutions to the navier stokes cahn hilliard phase field model numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

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

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

l stable parallel one block methods for ordinary differential equations in this contribution the author considers the one block methods designed by sommeijer couzy and van der houwen for the purpose of solving ordinary differential equations odes on a parallel computer the author also derives a new set of order conditions studies the stability and exhibits a new class of parallel methods which contains l stable schemes up to order eleven

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

a family of methods for abel integral equations of the second kind abstract a class of methods depending on some parameters are introduced for the numerical solution of the abel integral equations of the second kind some bounds on the parameters are determined so that the corresponding methods have infinite stability intervals

parallel iterative linear solvers for multistep runge kutta methods abstract this paper deals with solving stiff systems of differential equations by implicit multistep runge kutta mrk methods for this type of methods nonlinear systems of dimension sd arise where s is the number of runge kutta stages and d the dimension of the problem applying a newton process leads to linear systems of the same dimension which can be very expensive to solve in practice with a parallel iterative linear system solver especially designed for mrk methods we approximate these linear systems by s systems of dimension d which can be solved in parallel on a computer with s processors in terms of jacobian evaluations and lu decompositions the k steps stage mrk applied with this technique is on s processors equally expensive as the widely used k step backward differentiation formula on 1 processor whereas the stability properties are better than that of bdf a simple implementation of both methods shows that for the same number of newton iterations the accuracy delivered by the new method is higher than that of bdf

convergence of implicit monotone schemes with applications in multiphase flow in porous media phase based upstreaming which is a commonly used spatial discretization for multiphase flow in reservoir simulation naturally gives rise to implicit monotone schemes when implicit time stepping is used the nonlinear gauss seidel and jacobi algorithms are shown to converge to a unique bounded solution when applied to the resulting system of equations thus for one dimensional problems we obtain an alternate constructive proof that such schemes are well defined and converge to the entropy solution of the conservation law for arbitrary cfl numbers the accuracy of phase based upstream solutions is studied for various spatial and temporal grids revealing the importance of unconditional stability when nonuniform grids and or variable porosity is involved

pitfalls in fast numerical solvers for fractional differential equations we consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x t f t x t x 0 x0 where x is the derivative of x of order in the sense of caputo and 0 1 we review some of the existing methods and explain their respective strengths and weaknesses we identify and discuss potential problems in the development of generally applicable schemes

a fast iterative method for discretized volterra fredholm integral equations solving the nonlinear systems arising in the discretization in space and time of volterra fredholm integral equations by newton iteration leads to dense linear systems whose dimension depends on the spatial mesh the solution of these linear systems can hence be very costly here we try to reduce these costs by solving each newton iteration by a non stationary inner iteration process each inner iteration again requires the solution of a linear system however since the splitting matrix is diagonal now the components or sets of components can be computed in parallel the performance of this iteration method is illustrated by means of a few significative examples

implicit hybrid upwind scheme for coupled multiphase flow and transport with buoyancy abstract numerical simulation of coupled multiphase flow and transport in porous media is used in many scientific and industrial applications such as groundwater management and oil recovery to solve the governing partial differential equations we focus on the fully implicit backward euler finite volume method fim in this numerical scheme large algebraic systems must be solved at each time step using newton s method this can be quite expensive especially for highly nonlinear problems with buoyancy in this work we present a new first order approximation of the flux in the fim leading to improved nonlinear convergence rate and radius this approximation is based on implicit hybrid upwinding and relies on separate treatments of the viscous and buoyancy parts of the numerical flux moreover in the viscous part the total velocity discretization is key as it controls the coupling between the elliptic flow problem and the hyperbolic multiphase transport problem to achieve robustness and efficiency we propose a differentiable total velocity discretization that adapts to the balance of forces at each interface we analyze the theoretical properties of the resulting numerical scheme and prove that saturations remain between physical bounds challenging two and three phase test cases illustrate that our numerical scheme brings a significant reduction in the number of nonlinear iterations compared to a finite volume discretization relying on the widely used phase per phase upstream weighting our numerical scheme therefore reduces the simulation cost while providing similar accuracy

superconvergence phenomena in the finite element method abstract we give a brief survey of superconvergence phenomena in finding a numerical solution of differential equations by finite elements several new results and open problems are introduced