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Abstract: For each integer , a new uniparametric family of stiffly accurate, strongly -stable, -stage Runge–Kutta methods is obtained. These are collocation methods with a first internal stage of explicit type. The methods are based on interpolatory quadrature rules, with precision degree equal to , and all of them have two prefixed nodes, and . The amount of implicitness of our -stage method is similar to that involved with the -stage LobattoIIIA method or with the -stage RadauIIA method. The new family of Runge–Kutta methods proves to be of interest for the numerical integration of stiff systems and Differential Algebraic Equations. In fact, on several stiff test problems taken from the current literature, two methods selected in our 4-stage family, seem to be slightly more efficient than the -stage RadauIIA method and also more robust than the -stage LobattoIIIA method. [Copyright &y& Elsevier]

Abstract: As a result of recent resurgence of interest in exponential integrators a number of such methods have been introduced in the literature. However, questions of what constitutes an efficient exponential method and how these techniques compare with commonly used schemes remain to be fully investigated. In this paper we consider exponentialKrylov integrators in the context of large scale applications and discuss what design principles need to be considered in construction of an efficient method of this type. Since the Krylov projections constitute the primary computational cost of an exponential integrator we demonstrate how an exponential-Krylov method can be structured to minimize the total number of Krylov projections per time step and the number of Krylov vectors each of the projections requires. We present numerical experiments that validate and illustrate these arguments. In addition, we compare exponential methods with commonly used implicit schemes to demonstrate their competitiveness. [ABSTRACT FROM AUTHOR]

Abstract: Numerical aspects of computational modeling of chemical vapor deposition are discussed. Large sparse strongly nonlinear algebraic systems are to be solved per time step. For this, inexact Newton methods and preconditioned Krylov subspace methods are suitable. To ensure positivity of concentrations, we propose a novel approach, namely a projected inexact Newton method. Unlike the commonly used method of clipping, this conserves mass. Efficiency of several preconditionings is compared. Our numerical tests culminate in an unusually large computation, namely a three-dimensional case with 17 species and 26 reactions. [Copyright &y& Elsevier]

Recently a new class of Monte Carlo methods, called time relaxed Monte Carlo (TRMC), designed for the simulation of the Boltzmann equation close to fluid regimes has been introduced [L. Pareschi and G. Russo, SIAM J. Sci. Comput., 23 (2001), pp. 1253-1273]. A generalized Wild sum expansion of the solution is the basis of the simulation schemes. After a splitting of the equation, the time discretization of the collision step is obtained from the Wild sum expansion of the solution by replacing high order terms in the expansion with the equilibrium Maxwellian distribution; in this way speed-up of the methods close to fluid regimes is obtained by efficiently thermalizing particles close to the equilibrium state. In this work we present an improvement of such methods which allows us to obtain an effective uniform accuracy in time without any restriction on the time step and subsequent increase of the computational cost. The main ingredient of the new algorithms is recursivity [L. Pareschi and B. Wennberg, Monte Carlo Methods Appl., 7 (2001), pp. 349-358]. Several techniques can be used to truncate the recursive trees generated by the schemes without degrading the accuracy of the numerical solution. Techniques based on adaptive strategies are presented. Numerical results emphasize the gain of efficiency of the present simulation schemes with respect to standard DSMC (direct simulation Monte Carlo) methods. [ABSTRACT FROM AUTHOR]

We present the construction and preliminary tests of the exponential time integration schemes for the Euler equations. The equations are written in terms of potential temperature and Exner function. The time stepping is accomplished with the EPI2 and EPI3 schemes which have been used in the past for very accurate, long time step integration of the shallow water equations on the sphere. The stability of exponential integrators of this class is ensured by approximation of the oscillatory term using an absolutely stable exponential formula. Consequently, the schemes are not subject to the CFL condition imposed by the linear oscillatory part of the system and they also eliminate the phase errors associated with all the known implicit time stepping algorithms. All calculations are performed using the phipm algorithm based on the Krylov space methods. It is shown that the spectral radius of the Jacobian and consequently the Krylov space dimension are very sensitive to the presence of spatial noise. Efficiency and accuracy increase after the Shapiro filter is applied every two time steps. The method is investigated using several standardized and well accepted benchmark tests, including convective bubbles and a cold density current proposed by Straka and collaborators. Numerical experiments show that a stable integration is obtained for the wave Courant number of the order of 60 to 250. We present a detailed discussion of the development of a cold density current capturing the spiraling Prandtl vortex and the subsequent appearance of the Kelvin–Helmholtz billows. Our results from the exponential method compare very well with those obtained from the accurate schemes reported in the recent literature. The algorithm accurately handles both advection as well as acoustic and gravity waves. These results warrant the further development of the scheme for modeling atmospheric processes on a wide range of scales. • Exponential methods are applied to integrate Euler's equations for atmospheric convection. • The exponential time stepping eliminates the CFL restriction imposed by acoustic and gravity waves. • The calculations are performed using the PHIPM algorithm based on Krylov projection methods. • The numerical schemes are evaluated using convective tests and Straka density current. [ABSTRACT FROM AUTHOR]

A theorem is proved which states that the maximal order of accuracy in L-stable ( m, k)-methods with Jacobian matrix freezing is equal to four. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the variational iteration method is applied to solve systems of ordinary differential equations in both linear and nonlinear cases, focusing interest on stiff problems. Some examples are given to illustrate the accuracy and effectiveness of the method. We compare our results with results obtained by the Adomian decomposition method. This comparison reveals that the variational iteration method is easier to be implemented. In fact, the variational iteration method is a promising method to various systems of linear and nonlinear equations. [Copyright &y& Elsevier]

Stochastic chemical kinetics describes the time evolution of a well-stirred chemically reacting system in a way that takes into account the fact that molecules come in whole numbers and exhibit some degree of randomness in their dynamical behavior. Researchers are increasingly using this approach to chemical kinetics in the analysis of cellular systems in biology, where the small molecular populations of only a few reactant species can lead to deviations from the predictions of the deterministic differential equations of classical chemical kinetics. After reviewing the supporting theory of stochastic chemical kinetics, I discuss some recent advances in methods for using that theory to make numerical simulations. These include improvements to the exact stochastic simulation algorithm (SSA) and the approximate explicit tau-leaping procedure, as well as the development of two approximate strategies for simulating systems that are dynamically stiff: implicit tau-leaping and the slow-scale SSA. [ABSTRACT FROM AUTHOR]

Abstract: A new class of exponential propagation techniques which we call exponential propagation iterative (EPI) methods is introduced in this paper. It is demonstrated how for large stiff systems these schemes provide an efficient alternative to standard integrators for computing solutions over long time intervals. The EPI methods are constructed by reformulating the integral form of a solution to a nonlinear autonomous system of ODEs as an expansion in terms of products between special functions of matrices and vectors that can be efficiently approximated using Krylov subspace projections. The methodology for constructing EPI schemes is presented and their performance is illustrated using numerical examples and comparisons with standard explicit and implicit integrators. The history of the exponential propagation type integrators and their connection with EPI schemes are also discussed. [Copyright &y& Elsevier]

We introduce a new and very simple explicit numerical method for a system of stiff ordinary differential equations (ODEs) that arises from combustion. The main aim of this paper is to show that, contrary to popular belief, an almost trivial explicit method can solve a small class of large nontrivial stiff systems surprisingly efficiently. The algorithm, which is motivated by the theory of Markov jump processes, is very simple and easy to implement. Various numerical experiments are performed to assess the efficiency of the algorithm. For this the ignition of a stoichiometric mixture of n -decane and air at constant pressure and temperature is calculated using a mechanism containing 1218 species and 4825 reactions. For the considered system at moderate accuracy requirements, the new algorithm exhibits performance several orders of magnitude better than standard explicit methods and in the same order of magnitude as the implicit solver packages DASSL and LSODE. The numerical experiments also indicate that the approximate solution obtained from the new algorithm converges to the exact solution of the ODE. Furthermore, we address limitations of our numerical scheme, such as its restriction to the high-temperature regime and moderate accuracy. [ABSTRACT FROM AUTHOR]

A decoupled method is proposed to deal with time-domain simulation for power system dynamic analysis. Traditionally, there are two main categories of numerical integration methods: explicit methods and implicit methods. The implicit methods are numerically stable but require more computational time to solve the nonlinear equations, while explicit methods are relatively efficient but may cause a numerical stability problem. This paper proposes a new hybrid method to take advantage of both explicit and implicit methods based on the invariant subspace partition. The original power system equations are decoupled into two parts that correspond to the stiff and nonstiff subspaces. For the stiff invariant subspace, the implicit method is applied to achieve numerical stability, and the explicit method is employed to handle nonstiff invariant subspace for the computational efficiency. As a result, the new hybrid method is both numerically stable and efficient. The approach is demonstrated through New England 39-bus and IEEE 118-bus systems. [ABSTRACT FROM AUTHOR]

Abstract: Environmental modelling is done more and more by practising ecologists rather than computer scientists or mathematicians. This is because there is a broad spectrum of development tools available that allows graphical coding of complex models of dynamic systems and help to abstract from the mathematical issues of the modelled system and the related numerical problems for estimating solutions. In this contribution, we study how different modelling tools treat a test system, a highly non-linear predator–prey model, and how the numerical solutions vary. We can show that solutions (a) differ if different development tools are chosen but the same numerical procedure is selected; (b) depend on undocumented implementation details; (c) vary even for the same tool but for different versions; and (d) are generated but with no notifications on numerical problems even if these could be identified. We conclude that improved documentation of numeric methods used in the modelling software is essential to make sure that process based models formulated in terms of these modelling packages do not become “black box” models due to uncertainty in integration methods. [Copyright &y& Elsevier]

Abstract: In this paper, the decomposition method is applied to initial value problems for systems of ordinary differential equations in both linear and nonlinear cases, focusing our interest in stiff problems. [Copyright &y& Elsevier]

Presents an abstract of a study which examines two different sets of time scales arising in stiff systems of reaction-diffusion partial differential equations (PDEs). Explanation that the fastest time scales of each set are responsible for the development of a low dimensional manifold; Methodology used in the study; Discussion on the results of the study.

Abstract: The integrating factor (IF) method for numerical integration of stiff nonlinear PDEs has the disadvantage of producing large error coefficients when the linear term has large norm. We propose a generalization of the IF method, and in particular construct multistep-type methods with several orders of magnitude improved accuracy. We also consider exponential time differencing (ETD) methods, and point out connections with a particular application of the commutator-free Lie group methods. We present a new fourth order ETDRK method with improved accuracy. The methods considered are compared in several numerical examples. [Copyright &y& Elsevier]

In this paper, we establish several algorithms for parallel chaotic waveform relaxation methods for solving linear ordinary differential systems based on some given models. Under some different assumptions on the coefficient matrix A and its multisplittings we obtain corresponding sufficient conditions of convergence of the algorithms. Also a discussion on convergence speed comparison of synchronous and asynchronous algorithms is given. [Copyright &y& Elsevier]

This article follows [R. Djouad, B. Sportisse, Appl. Numer. Math. 43 (2002) 383] in which we have proposed an automatic method in order to generate reduced atmospheric chemical mechanisms. On the basis of the slow/fast behaviour of chemical kinetics the exact model is replaced by a differential–algebraic system. We propose here an easy to perform algorithm in order to integrate such systems with a low CPU cost. Comparison is made with some classical solvers such as Euler Backward Implicit (EBI), QSSA and the second-order Rosenbrock method ROS2. This proves the efficiency and the accuracy of the proposed algorithm. We also show that the classical QSSA-like methods cannot be used apart from the framework of reduction procedures, which explains their rather poor accuracy. [Copyright &y& Elsevier]

The time integration of Air Pollution Models is a difficult task due to numerical stiffness, nonlinearity and coupling between equations and implicit schemes are usually advocated. Based on the slow/fast behaviour of the chemical dynamics, reducing procedures are alternative efficient techniques leading to nonstiff systems. Reduced models are defined by a set of lumped species and algebraic constraints and their validity is related with the correct partitioning of dynamics (species and reactions). By using some axiomatic hypothesis and algebraic properties we propose in this paper an efficient partitioning technique and an algorithm in order to compute the lumping of species in an automatic way. This is applied to four atmospheric chemical mechanisms. [Copyright &y& Elsevier]

The coupling of subsystems in a hierarchical modelling approach leads to different time constants in the dynamical simulation of technical systems. Multirate schemes exploit the different time scales by using different time steps for the subsystems. The stiffness of the system or at least of some subsystems in chemical reaction kinetics or network analysis, for example, forbids the use of explicit integration schemes. To cope with stiff problems, we introduce multirate schemes based on partitioned Runge–Kutta methods which avoid the coupling between active and latent components based on interpolating and extrapolating state variables. Order conditions and test results for such a lower order MPRK method are presented. [ABSTRACT FROM AUTHOR]

While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side of the ODE, such as the Taylor series method, outperform them in many cases. Nevertheless, the Taylor series method is not well-suited for stiff problems since it is explicit and not A-stable. In our paper, we present a numerical-analytical method based on the rational approximation of the ODE solution, which is naturally A- and A (α) -stable. We describe the rational approximation method and consider issues of order, stability, and adaptive step control. Finally, through examples, we prove the superior performance of the rational approximation method when solving highly stiff problems, comparing it with the Taylor series and Runge–Kutta methods of the same accuracy order. [ABSTRACT FROM AUTHOR]

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The significance and possible applications of conditional “total” transition probability expressions tij, earlier defined and shown to be basic properties of physical kinetic systems, are reviewed and discussed. A program to carry out explicit evaluation of these quantities is presented, which may facilitate their use in applications. [ABSTRACT FROM AUTHOR]

We present a simulation language GASP/S, an extension of a well- known FORTRAN-based simulation language GASP IV. GASP IV language has been extended using an integration package for solving models of continuous and combined systems described by a set of stiff ordinary differential equations. The V Runge-Kutta- England integration procedure employed in standard GASP IV is unsuitable for problems of this nature. The integration package STIFFSOLVER-81 used in GASP/S has been developed primarily for use in the field of microbial engineering and chemical kinetics. [ABSTRACT FROM PUBLISHER]

This paper describes extensions of and improvements to an efficient numerical integration method, the Implicit Modeling Package (IMP), which was presented in the July 1974 issue of SIMULATION and subsequent ly in the July 1979 issue of SIMULATION. The exten sion described here is a simulation language, General Equation Modeling System (GEMS), for the solution of systems of algebraic and differential equations. GEMS provides a simple and convenient way to formulate complex mathematical models for simulation using the IMP algorithms. GEMS is well suited for the solution of sets of stiff, nonlinear algebraic and differential equations. The GEMS package is compared to ACSL (Advanced Continuous Simulation Language) using a set of nine benchmark problems. [ABSTRACT FROM PUBLISHER]

We describe steady two-dimensional flows of two immiscible fluids through an undulating porous medium of constant thickness, with impermeable or slightly permeable boundaries. Flows in the same or opposite directions are called, respectively, direct or counter flows. Three special classes of flow are determined:Numerical examples illustrate the three classifications above. For incompressible flows the interface and pressure equations uncouple. A stability analysis shows that the direction of integration of the differential equation for the interface must be opposite to the flow direction for direct flows; for counter flows the direction of integration depends on whether the interface is above or below a critical height. Direct flows through cyclic geometries are asymptotically cyclic upstream. If the reservoir is 'leaky', asymptotically self-similar flows result when the (small) permeability ratio is scaled to the dynamical flow parameters. [ABSTRACT FROM AUTHOR]

The stability of numerical methods used for finite difference thermal modelling of buildings is discussed. A known instability in a commonly used process is described and alternative numerical methods with suitable stability properties are identified. With a view to selecting the optimum numerical method, the building energy simulation problem is characterized mathematically and appropriate implicit solvers are compared on the basis of accuracy and computational effort using a building-related test problem prepared for this purpose. A recently developed numerical method with the necessary strong stability is found to possess higher computational efficiency than methods frequently used in this application and it is recommended for inclusion in building energy simulation software. [ABSTRACT FROM AUTHOR]

The purpose of this work is to introduce a new idea of how to avoid the factorization of large matrices during the solution of stiff systems of ODEs. Starting from the general form of an explicit linear multistep method we suggest to adaptively choose its coefficients on each integration step in order to minimize the norm of the residual of an implicit BDF formula. Thereby we reduce the number of unknowns on each step from n to O (1) , where n is the dimension of the ODE system. We call this type of methods Minimal Residual Multistep (MRMS) methods. In the case of linear non-autonomous problem, besides the evaluations of the right-hand side of ODE, the resulting numerical scheme additionally requires one solution of a linear least-squares problem with a thin matrix per step. We show that the order of the method and its zero-stability properties coincide with those of the used underlying BDF formula. For the simplest analog of the implicit Euler method the properties of linear stability are investigated. Though the classical absolute stability analysis is not fully relevant to the MRMS methods, it is shown that this one-step method is applicable in stiff case. In the numerical experiment section we consider the fixed-step integration of a two-dimensional non-autonomous heat equation using the MRMS methods and their classical BDF counterparts. The starting values are taken from a preset slowly-varying exact solution. The comparison showed that both methods give similar numerical solutions, but in the case of large systems the MRMS methods are faster, and their advantage considerably increases with the growth of dimension. Python code with the experimental code can be downloaded from the GitHub repository https://github.com/bfaleichik/mrms. [ABSTRACT FROM AUTHOR]

In this paper, new Locally Linearized Runge-Kutta formulas of Dormand and Prince are introduced with the purpose of integrating large systems of initial value problems. The new formulas are obtained by replacing the Padé approximation for matrix exponential by a Krylov-Padé approximation in the embedded formulas proposed in a previous work. Unlike other high order exponential integrators, these new formulas involve the approximation of just a single phi-function times vector, which makes them particularly attractive from a computational viewpoint. With these new formulas, adaptive schemes are constructed with novelties in the approximation of phi-functions times vectors by Krylov-Padé approximations, and in the strategies for controlling the approximation errors, for estimating the dimension of the Krylov subspaces, and for the reuse of the Jacobians. In addition, numerical experiments are performed to show the potential of the new embedded formulas and adaptive codes in the integration of known physical, biophysical, and physical-chemistry models. [ABSTRACT FROM AUTHOR]

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature. [ABSTRACT FROM AUTHOR]

We present a deterministic global optimization method for nonlinear programming formulations constrained by stiff systems of ordinary differential equation (ODE) initial value problems (IVPs). The examples arise from dynamic optimization problems exhibiting both fast and slow transient phenomena commonly encountered in model‐based systems engineering applications. The proposed approach utilizes unconditionally stable implicit integration methods to reformulate the ODE‐constrained problem into a nonconvex nonlinear program (NLP) with implicit functions embedded. This problem is then solved to global optimality in finite time using a spatial branch‐and‐bound framework utilizing convex/concave relaxations of implicit functions constructed by a method which fully exploits problem sparsity. The algorithms were implemented in the Julia programming language within the EAGO.jl package and demonstrated on five illustrative examples with varying complexity relevant in process systems engineering. The developed methods enable the guaranteed global solution of dynamic optimization problems with stiff ODE–IVPs embedded. [ABSTRACT FROM AUTHOR]

Systems of ordinary differential equations (ODE) or ordinary differential/algebraic equations (DAE) are well-known mathematical models. The numerical solution of such systems are discussed. For (ODE) we mention some available codes and stress the need of type insensitive versions. Further the term stiffness is redefined, and ideas on handling discontinuities are presented. The paper ends with a discussion of index for DAE.

We propose the S-leaping algorithm for the acceleration of Gillespie's stochastic simulation algorithm that combines the advantages of the two main accelerated methods; the [Formula: see text]-leaping and R-leaping algorithms. These algorithms are known to be efficient under different conditions; the [Formula: see text]-leaping is efficient for non-stiff systems or systems with partial equilibrium, while the R-leaping performs better in stiff system thanks to an efficient sampling procedure. However, even a small change in a system's set up can critically affect the nature of the simulated system and thus reduce the efficiency of an accelerated algorithm. The proposed algorithm combines the efficient time step selection from the [Formula: see text]-leaping with the effective sampling procedure from the R-leaping algorithm. The S-leaping is shown to maintain its efficiency under different conditions and in the case of large and stiff systems or systems with fast dynamics, the S-leaping outperforms both methods. We demonstrate the performance and the accuracy of the S-leaping in comparison with the [Formula: see text]-leaping and R-leaping on a number of benchmark systems involving biological reaction networks.