Error Estimates and Adaptive Time-Step Control for a Class of One-Step Methods for Stiff Ordinary Differential Equations

We develop a two-step hybrid block method for the solution of stiff and oscillatory first-order Ordinary Differential Equations (ODEs) using the Laguerre polynomial as our basis function via interpolation and collocation techniques. The paper further investigates the basic properties of the method and found it to be zero-stable, consistent and convergent. The method was also tested on some sampled stiff and oscillatory problems and found to perform better than some existing ones with which we compared our results.

In this paper, we derive new method for solving particular solution of linear second order ordinary differential equations whenever one solution of their associated homogeneous differential equations is given. Also, we construct second solution of the associated homogeneous differential equation from this new method. Moreover, we have general solution method of linear second order ordinary differential equations without applying the two famous methods, undetermined coefficients method and variation of parameters method, for solving their particular solution

Many classes of methods for numerical problems can be characterized by parameters. For example, Adams methods for integrating ordinary differential equations can be characterized by the step size and order. In many cases it is known that a method is stable for fixed values of those parameters, whereas many programs vary those parameters in order to reduce computation time. This paper examines the effects on stability of changes to two kinds of parameters—“discrete” and “continuous” — corresponding to the parameters of order and step size in the numerical solution of ordinary differential equations. Conditions are given that are sufficient to ensure that these changes do not affect stability. These results are applied to two classes of automatic methods for ordinary differential equations—one based on unequal interval quadrature formulas, the other based on interpolation and equal interval quadrature formulas.

Most methods used for the numerical solution of ordinary differential equations may be characterized by a pair of matrices $(A,B)$. In this article, local error estimates for such methods are obtained as linear combinations of computed values. (This is often equivalent to constructing pairs of embedded methods.) Although these local error estimates are obtained by assuming that exact solution values are used to start the step, it is shown that this assumption is not required for certain estimates. Some hybrid methods have such (computable) estimates, and also have simple step length change procedures.

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge---Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge---Kutta stability is also briefly discussed

This chapter deals with the numerical solution of initial value problems of ordinary differential equations. First, different approaches to obtain integration methods for ordinary differential equations are presented. Then the important properties consistency, stability, and convergence of integration methods are introduced and studied. This allows for a fruitful application of the results obtained in Chapter 12. Subsequently, one-step methods are considered in detail and their important properties are shown. Moreover, Runge–Kutta methods are thoroughly investigated. Finally, we deal with linear multistep methods and study the asymptotic behavior of integration methods and of stiff differential equations. In particular, we deal with Dahlquist’s root condition and derive the first Dahlquist barrier and the second Dahlquist barrier.

Tau leaping of stiff stochastic chemical systems via local central limit approximation

To develop and evaluate an automatic train control algorithm for high-speed trains aiming at energy efficiency, timetable accuracy and increased passenger safety and comfort. Methods: An analytical and computer-based approach was used to design an automatic control algorithm. The finite element method was used to simulate an electric train movement. The algorithm has been implemented in Python language using NumPy, Pandas and Matplotlib libraries. The Sapsan high-speed train operation data have been analysed and a comparative analysis of experimental and computational data has been carried out. Results: The automatic train control algorithm has been developed to improve energy efficiency and timetable accuracy, as well as to reduce the train driver impact on energy consumption. The effect of energy savings up to more than 11% in conditions of different traffic volumes due to the implementation of the system has been recorded. The results of the motion curve simulation compared with experimental operating data are presented. Practical significance: The implementation of the proposed automatic control algorithm will improve the energy efficiency of high-speed trains, enhance their operational characteristics, increase the train timetable accuracy and upgrade the level of passenger comfort. The developed system can be used on railways with high traffic intensity to increase railway capacity without upgrading the infrastructure.

On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D

This paper presents the development and analysis of a continuous class of A-stable Block Generalized Backward Differentiation Formulae (BBDF) for the numerical solution of stiff ordinary differential equations (ODEs). The proposed methods, denoted as 4SCBGBDF and 4SHBGBDF, extend the conventional BBDF framework by incorporating continuous interpolation functions, which facilitate the generation of dense output without incurring additional computational cost. The block structure of these methods enables the simultaneous computation of multiple solution points within a single step, thereby enhancing computational efficiency and solution accuracy. A rigorous stability analysis confirms the A-stability of both methods, affirming their suitability for stiff initial value problems. Numerical experiments conducted on standard benchmark stiff problems validate the theoretical properties and demonstrate the superior performance of the proposed methods in terms of stability, accuracy, and computational efficiency. The results underscore the potential of continuous A-stable BBDF schemes as robust and reliable tools for solving stiff systems arising in scientific and engineering applications.

A gradient-based approach to multidisciplinary design optimization enables efficient scalability to large numbers of design variables. However, the need for derivatives causes difficulties when integrating ordinary differential equations (ODEs) in models. To simplify this, we propose the use of the general linear methods framework, which unifies all Runge-Kutta and linear multistep methods. This approach enables rapid implementation of integration methods without the need to differentiate each one, even in a gradient-based optimization context. We also develop a new parallel time integration algorithm that enables vectorization across time steps. We present a set of benchmarking results using a stiff ODE, a non-stiff nonlinear ODE, and an orbital dynamics ODE, and compare integration methods. In a modular gradient-based multidisciplinary design optimization context, we find that the new parallel time integration algorithm with high-order implicit methods, especially Gauss-Legendre collocation, is the best choice for a broad range of problems.

An approximate solution method for ordinary fractional differential equations with the Riemann–Liouville fractional derivatives

A long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation, obtained at the mesh points which subdivide the problem interval. It is well known that the use of collocation at Gauss points leads to solution approximations at the mesh points for which the global error is O(h2k), where k is the number of collocation points used per subinterval and h is the subinterval size. This discrete solution is said to be superconvergent. The collocation solution also yields a C0 continuous solution approximation that has a global error of O(hk+1). In this paper, we show how to efficiently augment the superconvergent discrete collocation solution to obtain C1 continuous "superconvergent" interpolants whose global errors are O(h2k ). The key ideas are to use the theoretical framework of continuous Runge--Kutta schemes and to augment the collocation solution with inexpensive monoimplicit Runge--Kutta stages. Specific schemes are derived for k = 1, 2, 3, and 4. Numerical results are provided to support the theoretical analysis.

In this article, we consider diagonal-implicitly iterated Runge-Kutta (DIIRK) methods for the numerical solution of stiff ordinary differential equations (ODEs) and investigate their performance behavior on a modern cluster system using MPI. DIIRK methods are implicit methods and require the solution of non-linear equation systems in each iteration step. In particular, we are interested in the parallel execution behavior when using different basis Newton methods for solving the resulting non-linear equation systems of different versions of the DIIRK method. We explore the use of direct solution methods based on LU factorization for the resulting linear equation systems as well as the use of Krylov subspace methods and investigate the resulting performance and accuracy.

This paper describes some methods for the solution of ordinary differential equations based on the Lanczos "selected points" or collocation principle. Some properties of the methods are examined theoretically, and some examples of their use given.