Efficient classes of Runge-Kutta methods for two-point boundary value problems

In this paper we study a class of explicit pseudo two-step Runge-Kutta (EPTRK) methods for first-order ODEs for parallel computers. We investigate linear stability and derive methods with enlarged stability regions. In numerical experiments on a shared memory computer we compare a parallel variable step size EPTRK implementation with the effcient sequential Runge-Kutta method dopri5.

In the previous chapter, we introduced explicit Runge-Kutta (ERK) methods and demonstrated how they can be implemented as a hierarchy of Python classes. For most ODE systems, replacing the simple forward Euler method with a higher-order ERK method will significantly reduce the number of time steps needed to reach a specified accuracy. Furthermore, it often leads to reduced computation time, since the additional cost per time step is outweighed by the reduced number of steps. However, there exists a class of ODEs known as stiff systems, where all the ERK methods require very small time steps, and any attempt to increase the time step leads to spurious oscillations and possible divergence of the solution. Stiff ODE systems pose a challenge for explicit methods, and they are better addressed by implicit solvers such as implicit Runge-Kutta (IRK) methods. IRK methods are well-suited for stiff problems and can offer substantial reductions in computation time when tackling challenging problems.

In this paper, explicit Runge–Kutta methods are investigated for numerical solutions of nonlinear dynamical systems with conserved quantities. The concept, ε-preserving is introduced to describe the conserved quantities being approximately retained. Then, a modified version of explicit Runge–Kutta methods based on the optimization technique is presented. With respect to the computational effort, the modified Runge–Kutta method is superior to implicit numerical methods in the literature. The order of the modified Runge–Kutta method is the same as the standard Runge–Kutta method, but it is superior in preserving the conserved quantities to the standard one. Numerical experiments are provided to illustrate the effectiveness of the modified Runge–Kutta method.

This paper is aimed at analyzing the consistency and convergence of a derived explicit fourth-stage fourth-order Runge-Kutta method. The analysis revealed that the method is consistent and convergent. The implementation of this method on initial-value problems was done in previous paper, and it revealed that the method compared favorably well with the existing classical fourth stage fourth order explicit Runge Kutta method.

Stability of some continuously imbedded Runge-Kutta methods of Sarafyan

On the economization of explicit Runge-Kutta methods

We propose a unified theoretical framework to examine the energy dissipation properties at all stages of explicit exponential Runge-Kutta (EERK) methods for gradient flow problems. The main part of the novel framework is to construct the differential form of EERK method by using the difference coefficients of method and the so-called discrete orthogonal convolution kernels. As the main result, we prove that an EERK method can preserve the original energy dissipation law unconditionally if the associated differentiation matrix is positive semi-definite. A simple indicator, namely average dissipation rate, is also introduced for these multi-stage methods to evaluate the overall energy dissipation rate of an EERK method such that one can choose proper parameters in some parameterized EERK methods or compare different kinds of EERK methods. Some existing EERK methods in the literature are evaluated from the perspective of preserving the original energy dissipation law and the energy dissipation rate. Some numerical examples are also included to support our theory.

The stability region of an explicit and consistentm-stage Runge-Kutta method cannot contain the closed disk with diameter [−2m, 0] as a proper subset.

A methodology that can generate the optimal coefficients of a numerical method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a specific system of ordinary differential equations numerically. The case we are examining here concerns an explicit two-stage Runge-Kutta method for the numerical solution of the two-body problem. Following the implementation of the network, the latter is trained to obtain the optimal values for the coefficients of the Runge-Kutta method. The comparison of the new method to others that are well known in the literature proves its efficiency and demonstrates the capability of the network to provide efficient algorithms for specific problems.

When the standard Runge-Kutta method is applied to a certain system of linear differential equations, the numerical solution converges to the origin but the true solution diverges to infinity. This phenomenon is named the abnormal convergence of numerical solutions. Abnormal convergence has not been studied fully; however, many concepts involving A-stability have been established to treat the phenomenon that the numerical solution diverges even though the true one converges, as is apt to occur in integration of a stiff system. An example will be illustrated here in which the standard Runge-Kutta method generates abnormal convergence. Further, it will be proved that the abnormality is caused by the existence of an intersection of the absolute stability region and the right-half plane. Even if the 1/8 formula method is used instead of the standard one, the same phenomenon can be seen because all four-stage, fourth-order, explicit Runge-Kutta methods have the unique absolute stability region. This paper seeks for Runge-Kutta methods that are A-stable and never generate abnormal convergence, even though this might be an excessive requirement. Concretely, two types of methods will be proposed whose absolute stability region coincides with the left-half plane. Furthermore, it will be reported that Gauss-Legendre methods are typical ones.

In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). The methods considered herein incorporate only the first and second derivatives terms of ODEs. These methods possess large interval of stability when compared with other existing methods in the literature. The experiments have been performed on standard problems, and comparisons were made with some standard explicit Runge-Kutta methods in the literature.

The one-step finite-difference methods of the previous chapter, with the possible exception of the trapezoidal rule method, are basically rather too uneconomic for general use. Their local truncation error is of rather low order, with the result that to achieve good accuracy we need to use either: (a) a rather small step-by-step interval, involving many steps to cover a specified range; or (b) one of our correcting devices which involve some additional computation and more computer programming. The special one-step methods of the previous chapter, the Taylor series method and the explicit Runge-Kutta methods, do not share these disadvantages, but they involve extra computer storage, the Taylor series method involves a possibly large amount of non-automatic differentiation, and with the Runge—Kutta methods we have to compute possibly rather complicated expressions several times in each step. Moreover, both these methods have rather poor partial stability properties.

Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods

An adaptive finite volume method for the simulation of time-dependent, viscous flow is presented. The Navier–Stokes equations are discretized by central schemes on unstructured grids and solved by an explicit Runge–Kutta method. The essential topics of the present study are a new concept for a local Runge–Kutta time-stepping scheme, called multisequence Runge–Kutta, which reduces the severe stability restriction in unsteady problems, a common grid generation and adaptation procedure and the application of dynamic grids for capturing moving flow structures. Results are presented for laminar, separated flow around an aerofoil with a flap.

In the paper we consider the systems of delay differential equations, which are mathematical models of many technical processes with delay in time. An explicit fifth-order hybrid method for systems of delay differential equations for the variable step of numerical integration is constructed. This algorithm is based on the most commonly used explicit fifth-order Runge-Kutta methods for ordinary systems of differential equations and the construction of Newton polynomials for the prehistory of the model and Taylor formula. The basic principles of construction of such explicit hybrid methods of higher orders of convergence are given. An accurate estimate of the local error of numerical integration by this method is obtained. This algorithm allows the use of numerical integration steps greater than the magnitude of the delay or procedures for adjusting the magnitude of the step depending on the calculation error. Such a problem of numerical integration of time-delay dynamical systems occurs when the time interval is large enough compared to the delay. Implicit continuous Runge-Kutta methods were previously used for numerical integration such problems, which complicated the numerical algorithm, because nonlinear systems of equations have to be solved at each step of numerical integration. The explicit hybrid method constructed by us for systems of delay differential equations is convenient for programming, has a high speed of calculation of numerical solution, in comparison with implicit methods for such models of problems. Also, this method has no restrictions on the magnitude of the delay, in contrast to the hybrid methods with decomposition in Taylor series for delay. The method obtained by us can be used to construct maps of dynamic modes in the study of regular and chaotic behavior of time-delay dynamical systems.