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We note a fact that stiff systems or differential equations that have highly oscillatory solutions cannot be solved efficiently using conventional methods. In this paper, we study two new classes of exponential Runge–Kutta (ERK) integrators for efficiently solving stiff systems or highly oscillatory problems. We first present a novel class of explicit modified version of exponential Runge–Kutta (MVERK) methods based on the order conditions. Furthermore, we consider a class of explicit simplified version of exponential Runge–Kutta (SVERK) methods. Numerical results demonstrate the high efficiency of the explicit MVERK integrators and SVERK methods derived in this paper compared with the well-known explicit ERK integrators for stiff systems or highly oscillatory problems in the literature. [ABSTRACT FROM AUTHOR]

In this manuscript, a new class of high-order multistep methods on the basis of hybrid backward differentiation formulas (BDF) have been illustrated for the numerical solutions of systems of ordinary differential equations (ODEs) arising from semi-discretization of time dependent partial differential equations. Order and stability analysis of the methods have been discussed in detail. By using an off-step point together with a step point in the first derivative of the solution, the new methods obtained are A-stable for order p, (p = 4 , 5 , 6 , 7) and A (α )-stable for order p, (p = 8 , 9 , ... , 14). Compared to the existing BDF based method, i.e. class 2 + 1 , hybrid BDF methods (HBDF), super-future points based methods (SFPBM) and MEBDF, there is a good improvement regarding to absolute stability regions and orders. Some numerical examples are given in order to check the advantage of these methods in reducing the CPU time and thus in increasing accuracy of low and high order the new methods compared to those of SFPBM and MEBDF. [ABSTRACT FROM AUTHOR]

Implicit block approaches are used by a number of numerical analyzers to model mild, medium, and hard differential systems. Their excellent stability characteristics, self-starting nature, quick convergence, and large decrease in computing cost all contribute to their widespread application. With these numerical benefits in mind, a new one-step implicit block method with three intrastep grid points has been created. The major term of the local truncation error is minimized to determine which of these points is optimal. The reformulation of the suggested technique leads to a significant decrease in computing cost while maintaining the same consistency, zero-stability, A -stability, and convergence. Several sorts of error are calculated, together with CPU time and efficiency plot, to determine which is superior. Differential models from the fields of heat transfer, population dynamics, and chemical engineering show that the suggested method does a better job than some of the current hybrid block and implicit Radau methods with similar properties. [ABSTRACT FROM AUTHOR]

A hybrid second derivative three-step method of order 7 is proposed for solving first order stiff differential equations. The complementary and main methods are generated from a single continuous scheme through interpolation and collocation procedures. The continuous scheme makes it easy to interpolate at off-grid and grid points. The consistency, stability, and convergence properties of the block formula are presented. The hybrid second derivative block backward differentiation formula is concurrently applied to the first order stiff systems to generate the numerical solution that do not coincide in time over a given interval. The numerical results show that the new method compares favorably with some known methods in the literature. [ABSTRACT FROM AUTHOR]

A new class of hybrid BDF-like methods is presented for solving systems of ordinary differential equations (ODEs) by using the second derivative of the solution in the stage equation of class 2 + 1hybrid BDF-like methods to improve the order and stability regions of these methods. An off-step point, together with two step points, has been used in the first derivative of the solution, and the stability domains of the new methods have been obtained by showing that these methods are A-stable for order p, p = 3,4,5,6,7and A(α)-stable for order p, 8 ≤ p ≤ 14. The numerical results are also given for four test problems by using variable and fixed step-size implementations. [ABSTRACT FROM AUTHOR]

In many applications, large systems of ordinary differential equations with both stiff and nonstiff parts have to be solved numerically. Implicit-explicit (IMEX) methods are useful for efficiently solving these problems. In this paper, we construct IMEX second-derivative BDF methods with considerable stability properties. To show the efficiency of the introduced technique, numerical comparisons are given by solving some problems. [ABSTRACT FROM AUTHOR]

In Mitsui (Sci Eng Rev Doshisha Univ Jpn 51(3):181-190, 2010) the author introduced a new class of discrete variable methods known as look-ahead linear multistep methods (LALMMs) which consist of a pair of predictor-corrector (PC), including a function value at one more step beyond the present step for the numerical solution of ordinary differential equations. Two-step family of Look-Ahead linear multistep methods of fourth-order pair were derived and shown to be A( $$\theta $$ )-stable in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181-188, 2011). The derived integration methods are of low orders and unfortunately cannot cope with stiff systems of ordinary differential equations. In this paper we extend the concept adopted in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181-188, 2011) to construct second-derivative of high-order accuracy methods with off-step points which behave essentially like one-step methods. The resulting integration methods are A-stable, convergent, with large regions of absolute stability, suitable for stiff systems of ordinary differential equations. Numerical comparisons of the new methods have been made and enormous gains in efficiency are achieved. [ABSTRACT FROM AUTHOR]

Two-Step Second-Derivative High-Order Methods with off-step points suitable for the approximate numerical integration of stiff systems of first order initial value problems in ordinary differential equations are developed. The second derivative terms in the methods provide more freedom to derive a set of methods which are highly stable, convergent, with large regions of absolute stability. Although the cost of calculating the second derivatives may be higher than the first derivatives, the advantage gained makes them suitable for solving stable systems with large Lipschitz constants. Another requirement demands that the second derivative of high-order methods are to be solved iteratively rather than directly in the usual way of the linear multistep methods, however, efficient methods for doing this is provided from the associated continuous scheme of each method, by the evaluation of the continuous scheme at some off-step points. The derived methods are illustrated by the applications to some test problems of stiff systems and the numerical results obtained confirm the accuracy and efficiency of these methods. [ABSTRACT FROM AUTHOR]

In this article, the details of new hybrid methods have been presented to solve systems of ordinary differential equations (ODEs). These methods are based on backward differentiation formulae (BDF) where one additional stage point (or off-step point) and two step points have been used in the first derivative of the solution to improve the absolute stability regions compared with some existing methods such as BDF, extended BDF (EBDF) and modified EBDF (MEBDF). Stability domains of our new methods have been obtained showing that these methods, we say Class 2+1 Hybrid BDF-Like methods, are A-stable for order p, p=3,4, and A( α)-stable for order p, p=5, 6, 7, 8. Numerical results are also given for five test problems. [ABSTRACT FROM AUTHOR]

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]

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