A novel two‐parameter class of optimized hybrid block methods for integrating differential systems numerically

Abstract

In this article, a two-parameter class of hybrid block methods for integrating first-order initial value ordinary differential systems is proposed. The methods exhibit hybrid nature which helps in bypassing the first Dahlquist barrier existing for linear multistep methods. The approach used in the development of a class of methods is purely interpolation and collocation technique. The class of methods is based on four intra-step points from which two intra-step points have been optimized by using an optimization strategy. In this optimization strategy, the values of two intra-step points are obtained by minimizing the local truncation errors of the formulas at the points x n+1∕2 and x n+1 .The order of accuracy of the proposed methods is six. A method as a special case of this class of methods is considered and developed into a block form which produces approximate numerical solutions at several points simultaneously. Further, the method is formulated into an adaptive step-size algorithm using an embedded type procedure. This method which is a special case of this class of methods has been tested on six well-known first-order differential systems. K E Y W O R D Sblock methods, embedded-type procedure, hybrid methods, ordinary differential equations INTRODUCTIONNumerous numerical integrators are available for solving first-order initial value ordinary differential systems numerically. Runge-Kutta and linear multistep methods are the usual classes of methods that are being used for solving these systems numerically. These days many built-in computer codes based on these classes of methods are available in software like Matlab, Mathematica, and so forth. To develop more accurate and efficient codes for solving the differential systems is an ongoing process of research. This article is concerned with the development of a class of optimized codes for solving the following system:whereTo proceed further, we assume that the system (1) has a unique continuously differentiable solution that we denote by Y(x). As a first step, the interval of interest [x 0 , x N ] is discretized as follows:x n = x 0 + nh, n = 0, 1, 2, 3, … , N; h = x n+1 − x n and an approximate numerical solution of the system (1) at x n is denoted as Y n ≈ Y(x n ).

Development of a Higher-Order 𝒜-Stable Block Approach with Symmetric Hybrid Points and an Adaptive Step-Size Strategy for Integrating Differential Systems Efficiently

“…Step 2: In order to obtain optimized values for r 1 and r 3 , we evaluate expression (3) at x = x j+1 and x = x j+r 2 . This allows us to approximate the true solution at the final point and at the midpoint of the interval [x j , x j+1 ], denoted as x j+1 and x j+r 2 , respectively, in terms of r 1 and r 3 , as shown in [15]. The evaluation of z(x j+1 ) and z(x j+r 2 ) can be readily obtained through a CAS, although the resulting expressions can be quite lengthy, and so they are not presented here.…”

A Single-Step Modified Block Hybrid Method for General Second-Order Ordinary Differential Equations

“…This particular research is anchored by the benefit of single-step methods, which by themselves are self-starting, and the usage of block methods as a collection of simultaneous integrators without relying on any way to generate starting values. Additionally, the methodology used in the recent studies by Adee, Kumleng and Patrick (2022), and Singla, Singh, Ramos and Kanwar, (2022), in which block hybrid methods were implemented as a collection of numerical integrators for first-order IVPs of ODEs on non-overlapping subintervals, is employed in this study.…”