Second derivative general linear methods (SGLMs) as an extension of general linear methods (GLMs) have been introduced to improve the stability and accuracy properties of GLMs.The coefficients of SGLMs are given by six matrices, instead of four matrices for GLMs, which are obtained by solving nonlinear systems of order and usually Runge-Kutta stability conditions. In this paper, we introduce a technique for construction of an special case of SGLMs which decreases the complexity of finding coefficients matrices.

In this paper, we introduce Second derivative multistep collocation meth-ods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both multistep methods and col-location methods, using Second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the Second-order deriva-tives leads to high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the Second derivative of the solution in the colloca-tion methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expecta-tions. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods.

In the construction of efficient numerical methods for the stiff initial value problems, some Second derivative multistep methods have been introduced equipping with super future point technique. In this paper, we are going to introduce a formula for the stability functions of a class of such methods. This group of methods encompasses SDBDF methods and their extensions with advanced step-point feature. This general formula, instead of obtaining the distinct stability functions for each of methods, will facilitate stability analysis of the methods.

Variable-step (VS) Second derivativek -step 3-stage Hermite-Birkhoff-Obrechkoff (HBO) methods of orderp= (k+3), denoted by HBO (p) are constructed as a combination of lineark -step methods of order (p- 2) and a Second derivative two-step diagonally implicit 3-stage Hermite-Birkhoff method of order 5 (DIHB5) for solving stiff ordinary differential equations. The main reason for considering this class of formulae is to obtain a set of k -step methods which are L -stable and are suitable for the integration of stiff differential systems whose Jacobians have some large eigenvalues lying close to the imaginary axis with negative real part. The approach, described in the present paper, allows us to developL -stable k -step methods of order up to 10. Selected HBO (p) of order p, p=9; 10, compare favorably with existing Cash L-stable Second derivative extended backward differentiation formulae, SDEBDF (p), p=7; 8 in solving problems often used to test stiff ODE solvers.