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Abstract
Several real life problems that are modelled as stiff ordinary differential equations or differential algebraic equations cannot be solved exactly, and these have led to the development of numerical methods. Solving stiff ordinary differential equations and differential algebraic equations require numerical methods having good stability properties. In this study, new classes of highly stable general linear methods that are nested in their stages and mono-implicit in their output for the numerical integration of stiff ordinary differential equations and differential algebraic equations are presented as nested general linear methods. Nested general linear methods that preserve the asymptotic solution of differential equations, inherent Runge-Kutta stable members and algebraically stable members are derived herein. The methods have been derived by using the order conditions obtained by using Taylor series alongside its structural assumption. The order conditions are then simplified and these leave a number of free parameters which are chosen to achieve the desired stability. The first family of methods derived are strongly regular. Two families of nested general linear methods with inherent Runge-Kutta stability have been derived. The third family of methods constructed are algebraically stable. The nested general linear methods have also been extended to second derivative nested general linear methods and L-stable members derived. The proposed methods have been implemented using fixed step size and variable step size and compared with the results of other existing methods including that of MATLAB ode15s, MATLAB ode23tb and RADUA IIA. On implementation, the numerical order of the proposed nested general linear methods agrees with its theoretical order, this affirms that the proposed methods do not suffer from order reduction and are suitable for solving stiff ordinary differential equations and differential algebraic equations.
