Derivation of Five-Stage Implicit Runge-Kutta Method of Order 10 via Gauss-Legendre Quadrature for Ordinary Differential Equations

This paper presents developing and implementing a five-stage implicit Runge-Kutta method of order ten via Gauss-Legendre quadrature for stiff or oscillatory first-order initial value problems (IVPs) of ordinary differential equations (ODEs). Using continuous collocation and interpolation techniques, the implicit Runge-Kutta method was developed by combining Legendre polynomials and exponential functions as the basis function. The properties of the method were investigated, and it was shown that it is consistent and A-stable. The new method was evaluated on two sampled problems involving stiffness and oscillation. The numerical results demonstrate that the new implicit Runge-Kutta is computationally efficient and outperforms previous methods of similar derivations.&nbsp