Diagonally implicit two and three derivative Runge-Kutta methods for solving first order oscillatory ordinary and delay differential equations

In this study, Diagonally Implicit Two Derivative Runge-Kutta (DITDRK) methods and Diagonally Implicit Three Derivative Runge-Kutta (DIThDRK) methods are constructed for the numerical integration of first-order Initial Value Problems (IVPs). For DITDRK methods, the methods derived are also used in the solution of stiff Ordinary Differential Equations (ODEs) and Delay Differential Equations (DDEs). Three new methods with a minimum number of function evaluations are derived for DITDRK methods. Meanwhile for DIThDRK methods also, three new methods are constructed with a minimum number of function evaluations. Solving ODEs which have periodic or oscillatory solutions in nature are more convenient with the implementation of trigonometrically-fitted and phase-fitted and amplification-fitted techniques. Hence, taking this idea into account, we implemented these techniques into DITDRK and DIThDRK methods. Two new methods each for DITDRK and DIThDRK methods for both oscillatory techniques are derived. They are fourth and fifth-order for DITDRK methods and sixth and seventh-order for DIThDRK methods. The Local Truncation Error (LTE) for each method is computed. Stiff system of ODEs are solved using implicit formulae and required the use of Newton-like iteration, which needs a lot of computational effort. Here, we focused on the derivation of DITDRK methods for both constant and variable step-size. For constant step-size, three new methods of order three, four and six are constructed. For variable step-size, two new embedded methods of 3(2) and 4(3) DITDRK methods are derived. The stability of these methods are discussed along with their stability regions. A brief introduction on Delay Differential Equations (DDEs) is given. The stability properties of DITDRK methods when applied to DDEs, using Lagrange interpolation to evaluate the delay term are investigated. The P-stability and Q-stability of fourth and fifth-order DITDRK methods are discussed along with the boundary of the region. In solving first-order DDEs, Newton Divided Difference Interpolation (NDDI) is used to approximate the delay term. As for solving periodic DDEs, we use Trigonometric interpolation which is specially design to solve oscillatory problems due to its periodic properties. Hence, two methods of fourth and fifth-order Trigonometrically-Fitted DITDRK (TFDITDRK) methods are used to solve these types of problems. Numerical experiments show that the newly derived methods are more efficient and accurate in comparison with existing Diagonally Implicit Runge-Kutta (DIRK) methods of the same order and properties in the literature in terms of maximum global error, number of function evaluation per step and execution time.

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