Numerical Solution of Ordinary and Delay Differential Equations by Runge-Kutta Type Methods

Runge-Kutta methods for the solution of systems of ordinary differential equations (ODEs) are described. To overcome the difficulty in implementing fully implicit Runge-Kutta method and to avoid the limitations of explicit Runge-Kutta method, we resort to Singly Diagonally Implicit Runge-Kutta (SDIRK) method, which is computationally efficient and stiffly stable. Consequently, embedded SDIRK methods of fourth order five stages in fifth order six stages are constructed. Their regions of stability are presented and numerical results of the methods are compared with the existing methods. Stiff systems of ODEs are solved using implicit formulae and require the use of Newton-like iteration, which needs a lot of computational effort. If the systems can be partitioned dynamically into stiff and nonstiff subsystems then a more effective code can be developed. Hence, partitioning strategies are discussed in detail and numerical results based on two techniques to detect stiffness using SDIRK methods are compared. A brief introduction to delay differential equations (DDEs) is given. The stability properties of SDIRK methods, when applied to DDEs, using Lagrange interpolation to evaluate the delay term, are investigated. Finally, partitioning strategies for ODEs are adapted to DDEs and numerical results based on two partitioning techniques, interval wise partitioning and componentwise partitioning are tabulated and compared.

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