In the numerical solution of initial value problems in ODEs, a state-of-the-art method will now deliver an approximation to the true solution at any point x in the interval of interest, [a, b], and not just at the adaptively selected discrete meshpoints. For Runge-Kutta methods, these off-mesh approximations are usually generated using a continuous extension (CRK) of an underlying discrete RK formula. In the development of optimal-order CRK methods, the focus has been on developing CRKs for explicit discrete RK formulas. In this investigation we will consider and discuss in detail the issues that arise when developing and implementing reliable, optimal order CRKs for implicit RK methods. In particular we will derive and justify suitable error control and step selection stratagies for these methods. The ability of these new methods to effectively approximate the solution of stiff IVPs and delay differential equations will be demonstrated.

When solving differential delay equations (DDEs) with state- or time-dependent delays, the presence of vanishing or near vanishing delays can make it impossible or impractical to use step-sizes which are smaller than the delays. The phenomenon of the delay being smaller than the step-size is referred to as "overlapping". When overlapping occurs delays fall in the current computational step, and the corresponding solution values are computed by interpolating between the stage values. In this case even explicit methods become fully implicit. Runge-Kutta methods which remain explicit even in the case of overlapping were first introduced by Tavernini for Volterra functional differential equations, and more recently developed as functional continuous Runge-Kutta methods by Maset and co-authors. Diagonally implicit Runge-Kutta methods can have better stability than explicit methods, but were not previously explored for state-dependent DDEs because standard methods become fully implicit in the case of overlapping. In the current work we apply and extend the techniques of the Italian school to create diagonally implicit Runge-Kutta methods which retain their diagonally implicit character even in the case of overlapping. Various methods and order bounds will be presented, including fourth order methods.

The nonlinearity and stiffness incorporated in the turbulent mixing of atmospheric boundary layer models is dealt with in this study using optimal total-variation-diminishing (TVD) singly-diagonally-implicit Runge-Kutta (SDIRK) methods which have been proved to be effective for the mentioned equations in the literature. Various aspects of these schemes, including stability properties, linear stability analysis, and numerical experiments, are studied with regard to their application on the time integration of well-known nonlinear damping and diffusive systems in atmospheric boundary layer models. At the end, two schemes, which are modified versions of the optimal TVD SDIRK schemes, are proposed to improve their performance and features. They exhibit significant improvements with respect to the schemes studied.

Current electrophysiological models vary greatly in both complexity and accuracy. These models often involve a set of partial differential equations coupled with a large number of stiff ordinary differential equations (ODEs). The system of ODEs attempt to simulate the flow of ionic currents present in the cellular level of the heart and has been continually developed to provide an increasingly detailed description of cellular physiology. However, the stiffness has the effect of decreasing the speed of the solving process since the stability is often the limiting factor for the solution. An efficient method is needed to significantly decrease computational time in the numerical solution of two and three dimensional models of the electrical activity of the myocardium. In this presentation, we will be exploring the usage and efficiency of Gauss-type nested implicit Runge-Kutta technique to solve cardiac cell models. The method is of order 4 and has only explicit internal stage that leads to practical implementations. Comparison with other numerical methods employed in the context of electrocardiology will be presented.