Third- and fourth-order ESDIRK methods for stiff and differential-algebraic problems

Stiffly accurate singly diagonally implicit Runge–Kutta methods with an explicit first stage (ESDIRK) deigned for solving stiff ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) are considered. An advantage of these methods is that they are easy to implement, but they have only the second stage order, which limits the possibility of constructing efficient methods of high orders. ESDIRK methods are most efficient in computations of relatively low accuracy, which is sufficient for solving most practical problems. Accordingly, we consider third- and fourth-order methods, which produce solutions with low computational costs under moderate requirements for accuracy. New methods satisfying certain additional conditions are proposed, which effectively solve not only stiff ODEs, but also DAEs of indices 2 and 3. An implementation of the methods with automatic stepsize selection is discussed, and results of numerical experiments are presented.