Very High-Order A-Stable Stiffly Accurate Diagonally Implicit Runge-Kutta Methods with Error Estimators

A numerical search approach is used to design high-order diagonally implicit Runge-Kutta (DIRK) time stepping schemes equipped with embedded error estimators, some of which have identical diagonal elements (i.e., SDIRK) and explicit first stage (i.e., ESDIRK). In each of these classes, we present new A-stable schemes of order six (the highest order of previously known A-stable DIRK-type schemes) up to order eight. For each order, we include one scheme that is only A-stable as well as several schemes that are L-stable, stiffly accurate, and/or have stage order two. The latter types require more stages, but yield better convergence rates for differential-algebraic equations (DAEs), and particularly those which have stage order two result in better accuracy for moderately stiff problems. The development of the eighth-order schemes requires, in addition to imposing A-stability, finding highly accurate numerical solutions for a system of 200 equations in over 100 variables, which is accomplished via a combination of global and local optimization strategies. The accuracy, stability, and adaptive stepsize control of the schemes are demonstrated on diverse problems.