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A numerical search approach is used to design high-order diagonally implicitRunge-Kutta (DIRK) schemes equipped with embedded error estimators, some ofwhich have identical diagonal elements (SDIRK) and explicit first stage(ESDIRK). In each of these classes, we present new A-stable schemes of ordersix (the highest order of previously known A-stable DIRK-type schemes) up toorder eight. For each order, we include one scheme that is only A-stable aswell as schemes that are L-stable, stiffly accurate, and/or have stage ordertwo. The latter types require more stages, but give better convergence ratesfor differential-algebraic equations (DAEs), and those which have stage ordertwo give better accuracy for moderately stiff problems. The development of theeighth-order schemes requires, in addition to imposing A-stability, findinghighly accurate numerical solutions for a system of 200 equations in over 100variables, which is accomplished via a combination of global and localoptimization strategies. The accuracy, stability, and adaptive stepsize controlof the schemes are demonstrated on diverse problems.

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