New Multivalue Methods for Differential Algebraic Equations

Multivalue methods are slightly different from the general linear methods John Butcher proposed over 30 years ago. Multivalue methods capable of solving differential algebraic equations have not been developed. In this paper, we have constructed three new multivalue methods for solving DAEs of index 1, 2 or 3, which include multistep methods and multistage methods as special cases. The concept of stiff accuracy will be introduced and convergence results will be given based on the stage order of the methods. These new methods have the diagonal implicit property and thus are cheap to implement and will have order 2 or more for both the differential and algebraic components. We have implemented these methods with fixed step size and they are shown to be very successful on a variety of problems. Some numerical experiments with these methods are presented.