Invariants and numerical methods for ODEs

The preservation of some invariants is important during the numerical integration of ODEs. In some cases, failure to maintain certain invariants leads to physically impossible solutions, in other cases to instability. Some authors report that the long term macroscopic characteristics of a solution are better represented if certain invariants are maintained. This paper considers two types of invariants, point invariants which are functions along a particular trajectory, and differential invariants which are invariant relations between neighboring trajectories, such as symplectic invariants. Conventional numerical methods usually introduce O(h^p+1) errors in an invariant at each numerical step and these will accumulate over the integration interval. Two approaches to preventing this accumulation are (1) to find methods that maintain the invariants within round-off error, or (2) to find methods that maintain another invariant which is no more than O(h^q) different from the invariant satisfied by the ODE. These approaches will be illustrated with some examples. The major objective of this paper is to consider the potential of general methods (that is, methods that do not depend on the specific differential equation) to maintain the invariants.