Exactly Conservative Integrators

Traditional explicit numerical discretizations of conservative systems generically predict artificial secular drifts of any nonlinear invariants. In this work we present a general approach for developing explicit nontraditional algorithms that conserve such invariants exactly. We illustrate the method by applying it to the three-wave truncation of the Euler equations, the Lotka--Volterra predator-prey model, and the Kepler problem. The ideas are discussed in the context of symplectic (phase--space-conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.