This paper proposes a second order accurate, Adams- Bashforth type, asynchronous integration scheme for numerically solving systems of ordinary differential equations. The method has three aspects; a local integration rule with third order truncation error, a third order accurate model of local influencers, and local time advance limits. The role of these elements in the scheme?s operation are discussed and demonstrated. The time advance limit, which distinguishes this method from other discrete event methods for ODEs, is argued to be essential for constructing high order accuracy schemes.