Structure-preserving and Exponential Discretizations of Initial Value Problems

Specialized integration algorithms for initial value problems, obtained by applying conventional explicit discretizations in a transformed space, are described. One example, conservative integration, is motivated by a theorem of Ge Zhong and Marsden [17] that establishes that in the absence of explicit time dependence, one must in practice choose between preserving symplecticity or conserving the Hamiltonian. Another example, exponential integration, is well suited to highly stiff ordinary differential equations. Fully Lagrangian methods for advection are shown to be a special case of exponential integration.