Generalized energy integrals and energy conserving numerical schemes for partial differential equations (Q701912)

This paper concerns numerical methods for initial-value problems for the differential equation, \(\partial_t^2 u = \partial_x^2 V(u)\). With \(V(u) = e^u - 1\) this equation is the continuous limit of the Toda lattice. The authors derive an energy equation for the differential equation, and they propose explicit and implicit finite-difference schemes which conserve energy. The explicit scheme, however, displays grid-sized oscillations behind shocks.