A numerical scheme is said to be locally exact if after linearization (around any point) it becomes exact. In this paper, we begin with a short review on exact and locally exact integrators for ordinary differential equations. Then, we extend our approach on equations represented in the so called linear gradient form, including dissipative systems. Finally, we apply this approach to the Duffing equation with a linear damping and without external forcing. The locally exact modification of the discrete gradient scheme preserves the monotonicity of the Lyapunov function of the discretized equation and is shown to be very accurate.
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