The midpoint rule applied to the discretized nonlinear Schrödinger equation is shown to possess a modified Hamiltonian energy, uniformly in the spatial mesh, leading to long-time numerical stability.
Almost conservation of the harmonic actions for fully discretized nonlinear Klein--Gordon equations at low regularity
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abstract
Close to the origin, the nonlinear Klein--Gordon equations on the circle are nearly integrable Hamiltonian systems which have infinitely many almost conserved quantities called harmonic actions or super-actions. We prove that, at low regularity and with a CFL number of size 1, this property is preserved if we discretize the nonlinear Klein--Gordon equations with the symplectic mollified impulse methods. This extends previous results of D. Cohen, E. Hairer and C. Lubich to non-smooth solutions.
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Fully discrete backward error analysis for the midpoint rule applied to the nonlinear Schroedinger equation
The midpoint rule applied to the discretized nonlinear Schrödinger equation is shown to possess a modified Hamiltonian energy, uniformly in the spatial mesh, leading to long-time numerical stability.