The Lagrange multiplier approach for general Hamiltonian PDEs

A novel linearly implicit energy-preserving scheme is proposed for general Hamiltonian PDEs by introducing the Lagrange multiplier approach. Unlike the previous scalar auxiliary variable (SAV) method, the new approach does not require the nonlinear part of the energy to be bounded from below, and conserves the original energy in both continuous and discrete cases. Moreover, the price we pay for these advantages is that a nonlinear algebraic equation has to be solved to determine the Lagrange multiplier. Combined with numerical experiments, where the computational cost of the Lagrange multiplier is generally not dominant, we show that the new scheme is computationally efficient as the SAV method, and that it accurately preserves the original energy.