High-order Lagrange multiplier schemes for general Hamiltonian PDEs

TL;DR

This work introduces a novel Lagrange multiplier framework for Hamiltonian PDEs that yields linearly implicit, arbitrarily high-order, energy-preserving schemes while preserving the original energy exactly at both continuous and discrete levels. By reformulating the dynamics as $z_t=\mathcal{S}(\mathcal{L}z+\lambda(t)N'(z))$ and enforcing energy conservation through a multiplier, the authors construct the second-order LM-CN scheme and extend to high-order schemes via Gauss collocation with a prediction-correction strategy (LM-GAUSS). They provide rigorous proofs of energy conservation and convergence, and demonstrate efficiency comparable to SAV/IEQ methods while avoiding the lower-bound restriction on the nonlinear energy, with numerical tests on KdV, NLS, and sine-Gordon confirming accuracy, long-time energy preservation, and cost-effectiveness. The approach generalizes easily to other conservative or dissipative systems and benefits from FFT-accelerated spatial discretization. Overall, the paper offers a practical, high-fidelity alternative for structure-preserving simulations of complex Hamiltonian PDEs.

Abstract

In this paper, we introduce a Lagrange multiplier approach to construct linearly implicit energy-preserving schemes of arbitrary order for general Hamiltonian PDEs. Unlike the widely used auxiliary variable methods, this novel approach does not require the nonlinear part of the energy to be bounded from below, thereby offering broader applicability. Moreover, this approach preserves the original energy exactly at both the continuous and discrete levels, as opposed to a modified energy preserved by the auxiliary variable methods. Rigorous proofs are provided for the energy conservation and numerical accuracy of all derived schemes. The trade-off for these advantages is the need to solve a nonlinear algebraic equation to determine the Lagrange multiplier. Nevertheless, numerical experiments show that the associated computational cost is generally not dominant, indicating that the new schemes retain computational efficiency comparable to the auxiliary variable-based schemes. Numerical results demonstrate the efficiency, accuracy, and structure-preserving properties of the proposed schemes.

Figures (11)

• Figure 1: The KdV equation: CPU times under different spatial grids for the one-soliton wave solution with $N=128$ and $\Delta t=0.002$ until $T=50$.
• Figure 2: The KdV equation: iteration numbers required to solve the Lagrange multiplier at each time step for the one-soliton wave solution with $N=128$ and $\Delta t=0.002$ until $T=50$.
• Figure 3: The KdV equation: numerical results for the one-soliton wave solution with $N=128$ and $\Delta t=0.002$ until $T=50$.
• Figure 4: The KdV equation: numerical results for the two-soliton waves solution with $N=128$ and $\Delta t=0.002$ until $T=50$.
• Figure 5: The NLS equation: numerical results for the one-soliton solution with $N=128$ and $\Delta t=0.02$ until $T=50$.

Theorems & Definitions (13)

• Theorem 2.1
• Proof 1
• Theorem 2.2
• Proof 2
• Lemma 2.3
• Proof 3
• Theorem 2.4
• Proof 4
• Theorem 2.5
• Proof 5