Geometric numerical schemes for the KdV equation
For researchers in numerical analysis and wave modeling, this work shows that geometric schemes are viable alternatives to spectral methods for long-time KdV simulations, though the result is incremental as it confirms known advantages in a specific context.
The paper demonstrates that geometric numerical schemes (symplectic/multi-symplectic) achieve robustness and accuracy comparable to Fourier-type pseudo-spectral methods for long-time KdV dynamics, making them suitable for modeling complex nonlinear wave phenomena.
Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-symplectic schemes have been proposed to solve numerically the celebrated Korteweg-de Vries (KdV) equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier-type pseudo-spectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.