A class of exponential time-stepping mass-momentum-energy-preserving schemes for Korteweg-de Vries equations

The Korteweg-de Vries (KdV) equation, as a completely integrable equation, possesses infinite number of invariants. In this paper, building upon existing structure-preserving methodologies such as Fourier-Galerkin spectral methods, exponential integrators and discrete gradient-based projection technique, we develop a class of exponential time-stepping invariants-preserving schemes for the numerical integration of the KdV equation. First, a Fourier-Galerkin semidiscretization in spatial direction leads to a system of Ordinary Differential Equations (ODEs). The resulting semi-discrete system of ODEs is a stiff semi-linear system that inherits the invariants-preserving properties of the KdV equation. Then the exponential integrators are applied for the time integration to address stiffness while maintaining explicit efficiency. Meanwhile, a projection technique based on discrete gradient is accompanied to exactly conserve multiple invariants for the semi-discrete system. This yields a class of fully discrete schemes which can conserve the semi-discrete mass, momentum and energy to machine precision. Numerical experiments are carried out to confirm the accuracy order, the conservative property and efficiency of the proposed schemes.