A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg–de Vries equation

Abstract

The invariant preserving property is one of the guiding principles for numerical algorithms in solving wave equations, in order to minimize phase and amplitude errors after long time simulation. In this paper, we design, analyze and numerically validate a Hamiltonian preserving discontinuous Galerkin method for solving the Korteweg-de Vries (KdV) equation. For the generalized KdV equation, the semi-discrete formulation is shown to preserve both the first and the third conserved integrals, and approximately preserve the second conserved integral; for the linearized KdV equation, all the first three conserved integrals are preserved, and optimal error estimates are obtained for polynomials of even degree. The preservation properties are also maintained by the fully discrete DG scheme. Our numerical experiments demonstrate both high accuracy of convergence and preservation of all three conserved integrals for the generalized KdV equation. We also show that the shape of the solution, after long time simulation, is well preserved due to the Hamiltonian preserving property.

Energy-conserving Hamiltonian Boundary Value Methods for the numerical solution of the Korteweg–de Vries equation

“…Moreover, its dimension equals that of the continuous problem (17), i.e., 2N . Conversely, even using the approximation (43), the simplified Newton iteration (42) would require to factor the matrix…”

“…The last example is the famous Zabusky-Kruskal example [54] (see also [23,Ex. 5.5] or [43,Ex. 4.3]):…”

“…It has been the subject, for about half a century, of many investigations both from a theoretical point of view (see, e.g., [26,5,44,38,45,35,29,36]) and from its numerical approximation. In this regard, besides the first numerical approaches in [49,1,50,7], conservative methods have been developed by using various approaches [33,25], including Galerkin methods [53,41,52,6,34,22], finite difference schemes [2,55,42], operator splitting and exponential-type integrators [32,31], structure and energypreserving methods [51,23,39,43,37,48]. 1 Ideally, in the most favourable case where u is analytic, its n-th Fourier coefficient decays exponentially with n, whereas it decays as n −r if u ∈ C r .…”

Discontinuous Galerkin methods for short pulse type equations via hodograph transformations

“…In these four schemes, the integration DG scheme is most accurate one. 20,20] and N = 160 cells at time T = 10.…”

“…Most recently, a series of schemes which called structure-preserving schemes have attracted considerable attention. For some integrable equations like KdV type equations [9,18,20,42], Zakharov system [31], Schrödinger-KdV system [32], Camassa-Holm equation [41], etc., the authors proposed various conservative numerical schemes to "preserve structure". These conservative numerical schemes have some advantages over the dissipative ones, for example, the Hamiltonian conservativeness can help reduce the phase error along the long time evolution and have a more accurate approximation to exact solutions for KdV type equations [42].…”