Geometric numerical integrators for Hunter–Saxton-like equations

Abstract

We present novel geometric numerical integrators for Hunter-Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems. We consider the Hunter-Saxton equation, the modified Hunter-Saxton equation, and the two-component Hunter-Saxton equation. Multi-symplectic discretisations based on these new formulations of the problems are exemplified by means of the explicit Euler box scheme, and Hamiltonian-preserving discretisations are exemplified by means of the discrete variational derivative method. We explain and justify the correct treatment of boundary conditions in a unified manner. This is necessary for a proper numerical implementation of these equations and was never explicitly clarified in the literature before, to the best of our knowledge. Finally, numerical experiments demonstrate the favourable behaviour of the proposed numerical integrators.where κ ∈ {−1, 1} and ρ := ρ(x,t). These two partial differential equations (PDEs) also possess many interesting properties. The modified Hunter-Saxton equation is a model for short capillary waves propagating under the action of gravity [21]. An interesting feature of this modified version of the original problem is that it admits (smooth as well as cusped) travelling waves. This is not the case for the original problem (1). Moreover, this PDE is also bihamiltonian [21]. The two-component generalisation of the Hunter-Saxton equation is a particular case of the Gurevich-Zybin system which describes the dynamics in a model of non-dissipative dark matter, see [28] and also [24]. As the original equation, this system is integrable; has a Lax pair; is bihamiltonian; it is also the high-frequency limit of the two-component Camassa-Holm equation; has peakon solutions; its flow is equivalent to the geodesic flow on a certain sphere; etc. [33,18,26,31,22] and references therein.Despite the fact that the above nonlinear PDEs are relatively well understood in a more theoretical way, there are not much results on numerical discretisations of these problems. In fact, although we still continuously find a number of papers on theoretical aspects every year as of writing this paper, we are only aware of the numerical schemes from [12,34,35], the latest being proposed in 2010. All the three schemes are only for the original Hunter-Saxton equation. The work [12] proves convergence of some discrete finite difference schemes to dissipative solutions of the Hunter-Saxton equation on the half-line. The references [34,35] analyse local discontinuous Galerkin methods for the Hunter-Saxton equation and in particular, using results from [12], prove convergence of the discretisation scheme to the dissipative solutions.The reason for such a lack of numerical studies should be attributed to the following points. First, due to the mixed derivatives present in the above problems, standard spatial discretisations of these nonlinear partial differential equations become, in general, nontrivial. Second, partially in connection with this, the Hunt...

Stability and convergence of a conservative finite difference scheme for the modified Hunter–Saxton equation

“…On the other hand, as stated by Miyatake, Cohen, Furihata, and Matsuo [17], all solutions u of (5) (not (12)) satisfy H(u(t)) = H(u(0)) (see also Lenells [13]). To confirm this, we rewrite (5) as (13) u txx = A(u(t))u,…”

“…Derivation of a stable numerical scheme for mHS equation. To replicate the preservation of H, we employ the discretization of (13) proposed by Miyatake, Cohen, Furihata, and Matsuo [17]:…”

“…k . As mentioned in [17], the conservation of the discrete counterpart H d of H is ensured, where (15) H…”

“…However, numerical investigation of the mHS equation has not been conducted sufficiently thus far. The only related numerical study is that by Miyatake, Cohen, Furihata, and Matsuo [17], who considered another problem related to the mHS equation. Specifically, as their study was conducted before the local wellposedness result was obtained [15], they considered how to obtain the traveling wave solution (discovered by Lenells [14]) of the underdetermined form (5) u txx + 1 2…”

Linear gradient structures and discrete gradient methods for conservative/dissipative differential-algebraic equations

“…(subscripts t and x denote temporal and spatial partial differentiation), turn out to be DAEs [33]. As this class of PDEs covers numerous conservative systems, there have been several studies on conservative numerical methods [25,13,24,32] using the spirit of DVDM.…”