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We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation. The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate for the solution is obtained and the order is O((∆x) 2 + (∆t) 2 ). Numerical examples demonstrate the effectiveness of the proposed scheme. Mathematics Subject Classification (1991): 65M06

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We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension d ≤ 3. We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.2000 Mathematics Subject Classification. 60H15, 60H35, 65C30.

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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...