We prove that a very general smooth cubic fourfold containing a plane can be embedded into an irreducible holomorphic symplectic eightfold as a Lagrangian submanifold. We construct the desired irreducible holomorphic symplectic eightfold as a moduli space of Bridgeland stable objects in the derived category of the twisted K3 surface corresponding to the cubic fourfold containing a plane.

Finite difference method is an important methodology in the approximation of waves. In this paper, we will study two implicit finite difference schemes for the simulation of waves. They are the weighted alternating direction implicit (ADI) scheme and the locally one-dimensional (LOD) scheme. The approximation errors, stability conditions, and dispersion relations for both schemes are investigated. Our analysis shows that the LOD implicit scheme has less dispersion error than that of the ADI scheme. Moreover, the unconditional stability for both schemes with arbitrary spatial accuracy is established for the first time. In order to improve computational efficiency, numerical algorithms based on message passing interface (MPI) are implemented. Numerical examples of wave propagation in a three-layer model and a standard complex model are presented. Our analysis and comparisons show that both ADI and LOD schemes are able to efficiently and accurately simulate wave propagation in complex media.

This paper is devoted to the well-posedness of abstract Cauchy problems for quasi-linear evolution equations. The notion of Hadamard well-posedness is considered, and a new type of stability condition is introduced from the viewpoint of the theory of finite difference approximations. The result obtained here generalizes not only some results on abstract Cauchy problems closely related with the theory of integrated semigroups or regularized semigroups but also the Kato theorem on quasi-linear evolution equations. An application to some quasi-linear partial differential equation of weakly hyperbolic type is also given.