The inverse Lax-Wendroff (ILW) procedure is a numerical boundary treatment technique, which allows finite difference schemes and other schemes to achieve stability and high order accuracy when using cartesian meshes to solve boundary value problems defined on complex computational domain. In this short survey we summarize the main ingredients of the ILW procedure, discuss its applicability and stability properties, and provide possible directions of its future development.

In order to describe the impact of the different geometric structures and the constraints for the dynamics of a Hamiltonian system, in this paper, for a magnetic Hamiltonian system defined by a magnetic symplectic form, we drive precisely the geometric constraint conditions of the magnetic symplectic form for the magnetic Hamiltonian vector field, which are called the Type I and Type II Hamilton-Jacobi equations. Second, for the magnetic Hamiltonian system with a nonholonomic constraint, we can define a distributional magnetic Hamiltonian system, then derive its two types of Hamilton-Jacobi equations. Moreover, we generalize the above results to nonholonomic reducible magnetic Hamiltonian system with symmetry, we define a nonholonomic reduced distributional magnetic Hamiltonian system, and prove the two types of Hamilton-Jacobi theorems. These research reveal the deeply internal relationships of the magnetic symplectic structure, the nonholonomic constraint, the distributional two-form, and the dynamical vector field of the nonholonomic magnetic Hamiltonian system.

We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data. We show that for small values of the parameter the corresponding solutions decay to 0, while for large values the related solutions converge to 1 uniformly on compacts. Moreover, we prove that the transition from extinction (converging to 0) to propagation (converging to 1) is sharp. Numerical results are provided to verify the theoretical results.

In this paper, we give equivalent conditions for the factor rings of the polynomial ring $k[x,y]$ modulo monomial ideals to be Armendariz rings, where $k$ is a field. For an ideal $I$ with 2 or 3 monomial generators, or $n$ homogeneous monomial generators, such that $k[x,y]/I$ is an Armendariz ring, we characterize the minimal generator set $G(I)$ of $I.$

We develop in this paper a lifting method for Fokker-Planck equations with drift-admitting jumps, such that high-order finite difference schemes can be constructed directly based on grids with pure solution points. To illustrate the idea, we present as an example the construction of a fifth-order finite difference scheme. The validity of the scheme is demonstrated by conducting numerical experiments for the cases with drift admitting one jump and two jumps, respectively. Additionally, by introducing a splitting technique, we show that the lifting method can be extended to high dimensions. In particular, a two-dimensional case is studied in details to show the effectiveness of the extension.