Numerically solving the 2D Helmholtz equation is widely known to be very difficult largely due to its highly oscillatory solution. A very fine mesh size is necessary to deal with a large wavenumber leading to a severely ill-conditioned huge coefficient matrix. To understand and tackle such challenges, it is crucial to analyze how the solution of the 2D Helmholtz equation depends on boundary and source data for large wavenumbers. In this paper, we analyze and derive several new sharp wavenumber-explicit stability bounds for the 2D Helmholtz equation with inhomogeneous mixed boundary conditions: Dirichlet, Neumann, and impedance.

Iteratively computing a limiting function and its consecutive derivatives, a Hermite subdivision scheme is of particular interest and importance in CAGD for generating smooth subdivision curves and in numerical PDEs for constructing Hermite wavelets. This paper characterizes the convergence and smoothness of a Hermite subdivision scheme, provides simple factorization of Hermite masks through the normal form of matrix-valued masks, and presents an algorithm for constructing all masks for Hermite subdivision schemes.

Using newly modified cubic B-splines with accuracy order 4, we proposed a numerical scheme for solving the 1D and 2D hyperbolic telegraph equation using a differential quadrature method. Our modified cubic B-splines retain the tridiagonal structure and achieve the fourth order convergence rate. The solution of the associated ODEs is advanced in the time domain by the SSPRK scheme.

The elliptic interface problems with discontinuous and high-contrast coefficients have many applications but are very challenging to numerically solve, because their solutions often have low regularity or even discontinuity (which result in huge highly ill-conditioned linear systems). Thus, it is highly desired to construct high order schemes to solve the elliptic interface problems with discontinuous and high-contrast coefficients. To solve such elliptic interface problem with a smooth interface curve, we propose a fourth order compact finite difference method for computing the solution and its gradient by using uniform meshes. We showe that our proposed FDM numerically satisfies the discrete maximum principle when the mesh size is small and then we use it to prove their convergence of four order.