In this paper, an efficient and unconditionally stable rectangular domain decomposition algorithm is proposed. The order of accuracy of the prediction scheme of the new algorithm is second. Numerical experiments support efficiency, accuracy, and unconditional stability of the method.

This work overcomes the difficulty of the previous matched interface and boundary (MIB) method in dealing with interfaces with non-constant curvatures for optical waveguide analysis. This difficulty is essentially bypassed by avoiding the use of local cylindrical coordinates in the improved MIB method. Instead, novel jump conditions are derived along global Cartesian directions for the transver...

We present a new method, based on averaging, to simulate certain systems with multiple time scales efficiently and demonstrate its utility in the context of the shallow-water equations. We first develop the method in a simple linear setting and analytically prove its stability. This is followed by an extension to the full equations and a presentation of a computational model for it. In this pre...

A new, high-order, conservative, and efficient method for conservation laws on unstructured grids is developed. It combines the best features of structured and unstructured grid methods to attain computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finit...

We develop high order accurate source discretizations for hyperbolic wave propagation problems in first order formulation that are discretized by finite di↵erence schemes. By studying the Fourier series expansions of the source discretization and the finite di↵erence operator, we derive su cient conditions for achieving design accuracy in the numerical solution. Only half of the conditions in F...

We describe a strategy for detecting discontinuities and for limiting spurious oscillations near such discontinuities when solving hyperbolic systems of conservation laws by high-order discontinuous Galerkin methods. The approach is based on a strong superconvergence at the outflow boundary of each element in smooth regions of the flow. By detecting discontinuities in such variables as density ...

In this paper, we present first, second, and third order implicit finite-volume solvers for advectiondiffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of an uniformly accurate third-order advection-diffusion scheme is made trivial by the hyperbolic method while a naive construction of adding a third-order diffusion scheme t...

2 Determining the order of accuracy We are often faced with the problem of how to determine the order p given a sequence of approximations ũh1 , ũh2 , . . . This is can be a good check that a method is correctly implemented (if p is known) and also a way to get a feeling for the credibility of an approximation ũh (high p means high credibility). We can either be in the situation that the exact ...

A new unified theory underlying the theoretical design of linear computational algorithms in the context of time dependent first-order systems is presented. Providing for the first time new perspectives and fresh ideas, and unlike various formulations existing in the literature, the present unified theory involves the following considerations: (i) it leads to new avenues for designing new compu...

Theoretically and computationally, it is possible to demonstrate that the order of accuracy of a discontinuous Galerkin (DG) solution for linear hyperbolic equations can be improved from order k+1 to 2k+1 through the use of smoothness-increasing accuracy-conserving (SIAC) filtering. However, it is a computationally complex task to perform this in an efficient manner, which becomes an even great...