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We develop a new numerical technique for approximating solutions of the Navier-Stokes equations on moving domains. The method aims at simulating an incompressible fluid past an object whose motion is assigned a priori using a level-set…
Finite difference method was extended to unstructured meshes to solve Euler equations. The spatial discretization is made of two steps. First, numerical fluxes are computed at the middle point of each edge with high order accuracy. In this…
The goal of this work is to study the electromagnetic scattering problem of time-domain Maxwell's equations in an unbounded structure. An exact transparent boundary condition is developed to reformulate the scattering problem into an…
We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter $\varepsilon \in (0,1]$, which is inversely proportional to the acoustic speed. In the subsonic limit…
We present a uniformly accurate finite difference method and establish rigorously its uniform error bounds for the Zakharov system (ZS) with a dimensionless parameter $0<\varepsilon\le 1$, which is inversely proportional to the speed of…
In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for…
A simple and reliable finite difference approach is presented for solution of the Dirac equation eigenproblem for states confined in rotationally symmetric systems. The method sets the boundary condition for the spinor wave function…
We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the Difference Potentials framework. The main novelty of the developed schemes is the use of local…
Staggered grid finite difference scheme is widely used for the first order elastic wave equation, which constitutes the basis for least-squares reverse time migration and full waveform inversion. It is of great importance to improve the…
This paper introduces a new weak Galerkin (WG) finite element method for second order elliptic equations on polytopal meshes. This method, called WG-FEM, is designed by using a discrete weak gradient operator applied to discontinuous…
A novel finite element method for the approximation of Maxwell's equations over hybrid two-dimensional grids is studied. The choice of appropriate basis functions and numerical quadrature leads to diagonal mass matrices which allow for…
In this paper, an efficient algorithm is presented by the extrapolation technique to improve the accuracy of finite difference schemes for solving the fractional boundary value problems with non-smooth solution. Two popular finite…
We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced and is designed to handle independent…
Weak Galerkin methods refer to general finite element methods for PDEs in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and…
In this paper, we concentrate on solving second-order singularly perturbed Fredholm integro-differential equations (SPFIDEs). It is well known that solving these equations analytically is a challenging endeavor because of the presence of…
We assess the validity of a single step Godunov scheme for the solution of the magneto-hydrodynamics equations in more than one dimension. The scheme is second-order accurate and the temporal discretization is based on the dimensionally…
We propose a new practical adaptive refinement strategy for $hp$-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of…
A new fast multipole formulation for solving elliptic difference equations on unbounded domains and its parallel implementation are presented. These difference equations can arise directly in the description of physical systems, e.g.…
Topologically protected waves in the linearly polarized Maxwell's equations with gyrotropic, magneto-optic media were studied a decade ago both computationally and experimentally. This paper develops a robust tight-binding model for this…
Miniaturized optical resonators with spatial dimensions of the order of the wavelength of the trapped light offer prospects for a variety of new applications like quantum processing or construction of meta-materials. Light propagation in…