Related papers: Sixth Order Compact Finite Difference Method for 2…
We show that the properties of the lower part of the spectrum of the Helmholtz equation for an heterogeneous system in a finite region in $d$ dimensions, where the solutions to the homogeneous problems are known, can be systematically…
The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a…
This article aims to present a general study of the Helmholtz problem in slowly varying waveguides. This work is of particular interest at locally resonant frequencies, where a phenomenon close to the tunnel effect for Schr\"odinger…
We propose some new mixed finite element methods for the time dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known [16] that the pressure…
We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem)…
In this paper, a parameter-uniform fitted mesh finite difference scheme is constructed and analyzed for a class of singularly perturbed interior turning point problems. The solution of this class of turning point problem possess two outflow…
In this paper, we consider the Stokes problem with Dirichlet boundary conditions and the constant kinematic viscosity $\nu$ in an axis-aligned domain $\Omega$. We decouple the velocity $\bm u$ and pressure $p$ by deriving a novel biharmonic…
We present an improved high-order weighted compact high resolution (WCHR) scheme that extends the idea of weighted compact nonlinear schemes (WCNS's) using nonlinear interpolations in conjunction with compact finite difference schemes for…
In this paper, a class of high-order compact finite difference Hermite scheme is presented for the simulation of double-diffusive convection. To maintain linear stability, the convective fluxes are split into positive and negative parts,…
This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter $\epsilon$ multiplying the highest derivative. We…
Although high-order Maxwell integral equation solvers provide significant advantages in terms of speed and accuracy over corresponding low-order integral methods, their performance significantly degrades in presence of non-smooth…
This report addresses the boundary value problem for a second-order linear singularly perturbed FIDE. Traditional methods for solving these equations often face stability issues when dealing with small perturbation parameters. We propose an…
We apply the local discontinuous Galerkin (LDG for short) method to solve a mixed boundary value problems for the Helmholtz equation in bounded polygonal domain in 2D. Under some assumptions on regularity of the solution of an adjoint…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
Context: Calculating stellar pulsations requires a sufficient accuracy to match the quality of the observations. Many current pulsation codes apply a second order finite-difference scheme, combined with Richardson extrapolation to reach…
We propose a novel finite-difference time-domain (FDTD) scheme for the solution of the Maxwell's equations in which linear dispersive effects are present. The method uses high-order accurate approximations in space and time for the…
In this paper, a high-order exponential scheme is developed to solve the 1D unsteady convection-diffusion equation with Neumann boundary conditions. The present method applies fourth-order compact exponential difference scheme in spatial…
For the $h$-finite-element method ($h$-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth $h$ must decrease with the frequency $k$ to maintain accuracy as $k$ increases has been studied since the mid 80's.…
In this paper, we discuss the 2D convection-diffusion-reaction equation with variable smooth coefficients and the Dirichlet boundary condition on a complicated, thin, and curved domain. We propose the fourth-order compact FDM at every grid…
We present two approaches for enhancing the accuracy of second order finite difference approximations of two-dimensional semilinear parabolic systems. These are the fourth order compact difference scheme and the fourth order scheme based on…