Difference Method For Elliptic Equations Research Articles

A finite difference method for elliptic equations with the variable-order fractional derivative

A finite difference method for elliptic equations with the variable-order fractional derivative

The Crank–Nicolson hybrid difference method for the time wave equation on exterior domains with the discrete radial absorbing boundary condition

The hybrid difference method (HDM) combined with the discrete radial absorbing boundary condition (ABC) and the Crank–Nicolson time marching is presented. The HDM method is a finite difference version of the hybridized Galerkin method, and it consists of two types of finite difference approximations; the cell finite difference and the interface finite difference. The discrete radial ABC is a time dependent version of the discrete radial ABC for the elliptic equations, that has been proposed in [Y. Jeon, Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition, J. Sci. Comput. 75, 889–905 (2018)]. Through a change of variables the Bayliss–Turkel ABC is renewed. Moreover, it is partially proved that the discrete radial ABC is a non-standard finite difference approximation of the Bayliss–Turkel ABC. The discrete radial ABC is easy to implement and can be applied to an arbitrary convex fictitious domain including the rectangular box since it is degenerated to a quasi-Dirichlet condition for the time dependent problem. Two-dimensional numerical experiments confirming efficiency of our numerical scheme for various wave numbers and several artificial boundaries are presented.

Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition

The hybrid spectral difference methods (HSD) for the Laplace and Helmholtz equations in exterior domains are proposed. We consider the fictitious domain method with the absorbing boundary conditions (ABCs). The HSD method is a finite difference version of the hybridized Galerkin method, and it consists of two types of finite difference approximations; the cell finite difference and the interface finite difference. The fictitious domain is composed of two subregions; the Cartesian grid region and the boundary layer region in which the radial grid is imposed. The boundary layer region with the radial grid makes it easy to implement the discrete radial ABC. The discrete radial ABC is a discrete version of the Bayliss–Gunzburger–Turkel ABC without pertaining any radial derivatives. Numerical experiments confirming efficiency of our numerical scheme are provided.

Immersed Interface Method for elliptic equations based on a piecewise second order polynomial

In this paper, we develop finite difference methods for elliptic equations in a domain Ω∈Rd, d=1,2. Within the region Ω, we suppose there is an irregular surface Γ of codimension 1 (hereafter called an interface) across which the function u or some of its derivatives are known to be discontinuous. We use uniform grid and a piecewise second order polynomial to approximate u, then we get a second order method for these problems. At last, we give several examples to show the correctness and efficiency of the scheme.

Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension

A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second-order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper [SIAM J. Numer. Anal., 31(1994), pp. 1019--1044]. The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasi-Newton method is developed that allows reasonable time steps to be used.

The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources

The authors develop finite difference methods for elliptic equations of the form \[ \nabla \cdot (\beta (x)\nabla u(x)) + \kappa (x)u(x) = f(x)\] in a region $\Omega $ in one or two space dimension...

A fourth‐order difference method for elliptic equations with nonlinear first derivative terms

AbstractWe present a nine‐point fourth‐order finite difference method for the nonlinear second‐order elliptic differential equation Auxx + Buyy = f(x, y, u, ux, uy) on a rectangular region R subject to Dirichlet boundary conditions u(x, y) = g(x, y) on ∂R. We establish, under appropriate conditions O(h4)‐convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth‐order convergence.