On the Eigenvalues of Second Order Elliptic Difference Operators

A general scheme for tridiagonalizing differential, difference or q-difference operators using orthogonal polynomials is described. From the tridiagonal form the spectral decomposition can be described in terms of the orthogonality measure of generally different orthogonal polynomials. Three examples are worked out: (1) related to Jacobi and Wilson polynomials for a second order differential operator, (2) related to little q-Jacobi polynomials and Askey–Wilson polynomials for a bounded second order q-difference operator, (3) related to little q-Jacobi polynomials for an unbounded second order q-difference operator. In case (1) a link with the Jacobi function transform is established, for which we give a q-analogue using example (2).

Download Full-text

Weierstrass elliptic difference equations

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013022 ◽

1987 ◽

Vol 35 (1) ◽

pp. 43-48 ◽

Cited By ~ 10

Author(s):

Renfrey B. Potts

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Difference Equations ◽

Elliptic Function ◽

Order Differential Equation ◽

Second Order ◽

Elliptic Difference ◽

First Order ◽

Second Order Differential Equation ◽

Weierstrass Elliptic Function

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

Download Full-text

Numerical analysis of a narrow-angle, one-way, elastic-wave equation and extension to curvilinear coordinates

Geophysics ◽

10.1190/1.2236380 ◽

2006 ◽

Vol 71 (5) ◽

pp. T137-T146 ◽

Cited By ~ 3

Author(s):

D. A. Angus ◽

C. J. Thomson

Keyword(s):

Wave Equation ◽

Numerical Analysis ◽

Finite Difference ◽

Velocity Structure ◽

Dispersion Analysis ◽

Second Order ◽

Curvilinear Coordinates ◽

Propagation Path ◽

Difference Operators ◽

Narrow Angle

In this paper, we review the finite-difference implementation of a narrow-angle, one-way vector wave equation for elastic, 3D media. Extrapolation is performed in the frequency domain, where the second-order-accurate lateral spatial-difference operators are sufficiently accurate for narrow-angle propagation. We perform a numerical analysis of the finite-difference scheme to highlight the stability and dispersion characteristics. The von Neumann stability criterion indicates that extracting a reference phase during the extrapolation step noticeably improves the forward marching scheme, and dispersion analysis shows that numerical grid anisotropy is minimal for the propagation path lengths, source pulse spectral content, and angular range of forward propagation of interest. Although the algorithm is reasonable, its computational efficiency is limited by the second-order-accurate extrapolation step; therefore, the extrapolation scheme can be improved. We extend the Cartesian narrow-angle formulation to curvilinear coordinates, where the computational grid tracks the true wavefront in a reference medium and the wavefield derivative normal to the reference wavefront is evaluated locally using the Cartesian propagator. An example of curvilinear extrapolation for a simple model consisting of a high-velocity sphere within a homogeneous background velocity structure shows that the narrow-angle propagator is capable of modeling frequency-dependent geometric spreading and diffraction effects in curvilinear coordinates.

Download Full-text

Schauder estimates for fully nonlinear elliptic difference operators

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050000216x ◽

2002 ◽

Vol 132 (6) ◽

pp. 1395-1406 ◽

Cited By ~ 2

Author(s):

Hung-Ju Kuo ◽

Neil S. Trudinger

Keyword(s):

Elliptic Partial Differential Equations ◽

Difference Operators ◽

Pointwise Estimates ◽

Fully Nonlinear ◽

Schauder Estimates ◽

Elliptic Difference ◽

Nonlinear Elliptic ◽

Special Case ◽

General Meshes ◽

Hölder Estimates

In this paper, we are concerned with discrete Schauder estimates for solutions of fully nonlinear elliptic difference equations. Our estimates are discrete versions of second derivative Hölder estimates of Evans, Krylov and Safonov for fully nonlinear elliptic partial differential equations. They extend previous results of Holtby for the special case of functions of pure second-order differences on cubic meshes. As with Holtby's work, the fundamental ingredients are the pointwise estimates of Kuo-Trudinger for linear difference schemes on general meshes.

Download Full-text

Theory of Second Order Numerical Simulation Method of Enhanced Oil Production

Journal of Mathematics Research ◽

10.5539/jmr.v11n2p73 ◽

2019 ◽

Vol 11 (2) ◽

pp. 73

Author(s):

Yirang Yuan ◽

Aijie Cheng ◽

Danping Yang ◽

Changfeng Li

Keyword(s):

A Priori Estimates ◽

A Priori ◽

Oil Production ◽

Coupled System ◽

Simulation Method ◽

Second Order ◽

Optimal Order ◽

Difference Operators ◽

Order Error ◽

L2 Norm

A kind of second-order implicit upwind fractional steps finite difference method is presented in this paper to numerically simulate the coupled system of enhanced (chemical) oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and optimal order error estimates in l2 norm are derived.

Download Full-text