CONVERGENCE OF FINITE DIFFERENCE METHODS FOR THE WAVE EQUATION IN TWO SPACE DIMENSIONS

• When using a finite difference method to solve an initial-boundary-value problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many research works for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis.

Subject headings

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Annan matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Other Mathematics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

Keyword

Convergence rate

accuracy

two space dimensions

normal mode analysis

finite difference method

boundary-value-problems

by-parts operators

summation

order

approximations

discretization

stability

schemes

systems

Mathematics

Accuracy

Publication and Content Type

ref (subject category)

art (subject category)