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ON [Equation ommitted] -INSTABILITY AND OSCTLtATION AT DISCONTINUITIES IN FINITE DIFFERENCE SCHEMES

IP.com Disclosure Number: IPCOM000128195D
Original Publication Date: 1984-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 5 page(s) / 23K

Publishing Venue

Software Patent Institute

Related People

Lloyd N. Trefethen: AUTHOR [+3]

Abstract

It is known that even-order finite differ-ence models of hyperbolic partial differential equations generate spurious oscillations ar-ound discontinuities, and also that they are unstable in RP norms for p#2. This paper pre-sents an elementary argument involving disper-sion and dissipation which shows that these two phenomena are closely related, explains their physical basis, and reproduces the known estimates for the width of the region of os-cillations and for the strength of the insta-bility. 1. Introduction It is well known that finite difference formulas for hyperbolic partial differential equations often suffer from spurious oscilla-tions near discontinuities. As a model hyper-bolic equation it is customary to consider the simple linear first-order wave equation [Equation ommitted] Various precise results for the behavior of the region of oscillations generated by finite difference approximations to (1) have been ob-tained over the years by Apelkrans, Brenner, Chin, Hedstrom, Serdjukova, Thomee, and others [1,2,4,6,7]. In particular, Chin and Hedstrom [4] have shown by saddle-point analysis that the numerical solutions to (1) behave approxi-mately like integrals of generalized Airy functions. Another widely recognized fact, first proved by Thomee (1964, unpublished) and also mentioned on p. 100 of the book by Richtmyer and Morton (8], is that every finite differ-ence model of (I1 with even order of accuracy is unstable in the RP norm for every pe[1,-] with p#2. See also [5]. For example the leap frog, Lax-Wendroff, and Crank-Nicolson differ-ence formulas are all 0-unstable. The insta-bility is weak, but it may have undesirable consequences in extensions to nonlinear prob-lems, where both Rl and R have a natural sig-nificance. For this reason a number of mathe-maticians have studied 0-instability of fi-nite difference formulas during the past twen-ty-five years, including Brenner, Hedstrom, Serdjukova, Stetter, Strang, Thomee, and Wahl-bin. A wealth of results of this work are presented in the monograph [3]. Most of the proofs given there are based on techniques of Fourier multipliers, the specialization to constant coefficients of pseudodifferential operators. The purpose of this brief paper is to show that both Rp-instability and oscillations at discontinuities are caused by a single pro-cess of numerical dispersion. An even-order finite difference model is one for which dis-persion dominates dissipation at low wave num-bers, so that the wiggles introduced by dis-persion are not all rapidly damped. By con-sidering the highest wave number for which this remains true, and by estimating its as-sociated group velocity, we will reproduce quantitatively the main results alluded to above. The argument is only heuristic, and indeed we state it very loosely so as not to obscure the physical idea with details. How-ever, much of this can probably be made rigor-ous. In two or more space dimensions, an easily visualized geometrical focusing process ren-ders hyperbolic differential equations ill-posed in LP. The reason that finite differ-ence models are Rp-unstable even in one space dimension is that the dispersion introduced by discretization can bring about a similar kind of focusing. Thi is analogous to the situa-tion regarding (R~-) stability of initial boundary value problems: uncontrolled radia- (1) tion of waves from the boundary can cause ill-posedness of a differential equation only in two or more dimensions, but it can cause in-stability of a finite difference model even in one dimension [10]. Acknowledgment. Gerald Hedstrom has con-tributed as much as anyone else to the topics discussed here, and shares my physical view of their underlying explanation. I am grateful to him for valuable advice and assistance on repeated occasions. 2. Estimates of dissipation and dispersion Let Q be a consistent R2-stable finite difference model of (1) with constant real co-efficients. The solution to Q is a function [Equation ommitted] defined on a grid with space siep h and time step k=constxh. We will also [Equation ommitted] By substituting the wave [Equation ommitted] into Q, where ~ is the wave number and w is the frequency, one [Equation ommitted] e -numerical dispersa.oation (in general complex) relating ~ and w. Consisten- cy implies that for C and w near 0, this re- lation is a function with the expansion [Equation ommitted]

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

ON

(Equation Omitted)

-INSTABILITY AND OSCTLtATION AT DISCONTINUITIES IN FINITE DIFFERENCE SCHEMES

B y

Lloyd N. Trefethen*

April 1984

Report # 116

9' *Supported by an NSF Postdoctoral Fellowship and by the U.S. Department of Energy under contract DE-AG02-76-ER03077-V. ON LP-INSTABILITY AND OSCILLATION AT DISCONTINUITIES IN FINITE DIFFERENCE SCHEMES

Lloyd N. Trefethen Courant Institute of Mathematical Sciences New York University , New York, NY 10012

Abstract

It is known that even-order finite differ-ence models of hyperbolic partial differential equations generate spurious oscillations ar-ound discontinuities, and also that they are unstable in RP norms for p#2. This paper pre-sents an elementary argument involving disper-sion and dissipation which shows that these two phenomena are closely related, explains their physical basis, and reproduces the known estimates for the width of the region of os-cillations and for the strength of the insta-bility.

1. Introduction It is well known that finite difference formulas for hyperbolic partial differential equations often suffer from spurious oscilla-tions near discontinuities. As a model hyper-bolic equation it is customary to consider the simple linear first-order wave equation

(Equation Omitted)

Various precise results for the behavior of the region of oscillations generated by finite difference approximations to (1) have been ob-tained over the years by Apelkrans, Brenner, Chin, Hedstrom, Serdjukova, Thomee, and others [1,2,4,6,7]. In particular, Chin and Hedstrom
[4] have shown by saddle-point analysis that the numerical solutions to (1) behave approxi- mately like integrals of generalized Airy functions.

Another widely recognized fact, first proved by Thomee (1964, unpublished) and also mentioned on p. 100 of the book by Richtmyer and Morton (8], is that every finite differ-ence model of (I1 with even order of accuracy is unstable in the RP norm for every pe[1,-] with p#2. See also [5]. For example the leap frog, Lax-Wendroff, and Crank-Nicolson differ-ence formulas are all 0- unstable. The insta-bility is weak, but it may have undesirable consequences in extensions to

New York University Page 1 Dec 31, 1984

Page 2 of 5

ON [Equation ommitted] -INSTABILITY AND OSCTLtATION AT DISCONTINUITIES IN FINITE DIFFERENCE SCHEMES

nonlinear prob-lems, where both Rl and R have a natural sig-nificance. For this reason a number of mathe-maticians have studied 0-instability of fi-nite difference formulas during the past twen-ty-five years, including Brenner, Hedstrom, Serdjukova, Stetter, Strang, Thomee, and Wahl-bin. A wealth of results of this work are presented in the monograph [3]. Most of the proofs given there are based on techniques of Fourier multipliers, the specialization to constant coefficients of pseudodifferential operators. The purpose of this brief paper is to show that both Rp-instability and oscillations at discontinui...