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Mathematics of Computation

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Rate of convergence of finite difference approximations for degenerate ordinary differential equations

Author: Jianfeng Zhang
Journal: Math. Comp. 75 (2006), 1755-1778
MSC (2000): Primary 65L70; Secondary 60H10, 65L12
Published electronically: July 6, 2006
MathSciNet review: 2240634
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Abstract: In this paper we study finite difference approximations for the following linear stationary convection-diffusion equations:

$\displaystyle {1\over 2}\sigma^2(x)u''(x) + b(x)u'(x) - u(x) =-f(x),\quad x\in\mathbb{R}, $

where $ \sigma$ is allowed to be degenerate. We first propose a new weighted finite difference scheme, motivated by approximating the diffusion process associated with the equation in the strong sense. We show that, under certain conditions, this scheme converges with the first order rate and that such a rate is sharp. To the best of our knowledge, this is the first sharp result in the literature. Moreover, by using the connection between our scheme and the standard upwind finite difference scheme, we get the rate of convergence of the latter, which is also new.

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Additional Information

Jianfeng Zhang
Affiliation: Department of Mathematics, University of Southern California, 3620 Vermont Ave., KAP 108, Los Angeles, California 90089

Keywords: Degenerate convection-diffusion equations, finite difference approximations, probabilistic solutions, sharp rate of convergence
Received by editor(s): August 8, 2004
Received by editor(s) in revised form: June 27, 2005
Published electronically: July 6, 2006
Additional Notes: The author was supported in part by NSF grant DMS-0403575
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.