Error bound between monotone difference schemes and their modified equations

Author:
Zhen-Huan Teng

Journal:
Math. Comp. **79** (2010), 1473-1491

MSC (2000):
Primary 65M06, 65M15; Secondary 35L45, 35K15

DOI:
https://doi.org/10.1090/S0025-5718-09-02306-0

Published electronically:
September 14, 2009

MathSciNet review:
2630000

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Abstract: It is widely believed that if monotone difference schemes are applied to the linear convection equation with discontinuous initial data, then solutions of the monotone schemes are closer to solutions of their parabolic modified equations than that of the original convection equation. We will confirm the conjecture in this paper. It is well known that solutions of the monotone schemes and their parabolic modified equations approach discontinuous solutions of the linear convection equation at a rate only half in the -norm. We will prove that the error bound between solutions of the monotone schemes and that of their modified equations is *order one* in the -norm. Therefore the conclusion shows that the monotone schemes solve the modified equations more accurately than the original convection equation even if the initial data is discontinuous. As a consequence of the main result, we will show that the half-order rate of convergence for the monotone schemes to the convection equation is the best possible.

**1.**Amiram Harten,*The artificial compression method for computation of shocks and contact discontinuities. I. Single conservation laws*, Comm. Pure Appl. Math.**30**(1977), no. 5, 611–638. MR**0438730**, https://doi.org/10.1002/cpa.3160300506**2.**A. Harten, J. M. Hyman, and P. D. Lax,*On finite-difference approximations and entropy conditions for shocks*, Comm. Pure Appl. Math.**29**(1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR**0413526**, https://doi.org/10.1002/cpa.3160290305**3.**G. W. Hedstrom,*Models of difference schemes for 𝑢_{𝑡}+𝑢ₓ=0 by partial differential equations*, Math. Comp.**29**(1975), no. 132, 969–977. MR**0388797**, https://doi.org/10.1090/S0025-5718-1975-0388797-4**4.**N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation,*USSR Comp. Math. and Math. Phys.***16**(1976), 105-119.**5.**Peter D. Lax,*Weak solutions of nonlinear hyperbolic equations and their numerical computation*, Comm. Pure Appl. Math.**7**(1954), 159–193. MR**0066040**, https://doi.org/10.1002/cpa.3160070112**6.**Randall J. LeVeque,*Numerical methods for conservation laws*, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1990. MR**1077828****7.**Jian-Guo Liu and Zhou Ping Xin,*𝐿¹-stability of stationary discrete shocks*, Math. Comp.**60**(1993), no. 201, 233–244. MR**1159170**, https://doi.org/10.1090/S0025-5718-1993-1159170-7**8.**V. V. Petrov,*Sums of independent random variables*, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown; Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. MR**0388499****9.**T. Tang and Zhen Huan Teng,*The sharpness of Kuznetsov’s 𝑂(√Δ𝑥)𝐿¹-error estimate for monotone difference schemes*, Math. Comp.**64**(1995), no. 210, 581–589. MR**1270625**, https://doi.org/10.1090/S0025-5718-1995-1270625-9**10.**Zhen-Huan Teng,*Modified equation with adaptive monotone difference schemes and its convergent analysis*, Math. Comp.**77**(2008), no. 263, 1453–1465. MR**2398776**, https://doi.org/10.1090/S0025-5718-08-02061-9**11.**A. I. Vol′pert and S. I. Hudjaev,*Analysis in classes of discontinuous functions and equations of mathematical physics*, Mechanics: Analysis, vol. 8, Martinus Nijhoff Publishers, Dordrecht, 1985. MR**785938****12.**R. F. Warming and B. J. Hyett,*The modified equation approach to the stability and accuracy analysis of finite-difference methods*, J. Computational Phys.**14**(1974), 159–179. MR**0339526****13.**Xin Wen and Shi Jin,*Convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients. I. 𝐿¹-error estimates*, J. Comput. Math.**26**(2008), no. 1, 1–22. MR**2378582**

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Additional Information

**Zhen-Huan Teng**

Affiliation:
LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, China

Email:
tengzh@math.pku.edu.cn

DOI:
https://doi.org/10.1090/S0025-5718-09-02306-0

Keywords:
Monotone difference schemes,
modified parabolic equation,
discontinuous solutions,
$L^1$-error estimates,
linear convection equation

Received by editor(s):
February 2, 2009

Received by editor(s) in revised form:
May 20, 2009

Published electronically:
September 14, 2009

Additional Notes:
This work was supported by the National Science Foundation of China (No. 10576001)

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.