Large-time behavior of solutions of Lax-Friedrichs finite difference equations for hyperbolic systems of conservation laws

Author:
I-Liang Chern

Journal:
Math. Comp. **56** (1991), 107-118

MSC:
Primary 65M12; Secondary 35L65, 39A12, 76L05

MathSciNet review:
1052088

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Abstract: We study the large-time behavior of discrete solutions of the Lax-Friedrichs finite difference equations for hyperbolic systems of conservation laws. The initial data considered here are small and tend to a constant state at . We show that the solutions tend to the *discrete diffusion waves* at the rate in , , with being an arbitrarily small constant. The discrete diffusion waves can be constructed from the self-similar solutions of the heat equation and the Burgers equation through an averaging process.

**[1]**I-Liang Chern and Tai-Ping Liu,*Convergence to diffusion waves of solutions for viscous conservation laws*, Comm. Math. Phys.**110**(1987), no. 3, 503–517. MR**891950****[2]**I-Liang Chern and Tai-Ping Liu,*Erratum: “Convergence to diffusion waves of solutions for viscous conservation laws” [Comm. Math. Phys. 110 (1987), no. 3, 503–517; MR0891950 (88g:35127)]*, Comm. Math. Phys.**120**(1989), no. 3, 525–527. MR**981217****[3]**I.-L. Chern,*Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws*, preprint, MCS-P134-0290, Math. Comp. Div., Argonno National Lab., 1990.**[4]**Eberhard Hopf,*The partial differential equation 𝑢_{𝑡}+𝑢𝑢ₓ=𝜇𝑢ₓₓ*, Comm. Pure Appl. Math.**3**(1950), 201–230. MR**0047234****[5]**S. Kawashima,*Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics*, Doctoral thesis, Kyoto University, 1983.**[6]**Shuichi Kawashima,*Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications*, Proc. Roy. Soc. Edinburgh Sect. A**106**(1987), no. 1-2, 169–194. MR**899951**, 10.1017/S0308210500018308**[7]**Tai-Ping Liu,*Pointwise convergence to 𝑁-waves for solutions of hyperbolic conservation laws*, Bull. Inst. Math. Acad. Sinica**15**(1987), no. 1, 1–17. MR**947772****[8]**Akitaka Matsumura and Takaaki Nishida,*The initial value problem for the equations of motion of viscous and heat-conductive gases*, J. Math. Kyoto Univ.**20**(1980), no. 1, 67–104. MR**564670**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1991-1052088-8

Keywords:
Lax-Friedrichs scheme,
hyperbolic systems of conservation laws,
discrete diffusion waves,
asymptotic behavior,
numerical viscosity

Article copyright:
© Copyright 1991
American Mathematical Society