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A class of convergent finite difference schemes for certain nonlinear parabolic systems. (English) Zbl 0535.65063
The Cauchy problem \(v_ t-u_ x=\epsilon v_{xx}\), \(u_ t+p(v)_ x =\epsilon u_{xx}\) for \(x\in {\mathbb{R}}\), \(t>0\), \(v=v_ 0(x)\), \(u=u_ 0(x)\), \(x\in {\mathbb{R}}\), \(t=0\) is studied where \(\epsilon>0\) and \(p'<0\), \(p''>0\). Here the initial functions \(u_ 0\) and \(v_ 0\) are of bounded total variation and \(v_ 0\) is bounded from below. It is shown that the Lax-Friedrich difference scheme gives approximations which converge uniformly on finite time intervals to the unique classical solution.
Reviewer: U.Hornung

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
76N15 Gas dynamics, general
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References:
[1] Conley, Comm. Pure Appl. Math. 23 pp 867– (1970)
[2] Convergence of approximate solutions to conservation laws, preprint.
[3] and , Polynomial Approximation, Williams and Wilkins, Baltimore, 1974.
[4] An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed., J. Wiley, New York, 1957.
[5] Glimm, Comm. Pure Appl. Math. 18 pp 697– (1965)
[6] Hoff, Math. Comp. 33 pp 1171– (1979)
[7] Kanel’, Zh. Vychislit., Matem. i Matem Fiziki 6 pp 446– (1966)
[8] USSR Comp. Math. and Math. Phys. 6 pp 74– (1966)
[9] Lax, Comm. Pure Appl. Math. 7 pp 159– (1954)
[10] Lax, Comm. Pure Appl. Math. 10 pp 537– (1957)
[11] Liu, Adv. Appl. Math. 1 pp 345– (1980)
[12] Matsumura, Proc. Japan Acad. 55 pp 337– (1979)
[13] Nishida, Proc. Japan Acad. 44 pp 642– (1968)
[14] Nishida, Comm. Pure Appl. Math. 26 pp 183– (1973)
[15] Oleinik, Usp. Mat. Nauk. (N.S.) 12 pp 3– (1957)
[16] Amer. Math. Soc. Transl. Ser. 2 26 pp 95– (1963) · Zbl 0131.31803 · doi:10.1090/trans2/026/05
[17] Zhang, Acta Math. Sinica 15 pp 386– (1965)
[18] Chinese Math. 7 pp 90– (1965)
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