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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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