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Convergence of relaxation schemes for conservation laws. (English) Zbl 0872.65086
We study the stability and the convergence for a class of relaxing numerical schemes for conservation laws. Following the approach recently proposed by S. Jin and Z. Xin [Commun. Pure Appl. Math. 48, No. 3, 235-276 (1995; Zbl 0826.65078)], we use a semilinear local relaxation approximation, with a stiff lower order term, and we construct some numerical first and second order accurate algorithms, which are uniformly bounded in the $$L^\infty$$ and BV norms with respect to the relaxation parameter. The relaxation limit is also investigated.

##### MSC:
 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws
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##### References:
 [1] Brenier Y., J. Diff. Eq. 50 pp 375– (1983) · Zbl 0549.35055 [2] Caffisch R., Comm. Pure Appl. Math 32 pp 589– (1979) · Zbl 0438.76059 [3] Caflisch R., Uniformly accurate schemes for hyperbolic systems with relaxation (1994) [4] Cercignani C., The Boltzmann equation and its applications (1988) · Zbl 0646.76001 [5] Cercignani C., The mathematical theory of dilute gases (1994) · Zbl 0813.76001 [6] Chen G. -Q., Comm. Pure Appl. Math. 46 pp 755– (1993) · Zbl 0797.35113 [7] Chen G. -Q., Comm. Pure Appl. Math 47 pp 787– (1994) · Zbl 0806.35112 [8] Crandall M., Math. of Comp. 34 pp 1– (1980) [9] Doolen G. D., Physica D 47 (1991) [10] Giga Y., Duke Math. J. 50 pp 505– (1983) · Zbl 0519.35053 [11] Goodman J. B., SIAM J. Numer. Anal. 25 pp 268– (1988) · Zbl 0645.65051 [12] Harten A., SIAM Rev. 25 pp 35– (1983) · Zbl 0565.65051 [13] Hanouzet B., Weakly coupled systems of quasilinear hyperbolic equations · Zbl 0879.35093 [14] Jin S., Comm. Pure Appl. Math. 48 pp 235– (1995) · Zbl 0826.65078 [15] Kruskov S. N., Mat. Sb. 81 pp 285– (1970) [16] Kuznetsov N. N., , USSR Comp Math and Math Phys 16 pp 105– (1976) · Zbl 0381.35015 [17] Lions P. L., Journal A. M. S. 7 pp 169– (1994) [18] Liu T. P., Comm. Math. Phys. 108 pp 153– (1987) · Zbl 0633.35049 [19] Natalini R., to appear in Comm. Pure Appl. Math. [20] Pember R. B., SIAM J. Appl. Math. 53 pp 1293– (1993) · Zbl 0787.65062 [21] Pember R. B., SIAM J. Sci. Comp. 14 pp 824– (1993) · Zbl 0812.65083 [22] Perthame B., SIAM J. Num. Anal. 29 pp 1– (1992) · Zbl 0744.76088 [23] Perthame B., Comm. Math. Phys. 136 pp 501– (1991) · Zbl 0729.76070 [24] Platkowski T., SIAM Review 30 pp 213– (1988) · Zbl 0668.76087 [25] Smith H. L., SIAM Review 30 pp 87– (1988) · Zbl 0674.34012 [26] Whitham J., Linear and nonlinear waves (1974)
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