Instability and stability of numerical approximations to discrete velocity models of the Boltzmann equation
Author:
Robert Peszek
Journal:
Quart. Appl. Math. 54 (1996), 777-791
MSC:
Primary 82C40; Secondary 35Q99, 39A11, 65M12
DOI:
https://doi.org/10.1090/qam/1417239
MathSciNet review:
MR1417239
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Abstract: We study a standard, explicit finite difference approximation of the 2-D Broadwell model and construct a numerical solution with the sum-norm growing in time faster than any polynomial. Our construction is based on a structure of a self-similar fractal!
T. Płatkowski and R. Illner, Discrete velocity models of the Boltzmann equation: A survey of mathematical aspects of the theory, SIAM Review 30, 213–255 (1988)
R. Illner, Global existence results for discrete velocity models of the Boltzmann equation in several dimensions, J. Méch. Théor. Appl. 1, 611–622 (1982)
R. Illner, Examples of non-bounded solutions in discrete kinetic theory, J. Méch. Théor. Appl. 5, 561–571 (1986)
S. Kawashima, Global solution of the initial value problem for a discrete velocity model of the Boltzmann equation, Proc. Japan Acad. Ser. A. Math. Sci. 57, 1–24 (1981)
R. Peszek, Convergence of numerical solutions to systems of semilinear hyperbolic equations, manuscript (1993)
B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Company, New York, 1983
T. Płatkowski and R. Illner, Discrete velocity models of the Boltzmann equation: A survey of mathematical aspects of the theory, SIAM Review 30, 213–255 (1988)
R. Illner, Global existence results for discrete velocity models of the Boltzmann equation in several dimensions, J. Méch. Théor. Appl. 1, 611–622 (1982)
R. Illner, Examples of non-bounded solutions in discrete kinetic theory, J. Méch. Théor. Appl. 5, 561–571 (1986)
S. Kawashima, Global solution of the initial value problem for a discrete velocity model of the Boltzmann equation, Proc. Japan Acad. Ser. A. Math. Sci. 57, 1–24 (1981)
R. Peszek, Convergence of numerical solutions to systems of semilinear hyperbolic equations, manuscript (1993)
B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Company, New York, 1983
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Article copyright:
© Copyright 1996
American Mathematical Society
