Abstract
In this note we show that when a discontinuous initial value problem for a scalar hyperbolic equation in one space variable is approximated by a difference scheme that is more than first order accurate; it leads to overshoots analogous to the Gibbs phenomenon when discontinuous functions are approximated by sections of Fourier series. A hybrid scheme due to Harten and Zwass removes the overshoots. Similar phenomena occur when solving schemes of hyperbolic equations.
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Lax P.D., Mock M.S. (1978). The computation of discontinuous solutions of hyperbolic equations. CPAM 31, 423–430
Liu X.D., Lax P.D. (1996). Positive schemes for solving multidimensional hyperbolic conservation laws. Comp. Fluid. Dynamics J. 5, 133–156
Majda A., Osher S. (1977). Propagation of error into regions of smoothness for accurate difference approximations to hyperbolic equations. CPAM 30, 671–706
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To David Gottlieb, master of scientific computation, subtle numerical analyst, Mensch extraordinaire.
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Lax, P.D. Gibbs Phenomena. J Sci Comput 28, 445–449 (2006). https://doi.org/10.1007/s10915-006-9075-y
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DOI: https://doi.org/10.1007/s10915-006-9075-y