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Convergence of a large time step generalization of Godunov's method for conservation laws
Author(s) -
Leveque Randall J.
Publication year - 1984
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160370405
Subject(s) - conservation law , mathematics , riemann problem , riemann hypothesis , subsequence , godunov's scheme , sequence (biology) , limit of a sequence , scalar (mathematics) , mathematical analysis , maxima and minima , generalization , entropy (arrow of time) , convergence (economics) , limit of a function , courant–friedrichs–lewy condition , limit (mathematics) , numerical analysis , geometry , bounded function , genetics , physics , quantum mechanics , biology , economics , economic growth , discretization
A natural generalization of Godunov's method for Courant numbers larger than 1 is obtained by handling interactions between neighboring Riemann problems linearly, i.e., by allowing waves to pass through one another with no change in strength or speed. This method is well defined for arbitrarily large Courant numbers and can be written in conservation form. It follows that if a sequence of approximations converges to a limit u(x,t) as the mesh is refined, then u is a weak solution to the system of conservation laws. For scalar problems the method is total variation diminishing and every sequence contains a convergent subsequence. It is conjectured that in fact every sequence converges to the (unique) entropy solution provided the correct entropy solution is used for each Riemann problem. If the true Riemann solutions are replaced by approximate Riemann solutions which are consistent with the conservation law, then the above convergence results for general systems continue to hold.