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Discontinuous solutions to hyperbolic systems under operator splitting
Author(s) -
Roe P. L.
Publication year - 1991
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690070306
Subject(s) - discontinuity (linguistics) , mathematics , operator (biology) , eigenvalues and eigenvectors , oblique case , mathematical analysis , operator splitting , function (biology) , biochemistry , chemistry , physics , linguistics , philosophy , repressor , quantum mechanics , evolutionary biology , biology , transcription factor , gene
Two‐dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time‐steps exactly solves the component one‐dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to “pseudo acoustic waves” and a nonphysical stationary wave. The implications of these findings for the design of high‐resolution computing schemes are discussed.