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@ARTICLE{Breu:872637,
      author       = {Breuß, Michael and Kleefeld, Andreas},
      title        = {{I}mplicit {M}onotone {D}ifference {M}ethods for {S}calar
                      {C}onservation {L}aws with {S}ource {T}erms},
      journal      = {Acta mathematica Vietnamica},
      volume       = {45},
      number       = {3},
      issn         = {0251-4184},
      address      = {Singapore},
      publisher    = {Springer Singapore},
      reportid     = {FZJ-2020-00128},
      pages        = {709–738},
      year         = {2020},
      abstract     = {In this article, a concept of implicit methods for scalar
                      conservation laws in one or more spatial dimensions allowing
                      also for source terms of various types is presented. This
                      material is a significant extension of previous work of the
                      first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986
                      2005). Implicit notions are developed that are centered
                      around a monotonicity criterion. We demonstrate a connection
                      between a numerical scheme and a discrete entropy
                      inequality, which is based on a classical approach by
                      Crandall and Majda. Additionally, three implicit methods are
                      investigated using the developed notions. Next, we conduct a
                      convergence proof which is not based on a classical
                      compactness argument. Finally, the theoretical results are
                      confirmed by various numerical tests.},
      cin          = {JSC},
      ddc          = {510},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {511 - Computational Science and Mathematical Methods
                      (POF3-511)},
      pid          = {G:(DE-HGF)POF3-511},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:000565039800009},
      doi          = {10.1007/s40306-019-00354-1},
      url          = {https://juser.fz-juelich.de/record/872637},
}