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@ARTICLE{MartnVaquero:878664,
      author       = {Martín-Vaquero, J. and Kleefeld, A.},
      title        = {{S}olving nonlinear parabolic {PDE}s in several dimensions:
                      {P}arallelized {ESERK} codes},
      journal      = {Journal of computational physics},
      volume       = {423},
      issn         = {0021-9991},
      address      = {Amsterdam},
      publisher    = {Elsevier},
      reportid     = {FZJ-2020-02985},
      pages        = {109771},
      year         = {2020},
      abstract     = {There is a very large number of very important situations
                      which can be modeled with nonlinear parabolic partial
                      differential equations (PDEs) in several dimensions. In
                      general, these PDEs can be solved by discretizing in the
                      spatial variables and transforming them into huge systems of
                      ordinary differential equations (ODEs), which are very
                      stiff. Therefore, standard explicit methods require a large
                      number of iterations to solve stiff problems. But implicit
                      schemes are computationally very expensive when solving huge
                      systems of nonlinear ODEs. Several families of Extrapolated
                      Stabilized Explicit Runge-Kutta schemes (ESERK) with
                      different order of accuracy (3 to 6) are derived and
                      analyzed in this work. They are explicit methods, with
                      stability regions extended, along the negative real
                      semi-axis, quadratically with respect to the number of
                      stages s, hence they can be considered to solve stiff
                      problems much faster than traditional explicit schemes.
                      Additionally, they allow the adaptation of the step length
                      easily with a very small cost.Two new families of ESERK
                      schemes (ESERK3 and ESERK6) are derived, and analyzed, in
                      this work. Each family has more than 50 new schemes, with up
                      to 84.000 stages in the case of ESERK6. For the first time,
                      we also parallelized all these new variable step length and
                      variable number of stages algorithms (ESERK3, ESERK4,
                      ESERK5, and ESERK6). These parallelized strategies allow to
                      decrease times significantly, as it is discussed and also
                      shown numerically in two problems. Thus, the new codes
                      provide very good results compared to other well-known ODE
                      solvers. Finally, a new strategy is proposed to increase the
                      efficiency of these schemes, and it is discussed the idea of
                      combining ESERK families in one code, because typically,
                      stiff problems have different zones and according to them
                      and the requested tolerance the optimum order of convergence
                      is different.},
      cin          = {JSC},
      ddc          = {000},
      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:000588199000032},
      doi          = {10.1016/j.jcp.2020.109771},
      url          = {https://juser.fz-juelich.de/record/878664},
}