TY - JOUR
T1 - On Dissipation and Dispersion Errors Optimization, A-Stability and SSP Properties
AU - Arman Rokhzadi & Abdolmajid Mohammadian
JO - Communications in Computational Physics
VL - 1
SP - 268
EP - 285
PY - 2018
DA - 2018/03
SN - 24
DO - http://doi.org/10.4208/cicp.OA-2017-0015
UR - https://global-sci.org/intro/article_detail/cicp/10937.html
KW - Diagonally Implicit Runge-Kutta methods, dissipation and dispersion, optimization, numerical stability, steady state.
AB - <p style="text-align: justify;">In a recent paper (Du and Ekaterinaris, 2016) optimization of dissipation
and dispersion errors was investigated. A Diagonally Implicit Runge-Kutta (DIRK)
scheme was developed by using the relative stability concept, i.e. the ratio of absolute
numerical stability function to analytical one. They indicated that their new
scheme has many similarities to one of the optimized Strong Stability Preserving (SSP)
schemes. They concluded that, for steady state simulations, time integration schemes
should have high dissipation and low dispersion. In this note, dissipation and dispersion
errors for DIRK schemes are studied further. It is shown that relative stability is
not an appropriate criterion for numerical stability analyses. Moreover, within absolute
stability analysis, it is shown that there are two important concerns, accuracy and
stability limits. It is proved that both A-stability and SSP properties aim at minimizing
the dissipation and dispersion errors. While A-stability property attempts to increase
the stability limit for large time step sizes and by bounding the error propagations via
minimizing the numerical dispersion relation, SSP optimized method aims at increasing
the accuracy limits by minimizing the difference between analytical and numerical
dispersion relations. Hence, it can be concluded that A-stability property is necessary
for calculations under large time-step sizes and, more specifically, for calculation of
high diffusion terms. Furthermore, it is shown that the oscillatory behavior, reported
by Du and Ekaterinaris (2016), is due to Newton method and the tolerances they set
and it is not related to the employed temporal schemes.</p>