On Dissipation and Dispersion Errors Optimization, A-Stability and SSP Properties (Communications in Computational Physics, 2018)
In a recent paper (Du and Ekaterinaris, 2016) optimization of dissipationand dispersion errors was investigated. A Diagonally Implicit Runge-Kutta (DIRK)scheme was developed by using the relative stability concept, i.e. the ratio of absolutenumerical stability function to analytical one. They indicated that their newscheme has many similarities to one of the optimized Strong Stability Preserving (SSP)schemes. They concluded that, for steady state simulations, time integration schemesshould have high dissipation and low dispersion. In this note, dissipation and dispersionerrors for DIRK schemes are studied further. It is shown that relative stability isnot an appropriate criterion for numerical stability analyses. Moreover, within absolutestability analysis, it is shown that there are two important concerns, accuracy andstability limits. It is proved that both A-stability and SSP properties aim at minimizingthe dissipation and dispersion errors. While A-stability property attempts to increasethe stability limit for large time step sizes and by bounding the error propagations viaminimizing the numerical dispersion relation, SSP optimized method aims at increasingthe accuracy limits by minimizing the difference between analytical and numericaldispersion relations. Hence, it can be concluded that A-stability property is necessaryfor calculations under large time-step sizes and, more specifically, for calculation ofhigh diffusion terms. Furthermore, it is shown that the oscillatory behavior, reportedby Du and Ekaterinaris (2016), is due to Newton method and the tolerances they setand it is not related to the employed temporal schemes.
Keywords: Diagonally Implicit Runge-Kutta methods; dissipation and dispersion; optimization; numerical stability; steady state.