On Dissipation and Dispersion Errors Optimization,A-Stability and SSP Properties

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.