Strong-stability-preserving, Hermite–Birkhoff time-discretization based on k step methods and 8-stage explicit Runge–Kutta methods of order 5 and 4

Explicit strong stability preserving multistep Runge–Kutta methods

“…Once again, the effective SSP coefficient occasionally decreases with increasing stage number. Our (s, k) = (2, 5) method has an effective SSP coefficient that matches the one in [18], and our values for (s, k) = (8, 3), (8,4), and (8, 5) improve upon the values obtained in [30]. Our values for (s, k) = (7, 3), (7,4) match those of [27] and our (s, k) = (7, 5) value improves on that in [27].…”

“…Compared to the seven-step two-stage method in [18], which has C = 0.234 and C eff = 0.117, our five step methods with s ≥ 3, four step with k ≥ 5, three step with k ≥ 6 and two step with k ≥ 9 all have larger effective SSP coefficient. Our (7, 4), (7,5), (3,5) methods have SSP coefficients that match those in [27] and [26], while our (7, 3) and (8, 3), (8,4), (8,5) have larger SSP coefficients that those in [27] and [30]. Table 6: C eff for seventh order methods C eff for seventh order methods Table 7: C eff for eighth order methods C eff for eighth order methods s\k…”

“…Our (s, k) = (2, 4) and (s, k) = (2, 5) methods have effective SSP coefficients that match the ones in [18]. Our (s, k) = (8, 2), (3,3), (3,4), (3,5), and (7, 3) methods have effective SSP coefficient that match those in [30,26,27]. Effective SSP coefficients of optimized sixthorder methods are given in Figure 1(d) and Table 5.…”

“…Overall, the effective SSP coefficient tends to increase more quickly with the number of stages than with the number of steps. Where relevant, we compare the methods we found to those of Constantinescu [5], Huang [18], and Vaillancourt [26,27,30].…”

“…Where relevant, we compare the methods we found to those of Constantinescu [5], Huang [18], and Vaillancourt [26,27,30].…”