Optimized smoothing kernels for smoothed particle hydrodynamics

Open Access

Table B.2

Multipliers to calculate self-bias correction for ISPH from the values of Table B.1.

	f₁	f₁	f₁	f₄	f₅	f₆	f₇	f₈	f₉
O₃₂	1.007	1.003	1.003	1.0	0.999	0.999	0.996	0.996	0.995
O₆₄	1.005	0.999	1.0	0.999	0.999	0.998	0.996	0.997	0.995
O₁₂₈	1.004	0.998	0.999	0.999	0.998	0.996	0.995	0.997	0.996
O₂₅₆	1.004	1.0	0.999	0.999	0.998	0.998	0.998	0.998	0.998
O_I,32	1.019	0.979	0.992	0.970	0.886	0.803	0.714	0.341	0.088
O_I,64	1.007	1.001	1.002	1.001	1.0	0.999	0.996	0.997	0.996
O_I,128	1.005	1.0	1.0	1.0	1.0	1.041	0.996	0.997	0.996
O_I,256	1.005	1.0	1.0	1.0	1.0	0.999	0.997	0.998	0.996
C₂	1.005	0.996	0.995	0.994	0.988	0.998	0.998	0.996	0.997
C₄	0.963	0.997	0.997	0.999	0.999	0.998	1.0	0.999	0.999
C₆	0.973	0.997	0.998	1.0	1.0	1.0	1.0	1.0	1.0
C₈	0.974	0.998	0.999	0.999	1.0	1.0	1.0	1.0	1.0
WU₂	0.978	0.990	0.987	0.956	0.937	0.895	0.853	0.693	0.363
WU₄	1.003	0.998	0.998	0.997	0.997	0.993	0.994	0.995	0.993
^CM₀₄	0.981	0.975	0.910	0.806	0.719	0.557	0.395	0.1	x
CM₀₅	0.976	1.0	1.0	1.0	0.998	0.999	0.997	0.998	0.997
BUH	0.970	0.996	0.997	0.998	0.994	0.999	0.998	0.998	0.998

Notes. Correction to self-bias for ISPH is given by using (f_i ∙ c_i) instead of c_i in Eq. B.2. In general the kernels with ISPH have very similar self-bias coefficients to that of GDSPH, but one can see that some kernels in ISPH become increasingly unstable as N_smooth increases, such as CM₀₄, WU₂, and O_I,32, and their behavior diverges from the others, which otherwise converges to roughly 1.

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