An Improved Incompressible Smoothed Particle Hydrodynamics ...

Chapter A. The SPHysics Code
Also the particle density can be calculated from Eq. 2.6, which is
ρ =
mjωij. (A.14)
j
For the accuracy consideration, both of the expression suffer from the trun-
cated kernel error at the free surface. For the efficiency consideration, Eq.
A.13 needs the first derivative of the kernel function, but Eq. 2.41 the kernel
value, which means extra memory and computing cost will be put in because
the momentum equation needs only the first derivative.
Time marching
In code SPHysics, several different time marching schemes, Predictor-Corrector
(also known as the modified Euler method) [72], Verlet and Beeman [3], are
used. The Predictor-Corrector and Verlet scheme are introduced here. Oth-
ers can be found in [31].
Predictor-Corrector
In Predictor-Corrector time marching scheme [72], the variable fields are
first predicted with the half step n + 1/2 based on time step n, and then
corrected based on the previous predicted stage, n+1/2.
Predicting stage:
u n+1/2
i
ρ n+1/2
i
r n+1/2
i
= u n i
+ ∆t
2 Fn
= ρ n i + ∆t
2 Dn
= r n i
182
+ ∆t
2 un
(A.15)
(A.16)
(A.17)