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This is achieved using a half time step integration scheme.
In the first step of this scheme, a Lax Friedrichs scheme is applied at the midpoint for the half time step.
At each half time step, a nonlinear system is solved to compute the static frictional forces and states of touching particle-pairs.
4) Shift phase-space distribution functions in the velocity space by an electric force with the half time step Δt/2 by using the multi-dimensional conservative scheme (Umeda et al., 2009).
(12 where 6) Shift phase-space distribution functions in the velocity space by an electric force with the half time step Δt/2 by using the multi-dimensional conservative scheme (Umeda et al., 2009).
Shift phase-space distribution functions in the velocity space by an electric force with the half time step Δt/2 by using the multi-dimensional conservative scheme (Umeda et al., 2009).
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Consequently, multiple ion species may be easily treated (3) A free-streaming ionic current density is collected (velocities are collected at positions a half time-step ahead).
The particles are advanced by a leapfrog procedure which requires the electric field to be a half time-step ahead of the particle velocities.
2nd-order Runge-Kutta requires additional computation and memory because a half time-step intermediate calculation is required, with the previous time-step and the half time-step values used to solve for the next δt.
In each time interval, we first solve the values along x-direction by (32) at half-time step and then solve the values along y-direction by (33) at next half-time step.
A constant density approach for incompressible multi-phase SPH simulations was introduced by Hu and Adams [26], which corrects ([25]) intermediate density errors by adjusting the half-time-step velocity with exact projection.
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