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5

In situations like this, it is a good idea to adjust the time step based on the gradient of the force - because the whole concept of numerical integration is that "things don't change too much from now until the next time step", and that assumption is violated when you move rapidly through a region with fast-changing force. This has a risky side-effect: if ...

3

What matters here is how the value of $c$ compares to the value of $k$. Let us choose a $\zeta = \frac{c}{2\sqrt{mk}}$ One can show that when $\zeta =1$ the system is critically damped, and will not exhibited any oscillations and will return to the origin in the shortest possible time interval. When $\zeta > 1$ the system is over damped and will take ...

2

For an incompressible fluid $\dot\rho=0$. Then the continuity equation implies $$\nabla\cdot u = 0 .$$ We can now take the divergence of the Navier-Stokes equation and get $$-\nabla^2 P = \rho\nabla_j(u_i\nabla_i u_j).$$ This means that the pressure is instantaneously determined by the velocity field (the pressure is no longer an independent ...

1

For any PIC simulation, you are necessarily tying yourself to particles, particles that experience forces. Thus, we have the generic force law: \begin{align} m_i\frac{d\mathbf v_i}{dt}&=\mathbf F_i \tag{1a}\\ \frac{d\mathbf x_i}{dt}&=\mathbf v_i \tag{1b} \end{align} In the case of PIC, you are often considering the electromagnetic (Lorentz) force, ...

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