Time discretization

For the time discretization of the momentum equation, a general trapezoid rule can be adopted. Using this rule, the time derivative of a generic function $\phi$ can be approximated by following equation

$$[\phi(x,t)]^{n+\theta} = \theta\phi(x,t^{n+1})+(1-\theta)\phi(x,t^n)=\theta\phi^{n+1}+(1-\theta)\phi^n\;.$$ (2.40)

Rewriting the time derivative on the left hand side of the momentum balance (2.35) as a finite difference in time and applying the trapezoidal rule on the right hand side, we obtain

$$\rho \mbox{$\displaystyle\frac{\partial \,{u_i}}{\partial \,{t}}$} \approx \rho\frac{u^{n+1}_i-u^n_i}{\Delta t}= \left[ - \frac{\par... ..._j}\left(\frac{\partial u_i}{\partial x_j}\right)+\rho b_i\right]^{n+\theta}\;.$$ (2.41)

The parameter $\theta$ can take values from the interval $[0,1]$. The approximation is considered as a weighted average of the derivative values in the time step $n$ and $n+1$. Using a specific value of the $\theta$ parameter, well-known methods can be recovered: The explicit Euler method $\theta=0$, the backward Euler for $\theta=1$ or the Crank-Nicolson method $\theta=1/2$. The current implemantation of PFEM allows the use of explicit and backward (implicit) method.