13. Time discretization

Time integrators range from simple backward Euler or a second order three state scheme, BDF2.

A general time discretization approach can be written as,

$$\int \frac{\partial \rho \varphi }{\partial t}dV=\int \frac{({\gamma }_1{\rho }^{n+1}{\varphi }^{n+1}+{\gamma }_2{\rho }^n{\varphi }^n+{\gamma }_3{\rho }^{n-1}{\varphi }^{n-1})}{\mathrm{\Delta }t}dV$$

where ${\gamma }_i$ represent the appropriate factors for either Backward Euler or a three-point BDF2 scheme. In both discretization approaches, the value for density and other dofs are evaluated at the node. As such, the time contribution is a lumped mass scheme with the volume simply the dual volume. The topology over one loops to assemble system is simply the node. Although CVFEM affords the use of a consistent mass matrix, this scheme is not used at present.