The matrix vector equation seen in last lesson is a first order ordinary differential equation (ODE) and it already includes the boundary conditions. Initial conditions need to be written for this ODE. The derivation was done using consistent mass matrix, but the mass matrix can be lumped and global lumped mass matrix is written. Time discretization is used for solving these semi-discrete weak forms. Finite difference method is used for the time discretization. Space time finite element methods exist and Galerkin method is one of them. The ODE is written in terms of time discretized notation.
The time discretized ODE is written at two different time intervals, and this is called time discrete version of the ODE. At this point the special integration algorithms are invoked and this family of the algorithms is Euler family for first order ODEs. Euler family, forward and backward Euler, and Mid-point rule (also called Crank Nicholson Method) are discussed. The basis for Euler method is discussed and it will be used in next lesson.