Time integration is an integral part of DNS and LES. Both DNS and LES require time-dependent simulations due to the unsteady nature of turbulence. As shown in the governing equations in Chapter 1, temporal or time derivatives are included in those equations. Unsteady or time-dependent ows have a fourth coordinate, time, in addition to the three spatial coordinates, which must be discretized. Temporal derivatives are dierent from spatial derivatives, in that time derivatives are parabolic like-an event at a given instant aects the ow only in the future-whereas for spatial derivatives, an event at any space location may inuence the ow anywhere else. Discretization methods for spatial derivatives will be discussed in subsequent chapters when applications of DNS and LES are described. is chapter discusses temporal integration methods for the time derivative terms in the governing equations. ese methods for the governing equations of uid dynamics are very similar to those applied to initial value problems for ordinary dierential equations (ODEs). e basic problem is to nd the solution φ at a short time interval ∆t aer the initial point. e solution at t1 = t0 + ∆t can then be used as a new initial condition and the solution can be advanced to t2 = t1 + ∆t, t3 = t2 + ∆t, … etc.