The present lecture is mainly concerned with the implicit finite difference technique of Beam and Warming and some of the many variants of this popular and widely used implicit method. The Euler and Navier-Stokes equations for 2-D compressible flows and the implicit time differencing formulae are presented in sections 2 and 3. The implicit, noniterative, approximate factorization, central difference algorithm in delta form is derived in section 4 while section 5 deals with stability and convergence considerations. Next, some salient features of implicit upwind schemes based on flux-vector splitting, and TVD schemes are discussed in section 6. The unfactored implicit relaxation method due to Hanel et al is outlined in section 7. Finally, an implicit finite volume nodal point scheme is presented in section 8. Section 9 presents some examples and results for certain flow problems while section 10 contains some concluding remarks.13; 13
