ABSTRACT
Chapter 5 explores the finite volume method (FVM) applied to one-dimensional steady-state diffusion, convection-diffusion problems. Central differencing is used for discretising the diffusive fluxes. For a stable numerical scheme, the discretisation scheme should be consistent, conservative, bounded, stable and transportive. The central differencing of convective terms lacks transportiveness and leads to instability at a higher cell Peclet number. Upwind, hybrid and power-law differencing schemes guarantee conservativeness, boundedness and transportiveness and are highly stable. However, they are first-order accurate and suffer from false diffusion. Higher-order schemes can minimise false diffusion errors but suffer from a boundedness property. Higher-order bounded convective schemes, also called high-resolution schemes, have been developed for obtaining solutions free of oscillations. A flux limiter Ψ(r) is used for avoiding unphysical oscillations. All Total Variation Diminishing (TVD) discretisation schemes based on the flux limiter give second-order accuracy without any undesirable oscillations, which is essential for general-purpose CFD computation.
