Discretization of the Source Term, Relaxation, and Other Details

Abstract

This chapter is devoted to a number of “small” numerical details that may have “big” effects on the solution behavior. First the treatment of the source term in the general case when it is solution dependent (i.e., when $Q^{{\phi}} = Q^{{\phi}} \left({\phi}\right)$) is examined. The source is linearized in terms of the dependent variable and split into two parts, one treated explicitly and the second treated implicitly. This is followed by a discussion of explicit and implicit techniques for under-relaxing the algebraic equations. Several implicit under-relaxing approaches are presented, starting with the well known implicit under relaxation method of Patankar (Numerical heat transfer and fluid flow, 1980) [1], the E-factor method of van Doormaal and Raithby (Numerical Heat Transfer 7:147–163, 1984) [2], and the false transient approach of Mallinson and de Vahl Davis (Journal of Computational Physics 12(4):435–461, 1973) [3]. Then the residual form of the discretized algebraic equation is introduced. The chapter ends with the presentation of convergence indicators used to evaluate the solution convergence status.