Advanced Ocean Modelling: Using Open-Source Software

3.4 Nonhydrostatic Solver 29Fig. 3.5 Location used to define the coefficients a t , a b , a e ,anda w . The pressure part q and thepressure correction Δq are calculated at the same grid points3.4.4 The S.O.R. MethodThe pressure equation (Eq. 3.24) can be solved by an iterative method calledSuccessive Over-Relaxation (or S.O.R in short) that can be formulated as:Δq r+1i,k= (1 − ω) Δqi,k r − ω qi,k ∗ a +o+ ω (ae Δqi,k+1 l a + a wΔqi,k−1 l + a tΔqi−1,k l + a bΔq l )i+1,ko(3.27)where r = 0, 1, 2, ··· is the iteration index, the superscript l is given by eitherl = r + 1orl = r dependent on whether an update of Δq already exists, and theparameter ω determines the degree of over-relaxation. Typical values are in a rangebetween 1.2 and 1.4.Start values of Δq for the S.O.R. iteration can be set to zero, but the iteration isoften faster if we use the values of the previous time step instead; that is,Δq r=0i,k= Δq n i,kThe surface boundary value for dynamic pressure needs to be given at every stepof the S.O.R. iteration. How this is done is described in the following. First, velocitycomponents are updated within the S.O.R. iteration with:u r+1i,k= ui,k ∗ − Δt (Δqr+1i,k+1ρ o Δx− )Δqr+1 i,kw r+1i,k= w ∗ i,k − Δtρ o Δz(Δqr+1i−1,k − )Δqr+1 i,k(3.28)(3.29)