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32) i where the row sum mi represents the share of the domain Ω associated with node i mi = Ω ϕi dx = ∑ mi j . 20) over i, one obtains the global mass balance equation dm = ∑ ∑ ki j u j + ∑ ri .

53) If the underlying velocity field and/or the diffusion tensor depend on t, then so do the coefficients of the discrete operators C = {ci j } and D = {di j }, respectively. 52). Let t n+1 = t n + ∆ t and un+1 − un = θ F(un+1 ) + (1 − θ )F(un ), ∆t 0 ≤ θ ≤ 1. 56) and the first-order accurate backward Euler method (θ = 1) which yields un+1 = un + ∆ tF(un+1 ). 54) is a weighted average of F(u,t) evaluated at the old and new time level. Depending on the value of θ , the resulting time discretization can be explicit or implicit.

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