A Finite Difference Scheme for Compressible Miscible Displacement Flow in Porous Media on Grids with Local Refinement in Time

and Applied Analysis 3 The grid function y(x, t) is a function defined at the grid points of g. we denote the nodal values of a grid function y(x, t) between time levels t 0 and t 0 as y (x, t) = y (x 1 , x 2 , t l,j i ) = y l,j n1 ,n2 , (11) for x ∈ ω i , i > 0, j = 0, . . . , m i . For x ∈ ω 0 we define y (x, t) = y (x 1 , x 2 , t l+1 0 ) = y l+1 n1 ,n2 . (12) δ x1 , δ x1 and δ x2 , δ x2 are the divided forward and backward difference operators, respectively, in x 1 and x 2 direction. Also define the divided backward time difference by δ τ0 y l 0 (x) = y l 0 (x) − y l−1 0 (x)