We study the L1-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces.Here the interface condition is immersed into the upwind scheme.We prove that,for initial data with a bounded variation,the numerical solution of the immersed interface upwind scheme converges in L1-norm to the differential equation with the corresponding interface condition.We derive the one-halfth order L1-error bounds with explicit coefficients following a technique used in [25].We also use some inequalities on binomial coefficients proved in a consecutive paper[32].