Convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients I: L-error estimates

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 similar technique used in [25]. We use some inequalities on binomial coefficients proved in a consecutive paper [32].