In this chapter, an accurate method, using a novel immersed-boundary approach, is presented for numerically solving linear, scalar convection problems. As is standard in immersed-boundary methods, moving bodies are embedded in a fixed Cartesian grid. The essence of the present method is that specific fluxes in the vicinity of a moving body are computed in such a way that they accurately accommodate the boundary conditions valid on themoving body. To suppress wiggles, tailor-made limiters are introduced for these special fluxes. The first results obtained are very accurate, without requiring much computational overhead. It is anticipated that the method can readily be extended to real fluid-flow equations.