A Conservative Semi-Lagrangian Method for the Advection Problem

In the paper, a new discrete analogue of an initial-boundary value problem is presented for the two-dimensional advection equation arising from a scalar time-dependent hyperbolic conservation law. At each time level, an approximate solution is found as a bilinear function on a uniform rectangular grid. For the presented scheme, a discrete analogue of the local integral balance equation is valid between two neighboring time levels. The numerical experiments are discussed for a solution with strong gradients.