Abstract <p>High-order accurate bicompact schemes for the multidimensional convection–diffusion equation with constant coefficients are considered. A substitution of dependent variables and simplified boundary conditions are used to construct a new implementation of these schemes on regular Cartesian grids. In contrast to a previous implementation, this one is based on multidimensional space marching computations, which allow on-the-fly interpolation of the sought functions on cell edges and faces, i.e., interpolation in the course of traversing the cells. Due to this property, the new implementation is generalized to hierarchical Cartesian grids with solution-based local adaptive mesh refinement. The computational algorithm is tested in wide ranges of the Courant number and the number of adaptation levels, and the test results demonstrate a high (third) order of accuracy.</p>