Abstract. We present a fourth-order accurate algorithm for solving Poisson's equation, the heat equation, and the advection-diffusion equation on a hierarchy of block-structured, adaptively refined grids. For spatial discretization, finite-volume stencils are derived for the divergence operator and Laplacian operator in the context of structured adaptive mesh refinement and a variety of boundary conditions; the resulting linear system is solved with a multigrid algorithm. For time integration, we couple the elliptic solver to a fourth-order accurate Runge-Kutta method, introduced by Kennedy and Carpenter [Appl. Numer. Math., 44 (2003), pp. 139-181], which enables us to treat the nonstiff advection term explicitly and the stiff diffusion term implicitly. We demonstrate the spatial and temporal accuracy by comparing results with analytical solutions. Because of the general formulation of the approach, the algorithm is easily extensible to more complex physical systems. 1. Introduction. The advection-diffusion equation governs numerous physical processes. Morton [17] lists ten sample applications ranging from semiconductor simulation to financial modelling. He also observes that "Accurate modelling of the interaction between convective and diffusive processes is the most ubiquitous and challenging task in the numerical approximation of partial differential equations." This observation is partly due to the fact that algorithms and analysis tend to be very different in the two limiting cases of elliptic and hyperbolic equations. Also, even for very simple initial and boundary conditions, the true solution may contain multiple length-scales that vary drastically across the spatial domain; see (3.1) in [18] for such an example.A finite-volume (FV) formulation is often preferred for applications where conservation is a primary concern. In its simplest form, the FV formulation is derived by applying the divergence theorem over the cells of a regular computational grid: