We present a numerical method for solving Poisson's equation, with variable coefficients and Dirichlet boundary conditions, on two-dimensional regions. The approach uses a fhite-volume discretization, which embeds the domain in a regular Cartesian grid. We treat the solution as a cell-centered quantity, even when those centers are outside the domain. Cells that contain a portion of the domain boundafy use conservative differencing of second-order accurate fluxes, on each cell volume. The calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes. This allows us to use multi-grid iterations with a simple point relaxation strategy. We have, combined this with an adaptive mesh refinement (AMR) procedure. We provide evidence that the algorithm is second-order accurate on various exact solutions, and compare the adaptive and non-adaptive calculations.

We present an algorithm for solving the heat equation on irregular time-dependent domains. It is based on the Cartesian grid embedded boundary algorithm of Johansen and Colella (J. Comput. Phys. 147(2):60-85) for discretizing Poisson's equation, combined with a second-order accurate discretization of the time derivative. This leads to a method that is second-order accurate in space and time. For the case where the boundary is moving, we convert the moving-boundary problem to a sequence of fixed-boundary problems, combined with an extrapolation procedure to initialize values that are uncovered as the boundary moves. We find that, in the moving boundary case, the use of Crank-Nicolson time discretization is unstable, requiring us to use the L0-stable implicit Runge-Kutta method of Twizell, Gumel, and Arigu.

Numerical weather, climate, or Earth system models involve the coupling of components. At a broad level, these components can be classified as the resolved fluid dynamics, unresolved fluid dynamical aspects (i.e., those represented by physical parameterizations such as subgrid-scale mixing), and nonfluid dynamical aspects such as radiation and microphysical processes. Typically, each component is developed, at least initially, independently. Once development is mature, the components are coupled to deliver a model of the required complexity. The implementation of the coupling can have a significant impact on the model. As the error associated with each component decreases, the errors introduced by the coupling will eventually dominate. Hence, any improvement in one of the components is unlikely to improve the performance of the overall system. The challenges associated with combining the components to create a coherent model are here termed physics–dynamics coupling. The issue goes beyond the coupling between the parameterizations and the resolved fluid dynamics. This paper highlights recent progress and some of the current challenges. It focuses on three objectives: to illustrate the phenomenology of the coupling problem with references to examples in the literature, to show how the problem can be analyzed, and to create awareness of the issue across the disciplines and specializations. The topics addressed are different ways of advancing full models in time, approaches to understanding the role of the coupling and evaluation of approaches, coupling ocean and atmosphere models, thermodynamic compatibility between model components, and emerging issues such as those that arise as model resolutions increase and/or models use variable resolutions.

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:

Mountains are natural dams that impede atmospheric moisture transport and water towers that cool, condense, and store precipitation. They are essential in the western United States where precipitation is seasonal, and snowpack is needed to meet water demand. With anthropogenic climate change increasingly threatening mountain snowpack, there is a pressing need to better understand the driving climatological processes. However, the coarse resolution typical of modern global climate models renders them largely insufficient for this task, and signals a need for an advanced strategy. This paper continues the assessment of variable‐resolution in the Community Earth System Model (VR‐CESM) in modeling mountain hydroclimatology to understand the role of grid‐spacing at 55, 28, 14, and 7 km and microphysics, specifically the Morrison and Gettelman (, MG1, https://doi.org/10.1175/2008JCLI2105.1) scheme versus the Gettelman and Morrison (, MG2, https://doi.org/10.1175/JCLI-D-14-00102.1) scheme. Eight VR‐CESM simulations were performed from 1999 to 2015 with the F_AMIP_CAM5 component set, which couples the atmosphere‐land models and prescribes ocean data. Refining horizontal grid‐spacing from 28 to 7 km with the MG1 scheme did not improve the simulated mountain hydroclimatology. Substantial improvements occurred with the use of MG2 at grid‐spacings ≤28 km compared to MG1 as shown with subsequent statistics. Average SWE bias diminished by 9.4X, 4.9X, and 3.5X from 55 to 7 km. The range in minimum (maximum) DJF spatial correlations increased by 0.1–0.2 in both precipitation and SWE. Mountain windward/leeward distributions and elevation profiles improved across hydroclimate variables, however not always with model resolution alone. Disconcertingly, all VR‐CESM simulations exhibited a systemic mountain cold bias that worsened with elevation and will require further examination.