J. David Moulton

Convergence of mimetic finite difference discretizations of the diffusion equation

Abstract The main goal of this paper is to establish the convergence of mimetic discretizations of the first-order system that describes linear diffusion. Specifically, mimetic discretizations based on the support-operators methodology (SO) have been applied successfully in a number of application areas, including diffusion and electromagnetics. These discretizations have demonstrated excellent robustness, however, a rigorous convergence proof has been lacking. In this research, we prove convergence of the SO discretization for linear diffusion by first developing a connection of this mimetic discretization with Mixed Finite Element (MFE) methods. This connection facilitates the application of existing tools and error estimates from the finite element literature to establish convergence for the SO discretization. The convergence properties of the SO discretization are verified with numerical examples.

Managing complexity in simulations of land surface and near-surface processes

Increasing computing power and the growing role of simulation in Earth systems science have led to an increase in the number and complexity of processes in modern simulators. We present a multiphysics framework that specifies interfaces for coupled processes and automates weak and strong coupling strategies to manage this complexity. Process management is enabled by viewing the system of equations as a tree, where individual equations are associated with leaf nodes and coupling strategies with internal nodes. A dynamically generated dependency graph connects a variable to its dependencies, streamlining and automating model evaluation, easing model development, and ensuring models are modular and flexible. Additionally, the dependency graph is used to ensure that data requirements are consistent between all processes in a given simulation. Here we discuss the design and implementation of these concepts within the Arcos framework, and demonstrate their use for verification testing and hypothesis evaluation in numerical experiments. We describe a conceptual model for managing complexity in multiphysics software.A process tree describes how individual equations are coupled.A dependency graph describes how variables are dependent upon each other.The model is implemented within the Arcos software framework.Examples of code and model runs are shown to demonstrate the idea.

The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient

Numerical schemes for nonlinear parabolic equations based on the harmonic averaging of cell-centered diffusion coefficients break down when some of these coefficients go to zero or their ratio grows. To tackle this problem, we propose new mimetic finite difference schemes that use a staggered discretization of the diffusion coefficient. The primary mimetic operator approximates div ( k ? ) ; the derived (dual) mimetic operator approximates - ? ( ? ) . The new mimetic schemes preserve symmetry and positive-definiteness of the continuum problem which allows us to use algebraic solvers with optimal complexity. We perform detailed numerical analysis of the new schemes for linear elliptic problems and a specially designed linear parabolic problem that has solution dynamics typical for nonlinear problems. We show that the new schemes are competitive with the state-of-the-art schemes for steady-state problems but provide much more accurate solution dynamics for the transient problem.

An efficient matrix-free algorithm for the ensemble Kalman filter

In this work, we present an efficient matrix-free ensemble Kalman filter (EnKF) algorithm for the assimilation of large data sets. The EnKF has increasingly become an essential tool for data assimilation of numerical models. It is an attractive assimilation method because it can evolve the model covariance matrix for a non-linear model, through the use of an ensemble of model states, and it is easy to implement for any numerical model. Nevertheless, the computational cost of the EnKF can increase significantly for cases involving the assimilation of large data sets. As more data become available for assimilation, a potential bottleneck in most EnKF algorithms involves the operation of the Kalman gain matrix. To reduce the complexity and cost of assimilating large data sets, a matrix-free EnKF algorithm is proposed. The algorithm uses an efficient matrix-free linear solver, based on the Sherman–Morrison formulas, to solve the implicit linear system within the Kalman gain matrix and compute the analysis. Numerical experiments with a two-dimensional shallow water model on the sphere are presented, where results show the matrix-free implementation outperforming an singular value decomposition-based implementation in computational time.

An electrostatic Particle-In-Cell code on multi-block structured meshes

Abstract We present an electrostatic Particle-In-Cell (PIC) code on multi-block, locally structured, curvilinear meshes called Curvilinear PIC (CPIC). Multi-block meshes are essential to capture complex geometries accurately and with good mesh quality, something that would not be possible with single-block structured meshes that are often used in PIC and for which CPIC was initially developed. Despite the structured nature of the individual blocks, multi-block meshes resemble unstructured meshes in a global sense and introduce several new challenges, such as the presence of discontinuities in the mesh properties and coordinate orientation changes across adjacent blocks, and polyjunction points where an arbitrary number of blocks meet. In CPIC, these challenges have been met by an approach that features: (1) a curvilinear formulation of the PIC method: each mesh block is mapped from the physical space, where the mesh is curvilinear and arbitrarily distorted, to the logical space, where the mesh is uniform and Cartesian on the unit cube; (2) a mimetic discretization of Poissons equation suitable for multi-block meshes; and (3) a hybrid (logical-space position/physical-space velocity), asynchronous particle mover that mitigates the performance degradation created by the necessity to track particles as they move across blocks. The numerical accuracy of CPIC was verified using two standard plasma–material interaction tests, which demonstrate good agreement with the corresponding analytic solutions. Compared to PIC codes on unstructured meshes, which have also been used for their flexibility in handling complex geometries but whose performance suffers from issues associated with data locality and indirect data access patterns, PIC codes on multi-block structured meshes may offer the best compromise for capturing complex geometries while also maintaining solution accuracy and computational efficiency.

Efficient nonlinear solvers for Laplace-Beltrami smoothing of three-dimensional unstructured grids

The Laplace-Beltrami system of nonlinear, elliptic, partial differential equations has utility in the generation of computational grids on complex and highly curved geometry. Discretization of this system using the finite-element method accommodates unstructured grids, but generates a large, sparse, ill-conditioned system of nonlinear discrete equations. The use of the Laplace-Beltrami approach, particularly in large-scale applications, has been limited by the scalability and efficiency of solvers. This paper addresses this limitation by developing two nonlinear solvers based on the Jacobian-Free Newton-Krylov (JFNK) methodology. A key feature of these methods is that the Jacobian is not formed explicitly for use by the underlying linear solver. Iterative linear solvers such as the Generalized Minimal RESidual (GMRES) method do not technically require the stand-alone Jacobian; instead its action on a vector is approximated through two nonlinear function evaluations. The preconditioning required by GMRES is also discussed. Two different preconditioners are developed, both of which employ existing Algebraic Multigrid (AMG) methods. Further, the most efficient preconditioner, overall, for the problems considered is based on a Picard linearization. Numerical examples demonstrate that these solvers are significantly faster than a standard Newton-Krylov approach; a speedup factor of approximately 26 was obtained for the Picard preconditioner on the largest grids studied here. In addition, these JFNK solvers exhibit good algorithmic scaling with increasing grid size.

A multiscale multilevel mimetic (M3) method for well-driven flows in porous media☆

Abstract The multiscale multilevel mimetic (M 3 ) method was designed in [13] for the accurate modeling of two-phase flows in highly heterogeneous porous media on general polygonal meshes. In this article, it is extended to well-driven flows in porous media. We demonstrate its ability to treat accurately non-orthogonal locally-refined meshes and tensorial material properties.

A Multilevel Multiscale Mimetic (M3) Method for an Anisotropic Infiltration Problem

Modeling of multiphase flow and transport in highly heterogeneous porous media must capture a broad range of coupled spatial and temporal scales. Recently, a hierarchical approach dubbed the Multilevel Multiscale Mimetic (M3) method, was developed to simulate two-phase flow in porous media. The M3 method is locally mass conserving at all levels in its hierarchy, it supports unstructured polygonal grids and full tensor permeabilities, and it can achieve large coarsening factors. In this work we consider infiltration of water into a two-dimensional layered medium. The grid is aligned with the layers but not the coordinate axes. We demonstrate that with an efficient temporal updating strategy for the coarsening parameters, fine-scale accuracy of prominent features in the flow is maintained by the M3 method.

Advanced Simulation Capability for Environmental Management (ASCEM): Early Site Demonstration

The U.S. Department of Energy’s Office of Environmental Management (EM), Technology Innovation and Development (EM-32), is supporting development of the Advanced Simulation Capability for Environmental Management (ASCEM). ASCEM is a state-of-the-art scientific tool and approach for understanding and predicting contaminant fate and transport in natural and engineered systems. This modular and open-source, high-performance computing tool will facilitate integrated approaches to modeling and site characterization that enable robust and standardized assessments of performance and risk for EM cleanup and closure activities. As part of the initial development process, a series of demonstrations was defined to test ASCEM components and provide feedback to developers, engage end users in applications, and lead to an outcome that would benefit the sites. The demonstration was implemented for a sub-region of the Savannah River Site General Separations Area that includes the F-Area Seepage Basins. The physical domain included the unsaturated and saturated zones in the vicinity of the seepage basins and the Fourmile Branch. An unstructured mesh was used to fit the grid to the hydrostratigraphy and topography of the site. The calculations modeled variably saturated flow, and the resulting flow field was used in simulations of the advection of non-reactive species and the reactivetransport of uranium. As part of the demonstrations, data management, visualization, and uncertainty quantification tools were developed to analyze simulation results and existing site data. These new tools can be used to provide summary statistics, including information on which simulation parameters were most important in predicting uncertainty and visualizing the relationships between model input and output.

An intermediate-scale model for thermal hydrology in low-relief permafrost-affected landscapes

Integrated surface/subsurface models for simulating the thermal hydrology of permafrost-affected regions in a warming climate have recently become available, but computational demands of those new process-rich simu- lation tools have thus far limited their applications to one-dimensional or small two-dimensional simulations. We present a mixed-dimensional model structure for efficiently simulating surface/subsurface thermal hydrology in low-relief permafrost regions at watershed scales. The approach replaces a full three-dimensional system with a two-dimensional overland thermal hydrology system and a family of one-dimensional vertical columns, where each column represents a fully coupled surface/subsurface thermal hydrology system without lateral flow. The system is then operator split, sequentially updating the overland flow system without sources and the one-dimensional columns without lateral flows. We show that the app- roach is highly scalable, supports subcycling of different processes, and compares well with the corresponding fully three-dimensional representation at significantly less computational cost. Those advances enable recently developed representations of freezing soil physics to be coupled with thermal overland flow and surface energy balance at scales of 100s of meters. Although developed and demonstrated for permafrost thermal hydrology, the mixed-dimensional model structure is applicable to integrated surface/subsurface thermal hydrology in general.