Daniel T. Graves

Adaptive mesh, finite volume modeling of marine ice sheets

Continental scale marine ice sheets such as the present day West Antarctic Ice Sheet are strongly affected by highly localized features, presenting a challenge to numerical models. Perhaps the best known phenomenon of this kind is the migration of the grounding line - the division between ice in contact with bedrock and floating ice shelves - which needs to be treated at sub-kilometer resolution. We implement a block-structured finite volume method with adaptive mesh refinement (AMR) for three dimensional ice sheets, which allows us to discretize a narrow region around the grounding line at high resolution and the remainder of the ice sheet at low resolution. We demonstrate AMR simulations that are in agreement with uniform mesh simulations, but are computationally far cheaper, appropriately and efficiently evolving the mesh as the grounding line moves over significant distances. As an example application, we model rapid deglaciation of Pine Island Glacier in West Antarctica caused by melting beneath its ice shelf.

A cell-centered adaptive projection method for the incompressible Navier-Stokes equations in three dimensions

We present a method for computing incompressible viscous flows in three dimensions using block-structured local refinement in both space and time. This method uses a projection formulation based on a cell-centered approximate projection, combined with the systematic use of multilevel elliptic solvers to compute increments in the solution generated at boundaries between refinement levels due to refinement in time. We use an L_0-stable second-order semi-implicit scheme to evaluate the viscous terms. Results are presented to demonstrate the accuracy and effectiveness of this approach.

A survey of high level frameworks in block-structured adaptive mesh refinement packages

Over the last decade block-structured adaptive mesh refinement (SAMR) has found increasing use in large, publicly available codes and frameworks. SAMR frameworks have evolved along different paths. Some have stayed focused on specific domain areas, others have pursued a more general functionality, providing the building blocks for a larger variety of applications. In this survey paper we examine a representative set of SAMR packages and SAMR-based codes that have been in existence for half a decade or more, have a reasonably sized and active user base outside of their home institutions, and are publicly available. The set consists of a mix of SAMR packages and application codes that cover a broad range of scientific domains. We look at their high-level frameworks, their design trade-offs and their approach to dealing with the advent of radical changes in hardware architecture. The codes included in this survey are BoxLib, Cactus, Chombo, Enzo, FLASH, and Uintah. A survey of mature openly available state-of-the-art structured AMR libraries and codes.Discussion of their frameworks, challenges and design trade-offs.Directions being pursued by the codes to prepare for the future many-core and heterogeneous platforms.

A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions

We present a method for solving Poisson and heat equations with discon- tinuous coefficients in two- and three-dimensions. It uses a Cartesian cut-cell/embedded boundary method to represent the interface between materi- als, as described in Johansen & Colella (1998). Matching conditions across the interface are enforced using an approximation to fluxes at the boundary. Overall second order accuracy is achieved, as indicated by an array of tests using non-trivial interface geometries. Both the elliptic and heat solvers are shown to remain stable and efficient for material coefficient contrasts up to 106, thanks in part to the use of geometric multigrid. A test of accuracy when adaptive mesh refinement capabilities are utilized is also performed. An example problem relevant to nuclear reactor core simulation is presented, demonstrating the ability of the method to solve problems with realistic physical parameters.

Petascale block-structured AMR applications without distributed meta-data

Adaptive mesh refinement (AMR) applications to solve partial differential equations (PDE) are very challenging to scale efficiently to the petascale regime. We describe optimizations to the Chombo AMR framework that enable it to scale efficiently to petascale on the Cray XT5. We describe an example of a hyperbolic solver (inviscid gas dynamics) and an matrixfree geometric multigrid elliptic solver. Both show good weak scaling to 131K processors without any thread-level or SIMD vector parallelism. This paper describes the algorithms used to compress the Chombo metadata and the optimizations of the Chombo infrastructure that are necessary for this scaling result. That we are able to achieve petascale performance without distribution of the metadata is a significant advance which allows for much simpler and faster AMR codes.

Short Note: An efficient solver for the equations of resistive MHD with spatially-varying resistivity

We regularize the variable coefficient Helmholtz equations arising from implicit time discretizations for resistive MHD, in a way that leads to a symmetric positive-definite system uniformly in the time step. Standard centered-difference discretizations in space of the resulting PDE leads to a method that is second-order accurate, and that can be used with multigrid iteration to obtain efficient solvers.

Granularity and the cost of error recovery in resilient AMR scientific applications

Supercomputing platforms are expected to have larger failure rates in the future because of scaling and power concerns. The memory and performance impact may vary with error types and failure modes. Therefore, localized recovery schemes will be important for scientific computations, including failure modes where application intervention is suitable for recovery. We present a resiliency methodology for applications using structured adaptive mesh refinement, where failure modes map to granularities within the application for detection and correction. This approach also enables parameterization of cost for differentiated recovery. The cost model is built with tuning parameters that can be used to customize the strategy for different failure rates in different computing environments. We also show that this approach can make recovery cost proportional to the failure rate.