Our basic discretization approach is that of finite difference methods on rectangular grids. Regularity in space leads to regular data layouts, making it easier to optimize for various types of data locality. The regular geometric structure of the spatial distribution of unknowns leads to fast efficient iterative solvers for elliptic and parabolic problems based on multigrid iteration.

To represent engineering geometries, we need to compute solutions in domains whose whose shape (and trajectory, if they are moving) are known. we will use a volume-of-fluid representation of the discretized solution near the boundary. In this approach, the surface is represented by its intersection with an underlying rectangular grid. This leads to a natural, finite-volume discretization of the PDE on irregular control volumes adjacent to the boundary. This approach is variously referred to as a Cartesian grid, embedded boundary, or cut cell method (in this proposal, we will use the term embedded boundary (EB)). One of the principal advantages of the embedded boundary method is that there already exists software for generating the description of the geometry on the grid starting from surface tesselations produced, for example, by a CAD system [1]. addition, there has been considerable progress on the development of discretizations for this type of geometry representation [21,20,26,23]. In this approach, the primary dependent variables are centered at the centers of the rectangular grid cells, even if those cells correspond to irregular control volumes. Because the values are centered at regular points on the grid, it is routine to construct finite difference approximations to various partial differential operators on the irregular control volumes that are consistent, conservative, and stable/well-conditioned.

Dynamic adaptive mesh refinement is a critical requirement for all three of our stakeholder applications. The PIC codes being used by our stakeholders are all based on uniform discretizations of space for the fields. However, the grid resolution required in such calculations is a local function of the particle distribution, and, for example, one could use a far larger mesh spacing in regions where there are no particles [3]. This represents a substantial potential savings, since, in many problems, only a small fraction of the domain contains particles. It is also believed that there is an additional requirement for mesh resolution in the neighborhood of the source in heavy-ion fusion, to efficiently capture the fields near the emitting surface. For the laser-plasma accelerator problem, adaptivity is essential in resolving the time-dependent dynamics of the gas jet. In magnetic fusion, the macroscopic stability of the plasma often depends on the physical processes present in a narrow reconnection layer. In combustion, the grid resolution required in the neighborhood of the flame front is much greater than elsewhere in the flow. Finally, adaptive refinement could make the difference between success and failure in attempting to integrate the Vlasov equations using grid-based discretizations in six-dimensional phase space.

Our adaptive mesh methods are based on the block-structured adaptive mesh refinement (AMR) algorithms of Berger and Oliger [8,9,6]. In this approach, the regions to be refined are organized into rectangular patches of several hundred to several thousand grid cells per patch. Thus one is able to use the high-resolution rectangular grid methods described above to advance the solution in time. Furthermore, the overhead in managing the irregular data is amortized over a relatively large amounts or floating point work on regular arrays. For time-dependent problems, refinement is performed in time as well as in space. Each level of spatial refinement uses its own stable time step, with the time steps at a level constrained to be integer multiples of the time steps at all finer levels. AMR is a mature technology for problems without geometry, with a variety of implementations and applications for various nonlinear combinations of elliptic, parabolic and hyperbolic PDE's [3,25,14,2,19,30,10,18,12,22,29]. In particular, the collection of applications here address most of the major algorithmic issues in developing adaptive algorithms for the applications described above, such as adaptive multigrid solvers for Poisson's equation, the representation of non-ideal effects in MHD, and the coupling of particle methods to AMR field solvers. AMR has also been successfully coupled to the volume-of-fluid algorithms described above [11,7,28,24].