1. Cartesian Grid Embedded Boundary Methods for Partial Differential Equations APDEC ISIC: Phil Colella, Dan Graves, Terry Ligocki, Brian van Straalen (LBNL); Caroline Bono, Bjorn Sjogreen, David Trebotich (LLNL); Marsha Berger (NYU) UC Davis: Mike Barad (DOE CSGF Program), Greg Miller LBNL: Cameron Geddes, Eric Esarey, Wim Leemans (AFRD); Peter Schwartz, Thomas Deschamps (CRD); Adam Arkin, Matt Onsum (PBD). UCSF: David Saloner Univ. of North Carolina: David Adalsteinsson

2. Embedded Boundary Discretization of Conservation Laws Primary dependent variables approximate values at Cartesian cell centers. Divergence theorem over each control volume leads to “finite volume” approximation. Approximation of fluxes based on finite differences of cell-centered data (standard conservative differences in regular cells).

3. Grid Generation Volume fraction Nondimensionalized face area, boundary area Face centroids, boundary centroid Geometric quantities required for discretization: All quantities other than must be computed to second- order accuracy. Aftosmis, Berger, and Melton (1998): generate geometric quantities directly from intersections with surface triangulation of boundary.

4. Grid Generation from an Implicit Function Description Moment equations are derived using the divergence theorem: Overdetermined system solved using least-squares. Right-hand side is obtained from higher-order moments or lower- dimensional moments - bootstrap up from 1D intersection data and moments. Generalizes to arbitrarily high-order accuracy, any number of dimensions.

5. Grid Generation from Implicit Function Descriptions. Implicit function grid generator provides a general and flexible tool for analytic representations, image data, geophysical data.

6. Numerical Analysis of Embedded Boundary Methods If the fluxes at centroids are computed to second-order accuracy, then the truncation error \ satisfies at interior cells at the boundary Modified equation analysis indicates the expected relationship between the truncation error and the solution error. Formal consistency:

7. Embedded Boundary Methods for Elliptic Equations Fluxes are computed using linear interpolation of centered differences in 2D, bilinear interpolation in 3D Stability is nontrivial: matrices are not symmetric, nor M- matrices (linear interpolation in 3D is unstable, bilinear is stable) The smoothing properties of the Green’s function of elliptic operators turn the singular truncation error into a much smoother solution error: in max norm.

8. Embedded Boundary Methods for Hyperbolic Equations Small-cell stability: hybridize with nonconservative stable method, and redistribute the missing mass. increment to maintain conservation. The nonconservative method must be designed carefully to maintain stability, robustness, and accuracy. Modified equation arguments lead us to expect second-order accuracy in L 1, first-order accuracy in max norm. Convergence results in L 1 for a simple wave in a 3D circular tube. Graphical depiction of redistribution Shock diffraction over an ellipsiod

9. Embedded Boundary Software Infrastructure EB Chombo generalizes Chombo: rectangular grids become more general graphs that map into rectangular grids. Nodes of the graph correspond to control volumes, while arcs of the graph correspond to faces that connect adjacent control volumes. The Chombo parallel infrastructure is sufficiently general to support patch- based parallelism for data defined over unions of rectangles.

10. Multigrid and Adaptive Mesh Refinement AMR calculation of shock diffraction over an ellipsoid. Multigrid convergence history for EB discretization of Poisson’s equation on an N 3 grid for N=64,128,256. Embedded boundary methods extend naturally to nested grid hierarchies. Coarsening grid generation is done without reference to original geometric description by coarsening the graph directly, leading to well- defined discretizations of underresolved geometries. Geometric multigrid leads to high-performance, algorithmically scalable solvers.

11. Application: Gas Jet Simulation for Wakefield Accelerators Embedded boundary method to compute the unsteady propagation of a jet into a vacuum chamber. Inviscid EB AMR solvers for time-dependent flow through a nozzle in 2D (axisymmetric) and 3D, including grid generation capabilities. Currently implementing parabolic solvers, including tensor solvers, for compressible viscous terms, heat conduction.

12. Application: Viscous Incompressible Flow Hyperbolic: Parabolic: Elliptic: We solve the Incompressible Navier-Stokes equations using a projection method, splitting the equations into three parts: Each of these equations are solved using the EB algorithms and software described above, and coupled using a second- order accurate predictor- corrector method. Vortex shedding past a cylinder, Re = 200

13. Applications: Non-SciDAC collaborations Diffusion on a surface Can be represented as diffusion in the annular region surrounding the surface and solved using embedded boundary methods. The resulting method is second-order accurate and can be combined with implicit function grid generation methods on biological image data. Convergence study for diffusion on a sphere

14. Applications: Non-SciDAC collaborations Geometric detail Air flow in the trachea (LBNL, LLNL, UCSF): CT image Level set descriptionEmbedded boundary calculation 300  m x 60  m channel contraction flow 100 X Experimental channel Pressure [bar] X-velocity [cm/s] Embedded Boundary Flow Microfluidic MEMS (LBNL, LLNL, UCB)

15. Volume-of-Fluid Methods for Free Boundary Problems Entension of discretization methods, software to the case of sharp fronts. Generalizes formally consistent EB discretizations to case where solution is defined on both sides of a moving boundary. Leverages the EB software infrastructure. Potential applications: tracking flame fronts in premixed combustion, type 1A supernovae. Results using 1D algorithm for a tracked shock overtaking an expansion fan. Image of tracked-front data defined on AMR hierarchy.

16. Future Plans Complete initial implementations for SciDAC applications: compressible Navier-Stokes solver for plasma-wakefield accelerator project, incompressible Navier-Stokes solver for combustion (9/30/2005). Continue development of these algorithms in response to further applications requirements. EB software review: serial, parallel performance, documentation, in preparation for initial public release of EB Chombo (12/31/2005). Continue algorithm development for formally consistent volume-of-fluid front tracking. Proposed work: development of extension of EB infrastructure to dimensions > 3, with underlying mapped Cartesian mesh, in support of FSP edge plasma project.