The Applied Numerical Algorithms Group develops numerical methods for the solution of partial differential equations in complex geometries. The equations of particular interest are those that arise from compressible and incompressible fluid dynamics. The representation of complex geometries is based on an embedded boundary model, in which the boundary of a complex domain cuts away at a regular Cartesian grid. This allows the use of highly efficient and well understood methods in the regions of the grid away from the boundary, and confines the use of irregular grid methods to the small region of the domain near the boundary.

Embedded Boundary Methods

• Support Software

  • Support software for AMR calculations has been released on the web. See the Chombo: Adaptive Mesh Refinement Library. A user document and a reference manual are available online as well.

  • Support software for representing embedded boundaries is described. A reference manual is also available.

  • Simple geometry generation capabilities are included in the embedded boundary package. We also have an interface to Aftosmis and Berger's Cart3d system.

  • A software framework for adaptive mesh refinement (AMR) calculations of hyperbolic systems is under development. (Click here for an example AMR grid) This framework is general enough to support a variety of physical models written by the user. A single-fluid gas dynamics model is included. Multifluid tracking and capturing modules are available on request.

  • Visualization and post-processing software for embedded boundary and AMR data is under development.
  
  

• Numerical Solution Software Development Efforts

  • Compressible Gas Dynamics

    The compressible gas dynamics project is developing 2D and 3D adaptive methods to simulate time-dependent compressible flows in complex geometries using an embedded boundary representation. The small cells issue is addressed by redistribution. The basic gas dynamics solver is an operator-split second order Godunov method. Multifluid capability will also be incorporated.

  • Poisson's Equation

    We are developing 2D and 3D methods for the solution of Poisson's equation on complex geometries using Cartesian meshes with embedded boundaries and adaptive mesh refinement. The solution procedure is based on the multigrid-like AMRPoisson algorithm.

  • Incompressible Fluid Dynamics

    The embedded boundary Poisson solver is incorporated into a cell-centered projection method to simulate incompressible fluid flow. The advective parts of the method are similar to the embedded boundary gas dynamics method. This is a joint project with Gerry Puckett and Mark Sussman at UC Davis.

  • Low Mach Number Fluid Dynamics

    We are developing a projection method for low speed compressible flow, based on Karen Pao's all-speed method.

  
  

• High Performance Computing

  • We are implementing adaptive embedded boundary methods on large-scale parallel architectures.