Network and Grid Computing –Modeling, Algorithms, and Software

1. Network and Grid Computing–Modeling, Algorithms, and Software Mo Mu Joint work with Xiao Hong Zhu, Falcon Siu

2. Trend in High Performance and Supercomputing • Parallel Computing • Distributed Computing • Network Computing • Grid Computing

3. Applied Math and Scientific Computing • Single model applications • Multi-scale problems • Multi-domain problems • Multi-media problems • Multi-modeling problems

4. Multi-modeling Problems • Inviscid-viscous flows • Compressible-incompressible flows • Turbulent-laminar flows • Interface stability with different media • Composite materials • Complex systems

5. Formulation • Different models in local regions • Interface coupling conditions • Complexity across the interfaces • Physical • Discontinuity • Boundary layer • Geometrical • Topology • Moving interfaces

6. Applications • Originated from a underlying problem where a global model approximation might not be applicable – physically or mathematically • Reduced from a underlying global model • Computational efficiency • Approximation accuracy • Stiffness • Domain decomposition

7. Characteristics • Modeling complex physical systems • Sharp resolution of interface structures • Local solvers with mature methods and codes • Software integration • Grid computing • More accurate and efficient in some cases

8. Research Issues • Modeling • Algorithms • Software

9. Case Study: Inviscid-Viscous Flow • Hybrid hyperbolic and parabolic problem • Example: Euler/N-S coupling • Characteristics • Nonlinear • System of equations • 2D or 3D • Existing work (Q, Cai, etc.) • Linear • Scalar equation • 2D

10. Simplest Case • 1D • Scalar equation • Linear

11. Hybrid Model • Local models (boundary layer problem with small viscosity) • Interface condition • Initial condition consistent with the boundary and interface conditions

12. (1) Outflow on Γ • Local models: a>0, b>0 • Boundary conditions • Interface condition • Well-posed • Fully decoupled: inviscid -> viscous

13. Steady State • Exact solution • Boundary layer • Discontinuity at the interface

14. Numerical Solution: Steady State • Inviscid solver • Upwind scheme • Explicit computation • Viscous solver • Central difference plus upwind for the elliptic operator • Forward difference for interface condition with input from the inviscid solver • Thomas solver • Cheap inviscid computation with “large” spacing • Sharp boundary layer structure with few grid points

15. Numerical Solution: Unsteady State • Explicit scheme • The same spatial discretization as in the steady state • Explicit computation for both inviscid solver and viscous solver at each time step • CFL: • Different spacing, thus different time step

16. Full Viscous Model • n(x) > 0, could be constant or piecewise constants • Unified treatment for modeling, numerical methods, … • Interface condition implicitly imposed • Approximation to the hybrid model • Boundary layer is difficult to resolve • Numerical solution • Central difference plus upwind for viscous flow • Local refinement strategies required • Accuracy at the interface singularity ? • Global system solved

17. Higher Dimensional Problems • Mixed inflow and outflow on the interface • Coupled hybrid models • Decoupling iterative approaches • Domain decomposition (Q, Cai) • Interface relaxation (Mu, Rice) • Optimization-based interface matching (Du)

18. Nonlinear Case: Burger’s equation • Non-uniqueness • Shock/boundary layer interaction at the interface • More interface conditions required, e.g. R-H condition

19. Nonlinear Case: System of equations • Hybrid Euler/N-S models • Complicated interface structures • Slow convergence