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Network and Grid Computing –Modeling, Algorithms, and Software

This book explores multi-modeling problems in network and grid computing, focusing on complex system formulation and applications. It delves into characteristics, research issues, and case studies like inviscid-viscous flows, and nonlinear cases. It covers modeling, algorithms, and software in depth.

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Network and Grid Computing –Modeling, Algorithms, and Software

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  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

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