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

Trend in High Performance and Supercomputing Parallel Computing Distributed Computing Network Computing Grid Computing

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

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

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

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

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

Research Issues Modeling Algorithms Software

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

Simplest Case 1D Scalar equation Linear

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

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

Steady State Exact solution Boundary layer Discontinuity at the interface

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

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

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

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)

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

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