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Krylov Subspace Methods & Nullspace for Resistive MHD

Krylov Subspace Methods & Nullspace for Resistive MHD. Jin Chen Plasma Physics Lab Princeton University. Background I: Resistive MHD equation. Supplementary equations. represent fast thermal equilibration along field lines artificial wave speed sB/  1/2.

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Krylov Subspace Methods & Nullspace for Resistive MHD

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  1. Krylov Subspace Methods &Nullspace for Resistive MHD Jin Chen Plasma Physics Lab Princeton University Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  2. Background I: Resistive MHD equation Supplementary equations • represent fast thermal equilibration along field lines • artificial wave speed sB/1/2 extreme stiffness and anisotropy. Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  3. Numerical Challenges • Stiffness: Time-scales that impact nonlinear MHD evolution include • Parallel particle motion leading to parallel thermal equilibration • over flux surfaces in 1-10 nanoseconds (10^-9) . • MHD wave propagation over global scales in microseconds (10^-6) . • Magnetic fluctuations and tearing • in hundreds of microseconds to milliseconds (10^-3) . • Nonlinear profile modification and transport • in tens to hundreds of milliseconds (10^-3). • Global resistive diffusion over seconds. • Anisotropy: Magnetization of nearly collisionless particles leads to • Effective thermal diffusivity ratios, , reaching and exceeding 1010. • Shear wave resonance that allows nearly singular behavior of MHD modes. Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  4. Background II: Poisson Equation with Neumann Boundary Condition F equation in MHD (related to perturbed toroidal flux) • Solvability not every system of equation has a solution. • Unique if F is a solution, so is F + c. Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  5. Outlines • Introduction • Domain decomposition and grid topology • Lagrange finite element discretization • Numerical singularity for Neumann Boundary Condition and Nullspace • CG and GMRES for non-singular linear equation • Nullspace based CG and GMRES for singular linear equation • Numerical Experiments • Conclusion Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  6. Introduction I: Scalar Representation Each Time Step: Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  7. Introduction II: 1st and 2nd kind of Elliptic Equations g is the resistivity 13 elliptic solver calls at each time step. Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  8. Parallel Domain Decomposition • (structured) finite difference in the toroidal direction • (unstructured) finite elements in the poloidal planes Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  9. Parallel Grid Topology Parallel Ordering Global Ordering Local Ordering Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  10. Lagrange Finite Element • Linear element • 2nd order element • 3rd order element Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  11. GMRES and ICCG for Au=b 16p X 20,000 vertices X 13 unknowns A u (per time step) • Unsymmetrical: GMRES. • Reform the poisson operator to have symmetric structure: CG. Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  12. 1.Solvability: 2.Unique: Least square solution Mean zero Numerical Singularity for Poisson Equation with Neumann B.C. Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  13. Nullspace Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  14. Nullspace based CG for singular systems Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  15. If Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  16. If … Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  17. If … Re-orthogonlization To assure there exists a solution. Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  18. Nullspace based GMRES for singular systems Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  19. Result II: GMRES v.s. ICCG 16p X 20,000 vertices X 13 unknowns A u (per time step) From GMRES to ICCG, achieved A factor of 2 times speedup Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  20. Result III: Application of Nullspace • F equation, • Singular check: Ae=0, • Solvability check: (b,e)=0, • Re-orthogonalization: b=b(b,e)/(e,e), • Uniqueness check: (x,e)=0, • CG with nullspace, • GMRES with nullspace, Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  21. Eigenmode k=2 l=0 Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

  22. Conclusion • Achieved, • Making progress, • Numerical Challenges left for future. Krylov Subspace Methods & Nullspace for Resistive MHD, Jin Chen, PPPL, SIAM2004

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