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Numerical Methods for Solving Neumann Boundary Value Poisson Equations in M3D

Learn about advanced numerical techniques such as CG and GMRES for solving Poisson equations with Neumann boundary values in M3D, including the use of null space methods. Explore strategies for handling singular systems and unique solutions.

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Numerical Methods for Solving Neumann Boundary Value Poisson Equations in M3D

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  1. How To Solve Poisson Equation with Neumann Boundary Values Jin Chen CPPG

  2. F equation in M3D (Auxiliary quantities related to perturbed toroidal flux) Background

  3. Outlines • Characteristics of Neumann Boundary Values • Numerical singularity of such Boundary Values • Null Space method for such singularity • CG and GMRES for non-singular linear equation • Null Space based CG and GMRES for singular linear equation • Application to eigenvalue problem • Application to M3D

  4. Characteristics of Neumann Boundary Values • Solvability not every system of equation has a solution. • Unique if u is a solution, so is u + c.

  5. Eigenvalues of the Poisson Equation with Neumann BV

  6. Is there anything we can do? Let’s assume A is non-singular FIRST. • Direct solver • Iterative solver Krylov Subspace Methods.

  7. Iterative solver

  8. Krylov Subspace Methods… • Conjugate Gradient (CG) symmetric positive definite matrix • Generalized Minimal Residual (GMRES) non-symmetric indefinite matrix

  9. CG

  10. GMRES

  11. Numerical Complexity

  12. 1.Solvability 2.Unique Mean zero Least square solution If A is singular …

  13. Definition of Null Space

  14. Null Space

  15. CG for singular systems

  16. If

  17. If …

  18. If …

  19. If …

  20. If … Re-orthogonlization To assure there exists a solution.

  21. GMRES for singular systems

  22. GMRES for singular systems …

  23. Numerical Experiment

  24. Matrix Structure

  25. Singular and Solvability Check

  26. Spectrum with a zero eigenvalue

  27. Strategy I: Fix one point Spectrum shift

  28. Spectrum shift by one point fixing… You are solving an approximate problem !!!

  29. Eigenmode k=2, l=0

  30. Strategy II: null space

  31. Eigenmode k=2 l=0

  32. Eigenmode k=3, l=0

  33. Application in M3D • 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,

  34. If you want to try it… … I am happy to help you …

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