Alternative Way to Solve Nonlinear PDEs in Plasma Physics Modeling

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Alternative Way to Solve Nonlinear PDEsin Plasma
Physics Modeling

• Jin Chen
• Princeton Plasma Physics Laboratory
• Princeton University
• SIAM CSE 2007

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Contents

• 2D nonlinear anisotropic thermal conduction
• vary dt and t at every mesh point ij
• Initialize Precondition the nonlinear system
• Update stiffness matrix contributed by nonlinear
  operator
• conclusion

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2D Nonlinear anisotropic thermal conduction
?, B
T1
C
D
?
periodic
periodic
x
A
B
T50exp(x)

• Jacobi-Free Newton-Krylov method
• expensive to form and invert Jacobian
• related preconditioner at each iteration,
• Home-made nonlinear solver based on
• Pseudo-transient continuation.

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Pseudo-transient continuation based solve
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Variale timestep and damping factor
B

• Timestep depends on the conductivity and mesh
  size in both x and y direction
• ?1, isotropic. dt is a
  symmetric function of (dx,dy)
• maximized at dxdy.
• Large ?,anisotropic. the larger dx is,
• the bigger timestep the smaller
  dy is,
• the bigger timestep.

x
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Convergence behavior
cc(?_?)
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Physics based Initialization Precondition
B
8
Stiffness matrix v.s. preserving symmetric
structure
n collocation point on a triangle ij the (ij)th
mesh point
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Well preserved symmetric structure
Contour plot of preconditioner and solution
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Conclusion

• 2D nonlinear anisotropic thermal conduction
• in pasma physics modeling
• We developed an nonlinear solver Variable
  Dynamic Relaxation
• Use variable dt and t Powerful
• Initialize and Precondition the system by a
  linear anisotropic system Efficient
• Update stiffness matrix by scale the linear
  stiffness matrix from isotropic system using an
  nonlinear vector at every timestep Efficient and
  Reliable