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Multigrid for PDE Systems

Multigrid for PDE Systems. Achi Brandt. Compressible Navier-Stokes: 2D. B.C.: Non-elliptic BVP unknown. Non-conservative MHD equations. MG Reduction Principles. Separate: interior boundary. Scale separation: Local processing at each scale. Principal terms at each scale.

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Multigrid for PDE Systems

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  1. Multigridfor PDE Systems Achi Brandt

  2. Compressible Navier-Stokes: 2D B.C.: Non-elliptic BVPunknown

  3. Non-conservative MHD equations

  4. MG Reduction Principles • Separate: interior boundary • Scale separation: Local processing at each scale • Principal terms at each scale

  5. Principal Terms Principal terms on scale h: the larger coefficients in (or in replacing by , etc.) Important for: 1. Boundary conditions 2. Discretization (stability) 3. Relaxation forms 4. Interpolation

  6. Principal-Term Relaxation fixed in calculating _____ Relax as FOR << 1 MAY CHANGE ON COARSE LEVELS ! Quasi-linear UU + … = f x Relax as fixed in calculating LIKE LINEAR ! AT ANY SCALE !

  7. PDE Systems e.g., Stokes

  8. Non-principal terms of a matrix Equation N is non-principal part of A if the iterations converge fast. Or Or

  9. on scale h h Principal terms of a system L are all terms which contribute to the principal part of det L on scale h h

  10. Stokes m11=2 m12=0 m13=1 m21=0 m22=2 m23=1 m31=1 m32=1 m33=0

  11. Inter-Grid Transfers mj = order of of the j-th function mi = order of of the i-th eq. residual mij = order of differentiation (principal)of the j-th function in the i-th equation Full efficiency rule Border case One cycle euclidean mi > mij mi = mij Asymptotic mi + mj > mij mi + mj = mij

  12. Below the border: divergence Above it: local mode analysis efficiency guaranteed.Should always be used Border case: supplemental global mode analysis is needed.

  13. h-principal L Compressible Navier-Stokes(on the viscous scale) Central Cauchy-Riemann Central (Navier-) Stokes

  14. Resistive MHD Equations: Conservative Form

  15. Non-conservative MHD equations

  16. Potentially principal MHD matrix

  17. Potentially principal MHD matrix _____ 0 _________ Principal, assuming

  18. 0 Degeneracy! _____ 0 = O(h)? Principal, assuming

  19. Cauchy-Riemann Equations Uy – Vx = F Ux + Vy = G One boundary condition Difference equations: Eq.1 defined at grid nodes Eq.2 defined at cell centers (5-point Laplacian) v v v u u u u v v F v uGu u u v v v 1 2

  20. Central Differencing of Cauchy-Riemann Eq.

  21. 1 1 -4 1 1 a b a b a b c d c d c d a b a b a b c d c d c d a b a b a b unstable modes: Positive type but “quasi-elliptic”: subtle instability. Bad smoothing by any relaxation. Multigrid: separate transfers for a, b, c, d. Three of them are redundant. vvvvv vv uuuuuuu vvvvvvv uuuuuuu vvvvv vv

  22. Cauchy-Riemann Define Choose Gauss-Siedel (GS) for Smoothing factor Same for u Distributive GS (DGS) for

  23. Relaxation of Systems Distributive Relaxation: DGS: Gauss-Seidel for Generally:can be obtained for any choice of relaxation for det L Method: Mij = cofactor of Lji Guiding principle: Principal can be obtained. Often by finding

  24. Incompressible Navier Stokes : 2D two boundary conditions Differencing: v u p u h-ellipticity of depends on 1 2 3 ~ ~ 2 1 1 3 v 2

  25. Relaxation: Distribution cofactors of last row in , divided by their common divisor Relaxation of is reduced to that of Box GS on coarsest grids and near boundaries FAS FMG solution to in one Non staggered grids. Conservative schemes.

  26. Relaxation: GS for GS for DGS for chosen to satisfy 1 2 3

  27. MG Reduction Principles • Separate: interior boundary • Scale separation: Local processing at each scale • Principal terms at each scale • Design of discretizations, relaxation, interpolation, restriction in terms of the (scale-dependent) simple factors of the determinant of the principal quasi-linear operator scalar low-order factors, each featuring its own type (ellipticity measure, anisotropy,…) and characteristic directions • No linearizations • Design, debug - guided by quantitative predictions

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