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Astronum 2012 The 7th Annual Interna4onal Conference On Numerical Modeling of Space
Plasma Flows, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
A Solution Accurate, Efficient and Stable 3D Unsplit Staggered Mesh (USM) MHD
Solver in FLASH
Dongwook Lee
University of Chicago
The Flash Center for Computa2onal Science
Two USM papers
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
2D paper, JCP, 2009
3D paper, submi.ed to JCP, 2012
Astrophysical Applica2ons
HEDP Applica2ons
Outline
q Part 1: q Reduced/Full corner-‐transport-‐upwind (CTU) for 3D q 6 Riemann solves for USM; 3 for UHD in 3D q CFL stability limit reaches 1 in the full CTU algorithm
q Part 2: q Third order electric fields constrained-‐transport (CT) scheme for USM-‐
MHD (Lee and Deane, JCP, 2009) q A new upwind biased CT (Lee, JCP, 2012, under review)
q Summary
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Part 1
3D Unsplit Algorithms for USM & UHD: Reduced & Full CTU
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
MHD Governing Equa2ons
q MHD system of equa4ons:
q This can be wri[en in a simple matrix form:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
∂U∂t
+∂F∂x
+∂G∂y
+∂H∂z
= 0
q A primi4ve form:
where the coefficient matrix is
Linearized System
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
High-‐Order Corner Transport Upwind (CTU)
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
q Use a linearized system in primi4ve form
q High-‐order spa4al reconstruc4on & temporal evolu4on to obtain Riemann states at interfaces (e.g., MH)
q Solve Riemann problems using the Riemann states
q x-‐direc4on:
q y-‐direc4on:
q z-‐direc4on:
High-‐Order Corner Transport Upwind (CTU)
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
q Use a linearized system in primi4ve form
q High-‐order spa4al reconstruc4on & temporal evolu4on to obtain Riemann states at interfaces (e.g., MH)
q Solve Riemann problems using the Riemann states
q x-‐direc4on:
q y-‐direc4on:
q z-‐direc4on:
Normal predictor Transverse corrector
Usual CTU Algorithms take…
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Usual CTU Algorithms take…
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Usual CTU Algorithms take…
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
q Solve 1D high-‐order reconstruc4ons using characteris4c tracing in each
normal direc4on to get Riemann states
Usual CTU Algorithms take…
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Usual CTU Algorithms take…
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
F∗ Vi, j,En+1/2,Vi+1, j,W
n+1/2( )G∗ Vi, j,Nn+1/2,Vi, j+1,S
n+1/2( )
Usual CTU Algorithms take…
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
F∗ Vi, j,En+1/2,Vi+1, j,W
n+1/2( )G∗ Vi, j,Nn+1/2,Vi, j+1,S
n+1/2( )
+ Trans Fluxes in y & z
+ Trans Fluxes in x & y
+ Trans Fluxes in x & z
This Leads To 6-‐CTU & 12-‐CTU
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
q Depends on the number of Riemann problems per cell per 4me step: q 6-‐CTU (6 Riemann problems)
q Colella, JCP, 1990; Gardiner & Stone, JCP, 2008 q A simple direct extension of 2D CTU to 3D accoun4ng for a single
intermediate state n+1/2 q CFL < ½
q 12-‐CTU (12 Riemann problems) q Saltzman, JCP, 1992; Gardiner & Stone, JCP, 2008; Minia4 & Mar4n,
ApJS, 2011 q More expensive approach q Accoun4ng for two intermediate states of n+1/3 & n+1/2 q CFL < 1
q (Gardiner & Stone, JCP, 2008) CPU12−ctu
CPU6−ctu
≈1
Different Treatment for Transverse Fluxes in USM
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Normal predictor Transverse corrector q Use characteris4c tracing in BOTH normal and transverse flux
calcula4ons (Lee & Deane, JCP, 2009)
Different Treatment for Transverse Fluxes in USM
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Normal predictor Transverse corrector q Use characteris4c tracing in BOTH normal and transverse flux
calcula4ons (Lee & Deane, JCP, 2009)
! Normal predictor
Different Treatment for Transverse Fluxes in USM
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Normal predictor Transverse corrector q Use characteris4c tracing in BOTH normal and transverse flux
calcula4ons (Lee & Deane, JCP, 2009)
! Transverse corrector
! Normal predictor
Different Treatment for Transverse Fluxes in USM
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Normal predictor Transverse corrector q Use characteris4c tracing in BOTH normal and transverse flux
calcula4ons (Lee & Deane, JCP, 2009)
! Transverse corrector
! Normal predictor
Different Treatment for Transverse Fluxes in USM
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Normal predictor Transverse corrector q Use characteris4c tracing in BOTH normal and transverse flux
calcula4ons (Lee & Deane, JCP, 2009)
! Transverse corrector
! Normal predictor
Different Treatment for Transverse Fluxes in USM
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Normal predictor Transverse corrector q Use characteris4c tracing in BOTH normal and transverse flux
calcula4ons (Lee & Deane, JCP, 2009)
! Transverse corrector
! Normal predictor
! Monotonicity
! Stability
Characteris2c tracing for Transverse corrector
q A jump rela4onship:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
AyVl +m=1
m0−1
Σλmrm Δ~α =AyVr −
m=m0
7
Σ λmrm Δ~α
Characteris2c tracing for Transverse corrector
q A jump rela4onship:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
m=1
7
Σλmrm Δ~α =AyVr −AyVl
m=1
7
Σλmrm Δ~α =AyVr −AyVl
=AyΔ
=Gr −Gl
= Gi+1/2, j −Gi−1/2, j
Characteris2c tracing for Transverse corrector
q The summa4on of all waves becomes an upwind transverse flux gradient:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
m=1
7
Σλmrm Δ~α =AyVr −AyVl
=AyΔ
=Gr −Gl
= Gi+1/2, j −Gi−1/2, j
Characteris2c tracing for Transverse corrector
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
q The summa4on of all waves becomes an upwind transverse flux gradient:
Different Treatment for Transverse Fluxes in USM
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
! Transverse corrector
! Normal predictor
Single-‐Step Data Reconstruc2on-‐Evolu2on
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Reduced & Full CTU in UHD
q New approach of using characteris4c tracing for BOTH normal predictor and transverse corrector
q Reduced 3D CTU q A direct extension of 2D CTU to 3D q Requires 3 Riemann solves for 3D (6-‐CTU needs 6 Riemann solves) q Only including second cross deriva4ves q CFL limit ~ 0.5
q Full 3D CTU q Full considera4ons of accoun4ng for third cross deriva4ves q Requires 3 Riemann solves for 3D (12-‐CTU needs 12 Riemann solves) q CFL limit ~ 1.0
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
∂3 /∂x∂y∂z
Reduced & Full CTU in USM
q New approach of using characteris4c tracing for BOTH normal predictor and transverse corrector
q Reduced 3D CTU q A direct extension of 2D CTU to 3D q Requires 6 Riemann solves for 3D (6-‐CTU needs 6 Riemann solves) q Only including second cross deriva4ves q CFL limit ~ 0.5
q Full 3D CTU q Full considera4ons of accoun4ng for third cross deriva4ves q Requires 6 Riemann solves for 3D (12-‐CTU needs 12 Riemann solves) q CFL limit ~ 1.0
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
∂3 /∂x∂y∂z
Performance of Full-‐CTU
q Verifica4on tests for the reduced/full 3D CTU schemes:
q CFL=0.95 for all 3D simula4ons using the full CTU scheme
q CFL=0.475 for the reduced CTU scheme
q They both converge in 2nd order
q 20% performance gain in using the full CTU scheme: CPUF−ctu
CPUR−ctu
≈ 0.8
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
cf. CPU12−ctu
CPU6−ctu
≈1#
$%
&
'(
2nd order Convergence
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
3D Orszag-‐Tang (PPM+MC+Roe+cfl 0.95)
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Cloud-‐Shock (WENO5+VL+Roe+cfl 0.95)
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Part 2
Divergence-‐Free fields: Constrained Transport (CT) MHD
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Constrained-‐Transport by Balsara & Spicer
q CT scheme by Balsara and Spicer, JCP, 1999:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Lack of Upwind Considera2on in CT
q CT scheme by Balsara and Spicer, 1998:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
q Consider u>0, v"0:
Lack of Upwind Considera2on in CT
q CT scheme by Balsara and Spicer, 1998:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
q Consider u>0, v"0:
q is the only E term in the upwind direc4on!
Lack of Upwind Considera2on in CT
q CT scheme by Balsara and Spicer, 1998:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
q Consider u>0, v"0:
q is the only E term in the upwind direc4on! q Numerical Oscilla4ons!
q This is true in most CT schemes!
Standard-‐MEC
q 3rd order modified electric field construc4on (standard-‐MEC), Lee & Deane, JCP, 2009:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
New Upwind-‐MEC
q New upwind biased modified electric field construc4on(upwind-‐MEC), Lee, JCP, submi[ed, 2012
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
New Upwind-‐MEC
q New upwind biased modified electric field construc4on(upwind-‐MEC), Lee, JCP, submi[ed, 2012. Considering u>0, v"0:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
New Upwind-‐MEC
q New upwind biased modified electric field construc4on(upwind-‐MEC), Lee, JCP, submi[ed, 2012. Considering u>0, v"0:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
New Upwind-‐MEC
q New upwind biased modified electric field construc4on(upwind-‐MEC), Lee, JCP, submi[ed, 2012. Considering u>0, v"0:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
New Upwind-‐MEC
q New upwind biased modified electric field construc4on(upwind-‐MEC), Lee 2012:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
CT vs. Upwind-‐MEC
q Small angle advec4on of the 2D field loop:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
θ = tan−1(0.01) = 0.573
CT, Standard & Upwind MECs
q Small angle advec4on of the 3D field loop:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
θ = tan−1(0.01) = 0.573
Conclusion
q Direc4onally unsplit formula4ons for the USM-‐MHD (also valid for the unsplit hydro solver) in FLASH4
q Two new features: q The reduced and full 3D CTU algorithms q Upwind-‐MEC scheme for MHD
q Efficiency and accuracy in the full CTU scheme: q Stable solu4ons with 2nd order convergence with CFL=0.95 q 20% performance gain in the full CTU scheme over the reduced CTU
scheme
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
CPUF−ctu
CPUR−ctu
≈ 0.8
Thank You
Ques2ons?
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012
Standard & Upwind MEC
q Small angle advec4on of the 3D field loop:
The 7th Astronum, The Big Island, HI, U.S.A, Jun 25-‐29, 2012