Dongwook Astronum 2012

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


Part 1

3D Unsplit Algorithms for USM & UHD: Reduced & Full CTU


MHD Governing Equa2ons

q MHD system of equa4ons:

q  This can be wri[en in a simple matrix form:


∂U∂t

+∂F∂x

+∂G∂y

+∂H∂z

= 0

q  A primi4ve form:

where the coefficient matrix is

Linearized System


High-‐Order Corner Transport Upwind (CTU)


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)


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…






q  Solve 1D high-‐order reconstruc4ons using characteris4c tracing in each

normal direc4on to get Riemann states





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( )



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


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


Normal predictor Transverse corrector q  Use characteris4c tracing in BOTH normal and transverse flux

calcula4ons (Lee & Deane, JCP, 2009)





! Normal predictor





! Transverse corrector

! Normal predictor






! Normal predictor






! Normal predictor






! Normal predictor

! Monotonicity

! Stability

Characteris2c tracing for Transverse corrector

q A jump rela4onship:


AyVl +m=1

m0−1

Σλmrm Δ~α =AyVr −

m=m0

7

Σ λmrm Δ~α


q A jump rela4onship:


m=1

7

Σλmrm Δ~α =AyVr −AyVl

m=1

7


=AyΔ

=Gr −Gl

= Gi+1/2, j −Gi−1/2, j


q The summa4on of all waves becomes an upwind transverse flux gradient:


m=1

7


=AyΔ

=Gr −Gl

= Gi+1/2, j −Gi−1/2, j



q The summa4on of all waves becomes an upwind transverse flux gradient:




! Normal predictor

Single-‐Step Data Reconstruc2on-‐Evolu2on


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


∂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


∂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


cf. CPU12−ctu

CPU6−ctu

≈1#

$%

&

'(

2nd order Convergence


3D Orszag-‐Tang (PPM+MC+Roe+cfl 0.95)


Cloud-‐Shock (WENO5+VL+Roe+cfl 0.95)


Part 2

Divergence-‐Free fields: Constrained Transport (CT) MHD


Constrained-‐Transport by Balsara & Spicer

q  CT scheme by Balsara and Spicer, JCP, 1999:


Lack of Upwind Considera2on in CT

q  CT scheme by Balsara and Spicer, 1998:


q  Consider u>0, v"0:





q  is the only E term in the upwind direc4on!





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:


New Upwind-‐MEC

q  New upwind biased modified electric field construc4on(upwind-‐MEC), Lee, JCP, submi[ed, 2012


New Upwind-‐MEC

q  New upwind biased modified electric field construc4on(upwind-‐MEC), Lee, JCP, submi[ed, 2012. Considering u>0, v"0:


New Upwind-‐MEC



New Upwind-‐MEC



New Upwind-‐MEC

q  New upwind biased modified electric field construc4on(upwind-‐MEC), Lee 2012:


CT vs. Upwind-‐MEC

q  Small angle advec4on of the 2D field loop:


θ = tan−1(0.01) = 0.573

CT, Standard & Upwind MECs



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


CPUF−ctu

CPUR−ctu

≈ 0.8

Thank You

Ques2ons?


Standard & Upwind MEC



Dongwook Astronum 2012

Education

q cfl q

ctu algorithm q

big island

q ydirec4on

q zdirec4on

riemann problems q colella

riemann problems q saltzman

riemann states q xdirec4on