Gardiner and Stone

Corner Transport Upwind of Colella (CTU) Upwinded Constrained Transport (UCT) Modifications to Characteristic Tracing of

PPM scheme. Sweeping across a 3D grid.

CTU scheme of Colella Use first set of Riemann solves to update

perpendicular edges before second set of Riemann solves.

etc... , where

2

2

2

2

,21,,21,,21,

1,211,2121,,*

21,,

,21,2121,,*

21,,

21,121,1,21,*

,21,

21,21,,21,*

,21,

jiRxjiLxxx

jix

xji

xjijiRyjiRy

xji

xjijiLyjiLy

yji

yjijiRxjiRx

yji

yjijiLxjiLx

qqFF

FFx

tqq

FFx

tqq

FFy

tqq

FFy

tqq

jiRxq ,21, jiLxq ,21,

yjiF 2/1,

yjiF 2/1,

* *

Ec CT algorithm of Gardiner and Stone Use cell centered E as well as face (edge)

centered E to construct edge (corner) centered E in an upwinded fashion.

Interpolate from cell facesto cell edge using gradientcomputed by differencingthe cell center and cell facein the upwind direction.

The piece-wise parabolic method of Colella is a dimensionally split method and involves a one-dimensional parabolic spatial reconstruction and characteristic evolution of the primitive variables to get the time averaged edge values where the characteristic evolution step is calculated by solving

Where and

0

x

VA

t

V

z

y

z

y

x

B

B

v

v

v

P

V

xxz

xxy

xx

xx

zyx

x

x

vBB

vBB

Bv

Bv

BBv

Pv

v

A

0000

0000

00000

00000

0010

00000

00000

The resulting evolution equation for the magnetic field from the one-dimensional characteristic tracing is:

0

0

0

x

vBBv

xt

Bx

vBBv

xt

Bt

B

zxzx

z

yxyx

y

x

It is however missing some terms compared to the split form of the induction equation in 2D:

In 2D, the constraint on B can be incorporated into the split form of the evolution equation for Bz. This helps prevent erroneous growth in Bz for grid-aligned flow.

0

0

0

y

vBBv

yt

B

y

vBBv

yt

Bt

B

zyzy

z

xyxy

x

y

In 3D, it is not clear how to use the constraint on B to modify the split form of the induction equation. The goal is to allow for multidimensional source terms, but in a way that reduces to the 2D scheme for grid-aligned flow.

0,minmod

0,minmod

0

y

B

x

B

x

Bv

x

vBBv

xt

B

z

B

x

B

x

Bv

x

vBBv

xt

Bt

B

yxxz

zxzx

z

zxxy

yxyx

y

x

0,minmod

0,minmod

0

x

B

y

B

y

Bv

y

vBBv

yt

B

z

B

y

B

y

Bv

y

vBBv

yt

B

t

B

xyyz

zyzy

z

zyyx

xyxy

x

y

If xBx and yBy have the same sign, then the source terms reduce to the split form of the induction equation.

If they are equal and opposite however (as they will be in 2D since zBz = 0), then the evolution equation for Bz reduces to the 2D form.

First determine overall range of dependence for different stencil pieces:

3:2,3:2

1:0,1:01:0,1:0

2:1,1:01:0,2:1

1:1,1:01:0,1:1

1:1,:1:1,1:1

yx

yxy

yxx

yxy

yxx

yxy

yxx

yxy

yxx

NNw

NNqRNNqR

NNqLNNqL

NNfNNf

NNFNNF

Second, determine lag and lead for different stencil pieces. 411010101lag

311121101lead

Stencil wqRqLqRqLffFF yyxxyxyx