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### Application to inviscid compressible flow

Keh-Ming Shyue

Department of Mathematics National Taiwan University

Taiwan

(2)

### Objective

Describe simple Lagrangian-like moving grid approach for numerical resolution of nonlinear hyperbolic balance laws of the form

∂tq (~x, t) +

N

X

j=1

∂xj fj (q, ~x) = ψ (q, ~x)

with discontinuous initial data in general N ≥ 1 rectangular or non-rectangular geometry

~x = (x1, x2, . . . , xN): spatial vector, t: time q ∈ lRm: vector of m state quantities

fj ∈ lRm: flux vector, j = 1, 2, . . . , N, ψ ∈ lRm: source term Model equation is hyperbolic if PNj=1d αj (∂fj/∂q) is

diagonalizable with real eigenvalues, αj ∈ lR

(3)

### Outline

Mathematical model for general balance laws Eulerian formulation

Generalized Lagrangian formulation

Example to single component compressible flow Wave-propagation based finite volume methods

Generalized Riemann problem & approximate solver Flux-based wave decomposition method

Sample numerical examples

Extension to compressible two-phase flow Future work

(4)

### Mathematical Model

To begin with, consider a general non-rectangular domain Ω (N = 2 shown below) & introduce coordinate change

(~x, t) 7→ (~ξ, τ ) via

ξ = (ξ~ 1, ξ2, . . . , ξN) , ξj = ξj(~x, t), τ = t,

that maps a physical domain Ω to a logical one Ωˆ

−1 0 1

−1.5

−1

−0.5 0

−1 −0.5 0 0.5

−2

−1.5

−1

−0.5

x1 x2

ξ1

ξ2

−→ ˆ

mapping

ξ1 = ξ1(x1, x2) ξ2 = ξ2(x1, x2)

logical domain physical domain

(5)

### Mathematical Model (Cont.)

To derive hyperbolic balance laws in this generalized

coordinate (~ξ, τ ), using chain rule of partial differentiation, derivatives in physical space become

∂t =

∂τ +

N

X

i=1

∂ξi

∂t

∂ξi,

∂xj =

N

X

i=1

∂ξi

∂xj

∂ξi for j = 1, 2, . . . , N ,

yielding the equation

∂q

∂τ +

N

X

j=1

∂ξj

∂t

∂q

∂ξj +

N

X

i=1

∂ξi

∂xj

∂fj

∂ξi

!

= ψ(q)

Note this is not in divergence form, and hence is not conservative, in case the source term ψ is ignored.

(6)

### Mathematical Model (Cont.)

To obtain a strong conservation-law form as

∂ ˜q

∂τ +

N

X

j=1

∂ ˜fj

∂ξj = ˜ψ

for some q˜, f˜j, & ψ˜, we first multiply J = det

∂~ξ/∂~x−1

to the aforementioned non-conservative equations, and have

J ∂q

∂τ +

N

X

j=1

J ∂ξj

∂t

∂q

∂ξj +

N

X

i=1

∂ξi

∂xj

∂fj

∂ξi

!

= Jψ(q)

Then use differentiation by parts, u dv = d(uv) − v du, yielding

∂ ˜q

∂τ +

N

X

j=1

∂ ˜fj

∂ξj = ˜ψ + G with q = Jq˜ , f˜j = J 

q∂ξ∂tj + PN

k=1 fk ∂x∂ξj

k

, ψ = Jψ,˜ & G (see next)

(7)

### Mathematical Model (Cont.)

Here we have

G = q

∂J

∂τ +

N

X

j=1

∂ξj



J ∂ξj

∂t



+

N

X

j=1

fj

" N X

k=1

∂ξk



J ∂ξk

∂xj

#

Note with the use of basic grid-metric relations (see next), it is known that

∂J

∂τ +

N

X

j=1

∂ξj



J ∂ξj

∂t



= 0 (geometrical conservation law)

N

X

k=1

∂ξk



J ∂ξk

∂xj



= 0 ∀ j = 1, 2, . . . , N (compatibility condition)

and hence G = 0

(8)

### Basic Grid-Metric Relations

Assume existence of inverse transformation

t = τ, xj = xj(~ξ, t) for j = 1, 2, . . . , N ,

To find basic grid-metric relations between different coordinates, employ elementary differential rule

∂(τ, ~ξ)

∂(t, ~x) = ∂(t, ~x)

∂(τ, ~ξ)

−1

yielding in N = 3 case, for example, as

1 0 0 0

tξ1 x1ξ1 x2ξ1 x3ξ1

tξ2 x1ξ2 x2ξ2 x3ξ2

tξ3 x ξ3 x ξ3 x ξ3

= 1 J

J 0 0 0

J01 J11 J21 J31 J02 J12 J22 J32 J03 J13 J23 J33

(9)

### Grid-Metric Relations (Cont.)

Here

J =

∂(x1, x2, x3)

∂(ξ1, ξ2, ξ3)

= det ∂(x1, x2, x3)

∂(ξ1, ξ2, ξ3)

 , J11 =

∂(x2, x3)

∂(ξ2, ξ3)

, J21 =

∂(x1, x3)

∂(ξ3, ξ2)

, J31 =

∂(x1, x2)

∂(ξ2, ξ3)

, J12 =

∂(x2, x3)

∂(ξ3, ξ1)

, J22 =

∂(x1, x3)

∂(ξ1, ξ3)

, J32 =

∂(x1, x2)

∂(ξ3, ξ1)

, J13 =

∂(x2, x3)

∂(ξ1, ξ2)

, J23 =

∂(x1, x3)

∂(ξ2, ξ1)

, J33 =

∂(x1, x2)

∂(ξ1, ξ2)

,

J0j = −

Nd

X

i=1

Jijτxi, j = 1, 2, 3,

and so grid-metric relations between different coordinates

∇ξ = (∂ ξ , ∇ ξ ) = (∂ ξ , ∂ ξ , ∂ ξ , ∂ ξ ) = 1

(J , J , J , J )

(10)

### Grid-Metric Relations (Cont.)

Note in two dimensions N = 2, we have

 ∂ξ1

∂t , ∂ξ1

∂x1 , ∂ξ1

∂x2



= 1 J



∂x1

∂τ

∂x2

∂ξ2 + ∂x2

∂τ

∂x1

∂ξ2 , ∂x2

∂ξ2 , −∂x1

∂ξ2



 ∂ξ2

∂t , ∂ξ2

∂x1 , ∂ξ2

∂x2



= 1 J

 ∂x1

∂τ

∂x2

∂ξ1 ∂x2

∂τ

∂x1

∂ξ1 , −∂x2

∂ξ1 , ∂x1

∂ξ1



J = ∂x1

∂ξ1

∂x2

∂ξ2 ∂x1

∂ξ2

∂x2

∂ξ1

Thus to have G = 0 fulfilled, grid-metrics should obey

∂J

∂τ +

∂ξ1



J ∂ξ1

∂t



+

∂ξ2



J ∂ξ2

∂t



= 0

∂ξ1



J ∂ξ1

∂x1



+

∂ξ2



J ∂ξ2

∂x1



=

∂ξ1

 ∂x2

∂ξ2



+

∂ξ2



∂x2

∂ξ1



= 0

∂ξ



J ∂ξ1

∂x



+

∂ξ



J ∂ξ2

∂x



=

∂ξ

 −∂x1

∂ξ



+

∂ξ

 ∂x1

∂ξ



= 0

(11)

### Mathematical Model (Cont.)

As an example, with gravity effect included, Euler equations for single component compressible gas flow take

Cartesian coordinate case

∂t

ρ ρui

E

+

N

X

j=1

∂xj

ρuj

ρuiuj + pδij Euj + puj

=

0

−ρ∂x∂φ

i

−ρ~u · ∇φ

, i = 1, . . . , N

Generalized coordinate case

∂τ

ρJ ρJui

JE

+

N

X

j=1

∂ξj J

ρUj

ρuiUj + p∂ξ∂xj

i

EUj + pUj − p∂ξ∂tj

=

0

−ρJ ∂x∂φ

i

−ρJ~u · ∇φ

ρ: density, p = p(ρ, e): pressure , e: internal energy

E = ρe + ρPN

j=1 u2j/2: total energy, φ: gravitational potential

(12)

### Mathematical Model (Cont.)

Note that to complete the model, we must

1. make clear the transformation (~x, t) 7→ (~ξ, τ ) initially

Depending on how complex the geometry is, this can be done by various numerical means

(13)

### Mathematical Model (Cont.)

Note that to complete the model, we must

1. make clear the transformation (~x, t) 7→ (~ξ, τ ) initially

Depending on how complex the geometry is, this can be done by various numerical means

2. choose a moving grid strategy for ∂τ~x

(14)

### Mathematical Model (Cont.)

Note that to complete the model, we must

1. make clear the transformation (~x, t) 7→ (~ξ, τ ) initially

Depending on how complex the geometry is, this can be done by various numerical means

2. choose a moving grid strategy for ∂τ~x

When ∂τ~x = 0 or ∂τ~x = ~ub(t) (rigid-body motion)

tξ~ & ∇~xξ~ are time-independent; no need to have more additional condition

(15)

### Mathematical Model (Cont.)

Note that to complete the model, we must

1. make clear the transformation (~x, t) 7→ (~ξ, τ ) initially

Depending on how complex the geometry is, this can be done by various numerical means

2. choose a moving grid strategy for ∂τ~x

When ∂τ~x = 0 or ∂τ~x = ~ub(t) (rigid-body motion)

tξ~ & ∇~xξ~ are time-independent; no need to have more additional condition

While ∂τ~x 6= 0 (flow-dependent motion) (see next)

tξ~ & ∇~xξ~ would be time-dependent; require

additional conditions to determine ∇ξ~~x (N2 of them in total) over time (see below)

(16)

### Lagrangian-Like Moving Grid

For compressible flow, to improve numerical resolution of interfaces (material or slip lines), it is popular to take ∂τ~x as

Lagrangian case: ∂τ~x = ~u (flow velocity)

Lagrangian-like case: ∂τ~x = h0~u (pseudo velocity) h0 ∈ [0, 1] (fixed piecewise const.)

Unified coordinate case: ∂τ~x = h~u

h ∈ [0, 1] but is determined from a PDE constraint arising from such as grid-angle or grid-Jacobian preserving condition

ALE-like case: ∂τ~x = ~U (arbitrary velocity)

(17)

### Lagrangian-Like Moving Grid

For compressible flow, to improve numerical resolution of interfaces (material or slip lines), it is popular to take ∂τ~x as

Lagrangian case: ∂τ~x = ~u (flow velocity)

Lagrangian-like case: ∂τ~x = h0~u (pseudo velocity) h0 ∈ [0, 1] (fixed piecewise const.)

Unified coordinate case: ∂τ~x = h~u

h ∈ [0, 1] but is determined from a PDE constraint arising from such as grid-angle or grid-Jacobian preserving condition

ALE-like case: ∂τ~x = ~U (arbitrary velocity)

Here we will focus on the simple Lagrangian-like case

(18)

### Unified Coordinate System

Consider N = 2 case, for example, and use simplified

notation ~x = (x, y), ξ = (ξ, η)~ . At given time instance, free parameter h can be chosen based on

Grid-angle preserving condition (Hui et al. JCP 1999)

∂τ cos−1  ∇ξ

|∇ξ| · ∇η

|∇η|



=

∂τ cos−1

−yηxη − yξxξ qyξ2 + yη2q

x2ξ + x2η

= · · ·

= Ahξ + Bhη + Ch = 0 (1st order PDE )

with

A = q

x2η + yη2 (vxξ − uyξ) , B = q

x2ξ + yξ2 (uyη − vxη) C = q

x2ξ + yξ2 (uηyη − vηxη) − q

x2η + yη2 (uξyξ − vξxξ)

(19)

### Unified Coordinate System

Consider N = 2 case, for example, and use simplified

notation ~x = (x, y), ξ = (ξ, η)~ . Or alternatively, based on Grid-Jacobian preserving condition

∂J

∂τ =

∂τ (xξyη − xηyξ)

= xξτ yη + xξ yητ − xητ yξ − xη yξτ

= · · ·

= Ahξ + Bhη + Ch = 0 (1st order PDE )

with

A = uyη − vxη, B = vxξ − uyξ, C = uξyη + vηxξ − uηyξ − vξxη

(20)

### Lagrangian-Like Grid (Cont.)

Now with the temporal motion of the coordinate system governed by ∂τ~x = h0~u. We should impose conditions on grid metrics ∂tξ~ & ∇~xξ~ to have the fulfillment of geometrical conservation law

∂J

∂τ +

N

X

j=1

∂ξj



J ∂ξj

∂t



= 0

Here we are interested in an approach that is based on the compatibility condition of ∂τξjxi & ∂ξjτxi, i.e.,

∂τ

 ∂xi

∂ξj



+

∂ξj



∂xi

∂τ



= 0 for i, j = 1, 2, . . . , N.

for unknowns ∂xi/∂ξj, yielding easy computation of J & ∇ξj

(21)

### Mathematical Model (Cont.)

In summary, our Lagrangina-like model system for single component compressible flow problems consists of

Physical balance laws

∂τ

ρJ ρJui

JE

+

N

X

j=1

∂ξj J

ρUj

ρuiUj + p∂ξ∂xj

i

EUj + pUj − p∂ξ∂tj

=

0

−ρJ ∂x∂φ

i

−ρJ~u · ∇φ

Geometrical conservation laws

∂τ

 ∂xi

∂ξj



+

∂ξj



∂xi

∂τ



= 0 for i, j = 1, 2, . . . , N.

Moving grid condition ∂τ~x = h0~u & pressure law p(ρ, e)

(22)

### Axisymmetric Compressible Flow

Physical balance laws (ξ1: axisymmetric direction)

∂τ

ρJ ρJui

JE

+

2

X

j=1

∂ξj J

ρUj

ρuiUj + p∂ξ∂xj

i

EUj + pUj − p∂ξ∂tj

=

x1ρJu1

x1ρJuiuj − ρJ ∂x∂φ

i

x1J(E + p)u1 − ρJ~u · ∇φ

for i = 1, 2

Geometrical conservation laws

∂τ

 ∂xi

∂ξj



+

∂ξj



∂xi

∂τ



= 0 for i, j = 1, 2.

Moving grid condition τ~x = h0~u & pressure law p(ρ, e)

(23)

### Mathematical Models: Remarks

For single component compressible flow model mentioned above, it is known that under some thermodynamic stability conditions

when h0 = 0 (Eulerian case), the model is hyperbolic when h0 = 1 (Lagrangian case), the model is weakly hyperbolic (do not possess complete eigenvectors) when h0 ∈ (0, 1) (Lagrangian-like case), the model is hyperbolic

(24)

### Mathematical Models: Remarks

For single component compressible flow model mentioned above, it is known that under some thermodynamic stability conditions

when h0 = 0 (Eulerian case), the model is hyperbolic when h0 = 1 (Lagrangian case), the model is weakly hyperbolic (do not possess complete eigenvectors) when h0 ∈ (0, 1) (Lagrangian-like case), the model is hyperbolic

If a prescribed velocity ~ub for a rigid body motion is included in the formulation i.e., with ∂τ~x = h0~u + ~ub, we should be able to use the model to solve some moving body problems as well.

(25)

### Review of Previous Work

The work presented here is related to

W.H. Hui et al. (JCP 1999, 2001): Unified coordinated system for Euler equations

W.H. Hui (Comm. Phys. Sci. 2007): Unified coordinate system in CFD

C. Jin & K. Xu (JCP 2007): Moving grid gas-kinetic method for viscous flow

P. Jia et al. (Computers and Fluids 2006) Unified

coordinated system for compressible milti-material flow Z. Chen et al. (Int J. Numer. Meth Fluids 2007): Wave speed based moving coordinates for compressible flow equations

(26)

### Numerical Methods

Employ finite volume formulation of numerical solution

Qnijk 1

∆ξ1∆ξ2∆ξ3 Z

Cijk

q(ξ1, ξ2, ξ3, τn) dV

that gives approximate value of cell average of solution q over cell Cijk at time τn (sample case in 2D shown below)

i − 1 i − 1

i

i j

j

j + 1 j + 1

Cij Cˆij

ξ1 ξ2

mapping

∆ξ1

∆ξ2 logical domain physical domain

←−

x1 = x11, ξ2) x2 = x21, ξ2) x1

x2

(27)

### Numerical Methods (Cont.)

In three dimensions N = 3, equations to be solved take

∂τ q ~ξ, τ + XN

j=1

∂ξj fj 

q, ~ξ

= ψ 

q, ~ξ

A simple dimensional-splitting method based on f-wave approach of LeVeque et al. is used for approximation, i.e.,

Solve one-dimensional generalized Riemann problem (defined below) at each cell interfaces

Use resulting jumps of fluxes (decomposed into each wave family) of Riemann solution to update cell

averages

Introduce limited jumps of fluxes to achieve high

(28)

### Numerical Methods (cont.)

Basic steps of a dimensional-splitting scheme ξ1-sweeps: solve

∂q

∂τ + f1  ∂

∂ξ, q, ∇~ξ



= 0 updating Qnijk to Qijk

ξ2-sweeps: solve

∂q

∂τ + f2



∂ξ2, q, ∇~ξ



= 0 updating Qijk to Q∗∗ijk

ξ3-sweeps: solve

∂q

∂τ + f3



∂ξ3 , q, ∇~ξ



= 0 updating Q∗∗ijk to Qn+1ijk

(29)

### Numerical Methods (cont.)

Consider ξ1-sweeps, for example, First order update is

Qijk = Qnijk ∆τ

∆ξ1

h A+1 ∆Qn

i−1/2,jk + A1 ∆Qn

i+1/2,jk

i

with the fluctuations

(A+1 ∆Q)ni−1/2,jk = X

m:(λ1,m)ni−1/2,jk>0

(Z1,m)ni−1/2,jk

and

(A1 ∆Q)ni+1/2,jk = X

m:(λ1,m)ni+1/2,jk<0

(Z1,m)ni+1/2,jk

1,m)nι−1/2,jk & (Z1,m)nι−1/2,jk are in turn wave speed and f-waves

(30)

### Numerical Methods (cont.)

High resolution correction is

Qijk := Qijk ∆τ

∆ξ1

 ˜F1n

i+1/2,jk  ˜F1n

i−1/2,jk



with ( ˜F1)ni−1/2,jk = 1 2

mw

X

m=1



sign1,m)



1 − ∆τ

∆ξ1 1,m|



Z˜1,m

n

i−1/2,jk

Z˜ι,m is a limited value of Zι,m

It is clear that this method belongs to a class of upwind schemes, and is stable when the typical CFL (Courant-Friedrichs-Lewy) condition:

ν = ∆τ maxm 1,m, λ2,m, λ3,m)

min (∆ξ1, ∆ξ2, ∆ξ3) ≤ 1,

(31)

### Generalized Riemann Problem

Generalized Riemann problem for our model equations at cell interface ξi−1/2 consists of the equation

∂qi−1,jk

∂τ + f1



∂ξ1 , qi−1,jk



= 0 if ξ1 < (ξ1)i−1/2,

∂qijk

∂τ + f1



∂ξ1 , qijk



= 0 if ξ1 > (ξ1)i−1/2,

together with piecewise constant initial data

q(ξ1, 0) =

Qni−1,jk for ξ < ξi−1/2 Qnijk for ξ > ξi−1/2

qijk = q|(∂ξ2~x, ∂ξ3~x)ijk & f1(∂ξ1, qijk) = f1(∂ξ1, q)|(∂ξ2~x, ∂ξ3~x)ijk

(32)

### Generalized Riemann Problem

Generalized Riemann problem at time τ = 0

τqL + f1 (∂ξ1, qL) = 0 τqR + f1 (∂ξ1, qR) = 0 ξ1 τ

Qni−1,jk Qnijk

(33)

### Generalized Riemann Problem

Exact generalized Riemann solution: basic structure

q + f (∂ , q ) = 0 q + f (∂ , q ) = 0 ξ1 τ

Qni−1,jk Qnijk

(34)

### Generalized Riemann Problem

Shock-only approximate Riemann solution: basic structure

τqL + f1 (∂ξ1, qL) = 0 τqR + f1 (∂ξ1, qR) = 0 ξ1 τ

Qni−1,jk Qnijk

λ0

λ1 λ2

λ3 qmL qmL+

qmR

Z1 = fL(qmL ) − fL(Qni−1,jk) Z2 = fR(qmR) − fR(qmL+ )

Z3 = fR(Qnijk) − fR(qmR)

(35)

### Shock-only Riemann Solver

Rotate velocity vector in Riemann data normal to each cell interface Find midstate velocity υm and pressure pm by solving

φ(pm) = υmR(pm) − υmL(pm) = 0

derived from Rankine-Hugoniot relation iteratively, where

υmL(p) = υL p − pL

ML(p), υmR(p) = υR + p − pR MR(p)

Propagation speed of each moving discontinuity is determined by

1,1)i−1/2,jk =



(1 − h0m ML(pm) ρmL(pm)



X~ ξ1

i−1/2,jk

1,2)i−1/2,jk = (1 − h0m

X~ξ1

i−1/2,jk

) =



(1 − h + MR(pm) 

ξ

(36)

### Lax’s Riemann Problem

Ideal gas EOS with γ = 1.4 h0 = 0 Eulerian result

h0 = 0.99 Lagrangian-like result

sharper resolution for contact discontinuity

0 0.5 1

0 0.5 1 1.5 2

x

ρ

Exact h0=0 h0=0.99

0 0.5 1

0 0.5 1 1.5 2

x

u

0 0.5 1

0 1 2 3 4

x

p

(37)

### Lax’s Riemann Problem

Physical grid coordinates at selected times

Each little dashed line gives a cell-center location of the proposed Lagrange-like grid system

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.05 0.1 0.15

x

time

(38)

### Woodward-Colella’s Problem

Ideal gas EOS with γ = 1.4 h0 = 0 Eulerian result

h0 = 0.99 Lagrangian-like result

sharper resolution for contact discontinuity

0 0.5 1

0 2 4 6 8

ρ

0 0.5 1

−10 0 10 20

0 0.5 1

0 100 200 300 400

u p

t = 0.016 t = 0.016

t = 0.016

(39)

### Woodward-Colella’s Problem

h0 = 0 Eulerian result

h0 = 0.99 Lagrangian-like result

sharper resolution for contact discontinuity

0 0.5 1

0 5 10 15 20

ρ

Fine grid h0=0 h0=0.99

0 0.5 1

−5 0 5 10 15

0 0.5 1

0 200 400 600

u p

t = 0.032 t = 0.032

t = 0.032

(40)

### Woodward-Colella’s Problem

h0 = 0 Eulerian result

h0 = 0.99 Lagrangian-like result

sharper resolution for contact discontinuity

0 0.5 1

0 2 4 6 8

ρ

0 0.5 1

−5 0 5 10 15

0 0.5 1

0 200 400 600

u p

x x

x

t = 0.038 t = 0.038

t = 0.038

(41)

### Woodward-Colella’s Problem

Physical grid coordinates at selected times

Each little dashed line gives a cell-center location of the proposed Lagrange-like grid system

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.01 0.02 0.03 0.04

x

time

(42)

### 2 D Riemann Problem

With initial 4-shock wave pattern

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

 ρ u v p

=

 1.5

0 0 1.5

0.532 1.206

0 0.3

0.138 1.206 1.206 0.029

0.532 0 1.206

0.3

(43)

### 2 D Riemann Problem

With initial 4-shock wave pattern Lagrangian-like result

Occurrence of simple Mach reflection

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

Density Pressure Physical grid

(44)

### 2 D Riemann Problem

With initial 4-shock wave pattern Eulerian result

Poor resolution around simple Mach reflection

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8

Density Pressure Physical grid

(45)

0 0.2 0.4

0 0.1 0.2 0.3 0.4 0.5

0 0.2 0.4

0 0.1 0.2 0.3 0.4 0.5

0 0.2 0.4

0 0.1 0.2 0.3 0.4

Density Pressure 0.5 Physical grid a) h0 = 0.99

0.1 0.2 0.3 0.4 0.5

0.1 0.2 0.3 0.4 0.5

0.1 0.2 0.3 0.4

Density Pressure 0.5 Physical grid b) h0 = 0

(46)

0 0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1 1.2

one−d h0=0 h0=0.99

0 0.1 0.2 0.3 0.4 0.5

0 0.1 0.2 0.3 0.4

0 0.1 0.2 0.3 0.4 0.5

0 0.5 1 1.5

0 0.1 0.2 0.3 0.4 0.5

0.5 1 1.5 2

r(m) r(m)

ρ(Mg/m3 ) ¯u(km/s)

p(GPa) J

 If a DSS school charges a school fee exceeding 2/3 and up to 2 &amp; 1/3 of the DSS unit subsidy rate, then for every additional dollar charged over and above 2/3 of the DSS

• Emergent Z_k 1-form &amp; 2-form symmetry. BF theory

where L is lower triangular and U is upper triangular, then the operation counts can be reduced to O(2n 2 )!.. The results are shown in the following table... 113) in

Taking second-order cone optimization and complementarity problems for example, there have proposed many ef- fective solution methods, including the interior point methods [1, 2, 3,

Elsewhere the difference between and this plain wave is, in virtue of equation (A13), of order of .Generally the best choice for x 1 ,x 2 are the points where V(x) has

[r]

• A sequence of numbers between 1 and d results in a walk on the graph if given the starting node.. – E.g., (1, 3, 2, 2, 1, 3) from