6-3.pdf

Numerical relativistic hydrodynamics with ADER-WENO

adaptive mesh renement

Olindo Zanotti

[email protected]

University of Trento, Italy

6th East-Asian Numerical Astrophysics Meeting

1519 September 2014 - Suwon

with: M. Dumbser

Outline

1 Motivations

2 The numerical method (ADER approach)

Finite volume scheme

Local space-time Discontinuous Galerkin predictor

3 Adaptive Mesh Renement

AMR implementation

Local Timestepping

4 The RichtmyerMeshkov instability

1. Motivations

Motivations

Many physical problems show great disparities in the spatial and temporal

scales (multiscale problems), which a static grid approach cannot treat

eciently.

Turbulence

Magnetic instabilities

Magnetic reconnection

Binary mergers

Circumbinary discs

Relativistic jets

...

Adaptive Mesh Renement (AMR) , name-ly the possibility to change the computatio-nal grid dynamically in space and in time,becomes necessary.High Order Methods can also help substan-tially when very small details need to besolved.

Olindo Zanotti ADER-WENO Schemes with AMR 1 / 20

1. Motivations

Taken separately, AMR and High Order methods have long been used in RHD

AMR+High Order Methods (relativistic case)- FLASH code by Fryxell et al. (2000), ApJ Suppl., 131, 273- Zhang & MacFadyen (2006), ApJ Suppl., 164, 255- RENZO code by Wang et al. (2008), ApJ, 176, 467- PLUTO code by Mignone et al. (2012), ApJ Suppl., 198, 7

A particularly appealing implementation of High order methods is represented byADER schemes, which are one-step time-update schemes.- Original version: use the Lax-Wendro procedure [Toro et al. (2001); Titarevand Toro (2002) ...]- Modern version: use a weak integral formulation of the governing PDE [Dumbseret al. (2008) JCP, 227, 3971; Balsara et al (2009) JCP, 228, 2480]

AMR+ADER schemes.... presented very recently for the non-relativistic Eulerequations by- Dumbser, Zanotti, Hidalgo, Balsara (2013), JCP, 248, 257-286- and extended to the relativistic regime at this conference for the rst time


2. The numerical method (ADER approach) Finite volume scheme

Outline

1 Motivations





AMR implementation

Local Timestepping





We consider hyperbolic systems of balance laws in Cartesian coordinates

u

t+f

x+g

y+h

z= 0

We use a Standard Finite Volume discretization

un+1ijk = unijk

t

xi

(fi+ 1

2,jk fi 1

2,jk

) t

yj

(gi,j+ 1

2,k gi,j 1

2,k

) t

zk

(hij,k+ 1

2

hij,k 12

)+ tSijk ,

over the control volumes Iijk = [xi 12

; xi+ 12

] [yj 12

; yj+ 12

] [zk 12

; zk+ 12

], with

unijk =

1

xi

1

yj

1

zk

xi+ 1

2xi 1

2

yj+ 1

2yj 1

2

zk+ 1

2zk 1

2

u(x , y , z , tn)dz dy dx

fi+ 12,jk =

1

t

1

yj

1

zk

tn+1tn

yj+ 1

2yj 1

2

zk+ 1

2zk 1

2

~f(qh (xi+ 1

2

, y , z , t), q+h (xi+ 12

, y , z , t))dz dy dt, .



- Reconstruction: introduce a nodal basis of polynomials l () of order M dened withrespect to a set of Gauss-Legendre nodal points k , such that l (k) = lk



- Build onedimensional reconstruction stencils along each direction

Ss,xijk =i+R

e=iL

Iejk , Ss,yijk =j+R

e=jL

Iiek , Ss,zijk =k+R

e=kL

Iije ,

three stencils for even M

four stencils for odd M

M + 1 cells in each stencil



1 Perform an entire polynomial WENO reconstuction along x direction:

ws,xh (x , t

n) =Mp=0

p()wn,sijk,p := p()w

n,sijk,p S

s,xijk

Impose integral conservation on all elements of the stencil:

1

xe

xe+ 1

2

xe 1

2

p((x))wn,sijk,p dx = u

nejk , Iejk Ss,xijk

Perform a data-dependent nonlinear combination:

wxh(x , t

n) = p()wnijk,p, with w

nijk,p =

Nss=1

swn,sijk,p

s =sk k

, s =s

(s + )r

s = pmwn,sijk,pw

n,sijk,m pm =

M=1

10

p()

m()

d .

[see Dumbser & Kser, (2007), JCP, 221, 693]



2 Perform a second polynomial WENO reconstuction along y direction using asinput the M + 1 degrees of freedom wnijk,p:

ws,yh (x , y , t

n) = p()q()wn,sijk,pq .

Apply integral conservation is now in the y direction:

1

ye

ye+ 1

2

ye 1

2

q((y))wn,sijk,pq dy = w

niek,p, Iiek Ss,yijk .

=

wy

h(x , y , tn) = p()q()w

nijk,pq, with w

nijk,pq =

Nss=1

swn,sijk,pq,



3 Perform the last polynomialWENO reconstuction along z direction using as inputthe (M + 1)2 degrees of freedom wnijk,pq: We therefore have

ws,zh (x , y , z , t

n) = p()q()r ()wn,sijk,pqr .

with the integral conservation written as above,

1

ze

ze+ 1

2

ze 1

2

r ((z))wn,sijk,pqr dz = w

niek,pq, Iije Ss,zijk .

= The nal three-dimensional WENO polynomial

wh(x, tn) = p()q()r ()w

nijk,pqr , w

nijk,pqr =

Nss=1

swn,sijk,pqr .


2. The numerical method (ADER approach) Local space-time Discontinuous Galerkin predictor

Outline

1 Motivations





AMR implementation

Local Timestepping




Local space-time DG predictor [see Dumbser et al. (2008), JCP, 227, 3971]

An alternative to the Cauchy-Kovalewski procedure to obtain the time evolution of thereconstructed polynomials and build one-step time-update numerical schemes.In practice we need

qh = fi+ 12,jk =

1

t

1

yj

1

zk

tn+1tn

yj+ 1

2yj 1

2

zk+ 1

2zk 1

2

~f(qh (xi+ 1

2

, y , z , t), q+h (xi+ 12

, y , z , t))dz dy , dt,

u

+f

+g

+h

= S

with

f =

t

xif, g =

t

yjg, h =

t

zkh, S = tS.

We then multiply by the test function p(, ) and integrate in space-time

10

10

10

10

q

(u

+f

+g

+h

S

)dddd = 0.

where we choose a tensor product of the basis functionsp(, ) = p()q()r ()s().



Integration by parts in time yields...

10

10

10

q(, 1)u(, 1)ddd 1

0

10

10

10

(

q

)udddd

+

10

10

10

10

[q

(f

+g

+h

S

)]dddd

=

10

10

10

q(, 0)wh(, tn)ddd.

We introduce the discrete space-time solution qh

qh = qh(, ) = p (, ) qp,

and similarly for the uxes and sources

fh = p f

p = pf

(qp) , . . . Sh = pS

p = pS

(qp) .

Insert everything in Eq. above and obtain....



10

10

10

q(, 1)p(, 1)qp ddd 1

0

10

10

10

(

q

)pqp dddd

+

10

10

10

10

[q

(

p f

p +

pg

p +

ph

p pSp

)]dddd

=

10

10

10

q(, 0)wh(, tn) ddd ,

which is a nonlinear algebraic equation system for the unknown coecients qp.In a more compact form:

K1qpqp + K

qp fp + Kqpgp + Kqphp = MqpSp + F0qmwnm,

with the various matrices dened as

K1

qp =

10

10

10

q(, 1)p(, 1)d 10

10

10

10

(

q

)pdd,

Kqp =

(Kqp,K

qp,K

qp

)=

10

10

10

10

q

pdd,

Mqp =

10

10

10

10

qpdd, F0

qp =

10

10

10

q(, 0)m()d,



The product of the matrices with the vectors of degrees of freedom can be eciently

implemented in a dimensionbydimension manner. An iterative scheme can be adopted

[see Dumbser & Zanotti (2009), JCP, 228, 6991]

K1qpqk+1p MqpS

,k+1p = F

0qmw

nm K

qp f

,kp K

qpg,kp K

qph,kp

Once this is done, we have everything to write the scheme as

un+1ijk = unijk

t

xi

(fi+ 1

2,jk fi 1

2,jk

) t

yj

(gi ,j+ 1

2,k gi ,j 1

2,k

) t

zk

(hij ,k+ 1

2 hij ,k 1

2

)+ tSijk ,

Local Lax-Friedrichs ux

~f(qh , q

+h

)=

1

2

(f(qh ) + f(q

+h )) 1

2|smax|

(q+h q

h

), (1)

where |smax| denotes the maximum absolute value of the eigenvalues of theJacobian matrix A = f/u.

Oshertype ux

~f(qh , q

+h

)=

1

2

(f(qh ) + f(q

+h )) 1

2

( 10

|A((s))| ds)(

q+h q

h

), (2)


3. Adaptive Mesh Renement AMR implementation

Outline

1 Motivations





AMR implementation

Local Timestepping




AMR implementation

We have developed a cell-by-cell AMR technique in which the computational

domain is discretized with a uniform Cartesian grid at the coarsest level.

Adopt a renement criterion, marking a cell for renement if

m > ref , where

m =

k,l (

2/xkxl )2k,l [(|/xk |i+1 + |/xk |i )/xl + |(2/xkxl )|||]2

.

When a cell of the level ` is rened, it is subdivided as

x` = rx`+1 y` = ry`+1 z` = rz`+1 t` = rt`+1

Each cell Cm, at any level of renement, has one among three possiblestatus ags.

- active cell

- virtual child cell

- virtual mother cellOlindo Zanotti ADER-WENO Schemes with AMR 13 / 20


...AMR implementation

If the Voronoi neighbors of an active rened cell Cm are not themselves at thesame level of renement of Cm, they have virtual children at the same level ofrenement of Cm.In order to keep the reconstruction local on the coarser grid level, we have r M.The levels of renement of two cells that are Voronoi neighbors of each other canonly dier by at most unity.



The beauty of the local-spacetime DG predictor: it does not need

any exchange of information with neighbor elements, even if two

adjacent cells are on dierent levels of renement.

Projection

Projection is the typical AMR operation, by which an active mother

assigns values to the virtual children ( = 1) at intermediate times

um(tn` ) =

1

x`

1

y`

1

z`

Cm

qh(x, tn` )dx. (3)

Needed for performing the reconstruction on the ner grid level at

intermediate times.

Averaging

Averaging is another typical AMR operation by which a virtual mother

cell ( = 1) obtains its cell average by averaging recursively over thecell averages of all its children at higher renement levels.

um =1

rd

CkBm

uk . (4)


3. Adaptive Mesh Renement Local Timestepping

Outline

1 Motivations





AMR implementation

Local Timestepping



3. Adaptive Mesh Renement Local Timestepping

Local Timestepping

Every renement level is advanced in time with its local timestep

t` = rt`+1. Update criterion:

tn+1` tn+1`1 , 0 ` `max, (5)

Starting from the common initial time t = 0, the nest level of renement`max is evolved rst and performs a number of r sub-timesteps before thenext coarser level `max 1 performs its rst time update.

= A total amount of r` sub-timesteps on each level are performed inorder to reach the time tn+10 of the coarsest level.


4. The RichtmyerMeshkov instability

RHD equations

tu + i fi = 0 ,

The conservative variables and the corresponding uxes in the i direction

are

u =

DSjE

, f i = v iDW ij

S i

. (6)D = ,

Si = h2vi ,

E = h2 p,

where = (1 v2)1/2 is the Lorentz factor of the uid with respect tothe laboratory observer and

Wij h2vivj + pij



Just one test...

Figura : Standard Riemann problem: the shock wave is resolved within one single

cell!!



The RichtmyerMeshkov instability

The RichtmyerMeshkov (RM) instability is a typical uid instability which

develops when a shock wave crosses a contact discontinuity within a uid,

or between two dierent uids.

Relevant in

Inertial Connement Fusion

Supernovae remnant formation

Relativistic jets

Figura : Schematic representation of the initial conditions in a representative

model with Atwood number A > 0.


Conclusions

We have presented the rst ADER-WENO nite volume scheme for relativistichydrodynamics on AMR grids

Compared to RungeKutta time stepping, the use of a high order onestep schemein time reduces the number of nonlinear WENO reconstructions and the number ofnecessary MPI communications.

High Order in combination with AMR is extremely benecial, in particular whensmall scale structures need to be solved

When applied to the study of the RichtmyerMeshkov instability, new interestingresults are obtained

Conclusions

We have presented the rst ADER-WENO nite volume scheme for relativistichydrodynamics on AMR grids

Compared to RungeKutta time stepping, the use of a high order onestep schemein time reduces the number of nonlinear WENO reconstructions and the number ofnecessary MPI communications.

High Order in combination with AMR is extremely benecial, in particular whensmall scale structures need to be solved

When applied to the study of the RichtmyerMeshkov instability, new interestingresults are obtained

MotivationsThe numerical method (ADER approach)Finite volume schemeLocal space-time Discontinuous Galerkin predictor

Adaptive Mesh RefinementAMR implementationLocal Timestepping

The RichtmyerMeshkov instability

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