Download ppt - Time Discretisation - Taylor-Galerkin Schemes V. Selmin Multidisciplinary Computation and Numerical Simulation

Time Discretisation - Taylor-Galerkin Schemes

V. Selmin

Multidisciplinary Computation and Numerical Simulation

Outline

Outline

• Spatial discretisation: summary

• Basic properties of numerical schemes

• Time discretisation

• Taylor-Galerkin schemes - Basic Taylor-Galerkin schemes - Extension to non-linear problems - Extension to multi-dimensional problems - Two-steps Taylor-Galerkin schemes

• Multi-stages algorithms

Spatial Discretisation

Spatial discretisation-Summary

Structured Grids versus Unstructured Grids

Structured grids:Same number of cells around a node

Unstructured grids:The number of cells around a node is not the same


Finite Difference Discretisation:

Finite Volume Discretisation:

Taylor-series expansion ),()()()()(

)( 221,1,,1,1, kj

k

kjkj

j

kjkjkj

WFWFWFWFWF

Integral formulationDivergence theorem

eednWWFdWWF ),(),(


Spatial discretisation-Summary

Reference element Physical element

0)()()( dVfuLdVuRuWV VV

)()(1

r

iii fwdf ξξξ

ξ

Finite Element Discretisation:

Physical space Reference space

Physical element Reference element

Function approximation

Integral method Integration by parts

Weighted residuals Galerkin method

PDE discretisation method

Numerical integration Gauss method

Numerical integration

Basic properties

Basic Properties

Truncation errorDifference between the original partial differential equation (PDE) and the discretised equation (DE).

ConsistencyConsistency deals with the extent to which the discretised equations approximate the partial differential equations. A discretised representation of the PDE is said to be consistent if it can be shown that the difference between the PDE and its discretised representation vanishes as the mesh is refined:

StabilityNumerical stability is a concept applicable in a strict sense only to marching problems.A stable numerical scheme is one for which errors for any source (round-off, truncation, …) are not permitted togrow in the sequence of numerical procedures as the calculation proceeds from one marching step to the next.

Convergence of Marching ProblemsLax’s Equivalence Theorem: Given a properly posed initial value problem and a discretised approximation to it that satisfies the consistency conditions, stability is necessary and sufficient condition for convergence.

),( ht

0)(lim)(lim0,0,

TEDEPDE

thth

DEPDETE

Discretisation in Time

Discretisation in time

Model equation:

Unsteady/steady problems

If the solution u is steady, u solution of

is also solution of the following pseudo-unsteady problem:

Finite Difference:

1- FD approximation of the time derivative

Spatial discretisation →

Time discretisation →

2

2

x

u

x

f

t

u

2

2

x

u

x

f

)(2

2

x

u

x

f

t

ulimt

Steady Euler equations: elliptic for subsonic flowshyperbolic for supersonic flows

j

jjj

x

f

x

uRR

t

u

2

2

,

)(2

43

,)(2

,)(

211

211

1

tt

uuu

t

u

tt

uu

t

u

tt

uu

t

u

nj

nj

njj

nj

njj

nj

njj

),(

),,()(

tnxiuu

tnxuxuni

n


2- Taylor-series expansion

Taylor-series expansion of →

Replace the time derivatives by using the equation

That leads to the following equation

which has to be discretised in space

...!

)(1

1

1

n

ll

nllnn

t

u

l

tuu1nu

0

uaf

,)()(

,

2

22

2

2

x

ua

t

u

xa

x

ua

tt

u

x

ua

t

u

nnnn

nn

...)(!

)(1

1

1

n

ll

nll

lnn

x

ua

l

tuu

Taylor series expansion


The equality

may be rewritten in the more concise form

A family of temporal schemes may be buit by using the Padé polynomials approximation of the exponential function.It consists to approximate the function H(v) by the ratio of two polynomials of order p and q, respectively, with an

error of

...!

)(1

1

1

n

ll

nllnn

t

u

l

tuu

nn ut

tu )(exp1

Padé Polynomials

)( 1qpv)(

)(

)()( 1 qp

q

p

vvQ

vPVH u

ttv )(

Explicit schemes

Implicit schemes


Taylor Galerkin Schemes The Taylor-Galerkin schemes may be considered as a generalisation of the explicit Euler scheme (Padé polynomals with q=0):

The time derivatives are replaced by the expressions obtained by using successive differentiation of the originalequation:

The third order derivative is expressed in terms of a mixed space-time form in order to allow the use of finite element for the spatial discretisation. In this term the time derivative is replaced by a finite difference approximation that maintains the global troncature error:

The time discretised equation is written according to:

where

)(62

43

33

2

221 t

t

ut

t

ut

t

utuu

nnnnn

t

u

xa

t

u

x

ua

t

u nnnn

2

22

3

3

2

22

2

2

,

t

uu

t

u nnn

1

2

221

2)1(

x

uta

x

ua

t

uuL

nnnn

x

36

20

2

222

px

ta

p

Lx

0C


Taylor Galerkin Schemes

If the convention is adopted for the scalar product on the computational domain, the Galerkin

equation at node j corresponds to

Explicitley, we got after integration by parts of the second derivatives terms

In the case of piecewise linear shape function, ETG2 and ETG3 schemes take the form

where is the Courant number,

dxwLwL ,

k

nkk

nj

nn UxNUNUUL )(,0),,( 1

2

22222

12

12

1

221

6,

2,

2,

)(

6,)(

6,

0,2

,2

,,

0

0

0

x

taL

x

UaN

t

x

N

x

Ua

tN

x

Ua

x

UUaN

t

x

N

x

UUa

tN

t

UU

Lx

UaN

t

x

N

x

Ua

tN

x

UaN

t

UU

x

x

x

n

jj

n

j

n

x

x

nn

jj

nn

j

nn

x

x

x

n

jj

n

j

n

j

nn

n

n

n

3,2

)1(6

11

2,26

11

22

0122

22

012

TGUc

UcUUc

TGUc

UcUU

nj

nj

nj

nj

nj

nj

nj

nj

x

tac

.2),(2

111

2110 jjjjjjj UUUUUUU

2TG

3TG


Taylor Galerkin Schemes

In the right-hand side of the discretised equation we may recognize the same term as the Law-Wendroff scheme

In addition, in the left-hand side of those equations, we may regognize the classical consistent mass of the

finite element theory which corrisponds to to the operator . In the TG3 scheme, it is modified by the

additional term that appears in the time discretised equation.

LW nj

nj

nj

nj U

cUcUU 2

2

01

2

6/1 2

Remarks:Due to the coupling terms, the presence of the mass matrix represent a disadvantage from the point of view of the computational time. Nevertheless, it is possible to exploit its effect in an explicit context.The following iterative procedure may be used

where

)1(1)1(1)(1

)1(1)1(1)2(1

220

)1(1

)()()()(

)()()()(

2

1)(

mnj

nj

nj

nj

mnj

nj

nj

nj

nj

nj

nj

nj

nj

nj

nj

nj

UUIMUUUU

UUIMUUUU

UcUcUU

3)1(6

11

26

11

22

2

ETGc

ETGM


Numerical Schemes Property 1- Von Neuman analysis methodThe Von Neumann procedure consits in replacing each term of the discretised equation by the Fourier

component of order k of an harmonic decomposition of :

where is the Fourier component of order k.The amplification factor G is defined by the equality:

In general, it is a complex number which may be written on the following form

where and are respectively the module and the phase of G .The stability condition of von Neuman states that, for each Fourier mode, the amplification factor must have a module limited by a quantity enough close to unity for all value of and .The explicit expression of this criteria is

The term emphasizes that in some physical process, the modes may increase exponentially and this divergence does not be confused with an unstability of the numerical method

njU

njU

)exp( xjkivU nnj

nv

nn vGv 1

ieG

t x

0)(1 tt

)( t


Numerical Schemes Property For the previous numerical schemes, the amplification factor takes the form

where and is a real number:

The stability condition for the three schemes is

The reduction of stability for TG2 is due to the consistent mass matrix . The correction contained in the TG3 scheme allows to recover the stability condition and the unit CFL property that states that the signal propagates without distorsions when .

2sin2sin),(11),( 221 ccicLcG x

xk xL

)(32

sin)1(3

2

)(22

sin3

2

0

),(

22

2

TGLWTGc

FELWTG

LW

cLx

)(31

)(23

3

1

TGLWTG

FELWTG

LW

c

1c1c


Numerical Schemes Property


Numerical Schemes Property In the case the spatial discretisation is performed by maintaining the time continuous, the following schemes are obtained:

for the finite differences, and

for piecewise linear elements

The consistent mass matrix is responsable of the better acurracy on the phase.

FEUt

c

t

U nj

j ,6

11 0

2

FDUt

c

t

U nj

j ,0

)(32

sin)1(3

2

)(22

sin3

2

0

),(

22

2

TGLWTGc

FELWTG

LW

cLx



2- Modified equation methodThe Modified Equation method consistsa- To perform a Taylor series expansion about of all the terms of the discretised equation.b- To replace all the time derivatives of order greater to one and the mixted space-time by using the equation obtained at the previous stepFollowing this procedure, we obtain the partial differential equation of infinite order genuinely solved by thenumerical scheme

The modified equation may be written according to

where the are real coefficients.

Let consider a elementary solution:

where k is real and is a complex number, the and have to satisfy the following relations:

njU

12

2

20

12

12

12p

p

p

pp

p

p

px

u

x

u

t

u

ikxteetxu ),(

21 i 1 2

012

122

12

21

)1(

,)1(

pp

pp

pp

pp

k

k



In the limit case where (large wave lenghts), we can negelect all the terms except the non-zero coefficientsof the lowest order which will be denoted by r . In this case

The necessary stability condition: becomes

In addition,

0k

rrr

rrr

k

k

212

2

22

1

)1(

,)1(

01

0)1( 221

rrr k

)1(212

2

)()1(

1

r

r

rr

e

xkc

k



xh

Discretisation

Time Space

FD

FD

FE

FE

Timecontinuous

Eulerscheme

LW

LW-FE

LW-TG




Propagation of a cosine profile

LW

LW-FE

LW-TG

To illustrate and compare the performance of the schemes discussed so far, consider the advection problem over the interval [0,1] and defined by the following initial and boundary conditions:

00),0(

0

]/)(cos[12

1)0,(

0

00

ttu

xx

xxxxxu

Extension to non linear convection


Let consider the following hyperbolic equation

Written in the quasi-linear form

it may be interpreted as a non linear convection equation for which each point of the solution propagates with a different velocity.As in the previous case, the equation is discretised in time by using the series expansion

in which the time derivatives are replaced by using the original equation and its successive differentiation

0)(

x

uf

t

u

u

ufua

x

uua

t

u

)(

)(,0)(

)(62

43

33

2

221 t

t

ut

t

ut

t

utuu

nnnnn

)()()(

)()(

222

22

3

3

22

2

t

u

xa

x

u

t

aa

x

ua

t

a

xt

u

xa

x

u

t

a

xx

ua

xtt

u

x

ua

xt

ua

xt

u

Extension to non linear convection


By using the following identities

the expression of the third derivative in time is equivalent to the following form

Then, in the nonlinear case, the equation discretised in time may be written according to

where

x

ua

u

a

t

u

u

a

t

a

t

u

x

a

x

u

t

a

,tx

ua

t

u

x

a

t

ua

xtx

f

xt

ua

x

u

t

a

x

ua

txt

f

22

22

)(

)(

)()( 223

3

t

u

xa

t

u

x

aa

t

u

x

ua

u

a

xt

u

xa

x

u

t

aa

x

ua

t

a

xt

u

x

ua

x

t

x

f

t

uuL

nn

nnn

x2

1

)(2

)1(

TGLW

xa

x

aa

x

ua

u

a

x

t

FELW

L nn

nn

nn

x 22

)(6

0

Remarks:In the case of a scalar equation (and only) the third order time derivative may be written in the following compact form:

t

ua

xt

u 22

2

3

3

The consistent massmatrix depends of theunkown

Extension to multi dimensional problems


Let consider the following hyperbolic equation

The time derivatives may be expressed according to

By using the following identities

the expression of the third derivative in time is equivalent to the following form

Then, in the case of multidimensional problems, the equation discretised in time may be written according to

where

0,0

uat

uf

t

u

)()()(

)(

3

3

2

2

t

uaau

t

aaua

t

a

t

u

uaat

u

)(,)( uau

a

t

a

t

uau

t

a

)()()(3

3

t

uaa

t

uaa

t

uua

u

a

t

u

)(2

)1(1

nnnnnn

uaat

ft

uuL

x

TGLWaaaaua

u

at

FELW

L nnnnnnn

)()()(6

02

x

Extension to multi dimensional problems


Multidimensional discretisation property

),(),(

,),(),(

),

ykxk

hyxy

ta

x

ta

aa

yxyx

yxyx

yx

λ

ν

(a

x

y

x

y

2.0ν 3.0ν

Domain of numerical stability of the LW schemes Phase velocity error of the LW schemes


Convection of a product cosine hill To illustrate and compare the performance of the schemes discussed so far, the advection of a product cosine hill in a pure rotation velocity field is considered. The initial condition is

The unknown has to be prescribed on the inflow boundary and leave free on the outflow boundary

0

000

0

]/)(cos[1]/)(cos[14

1)0,(

xx

xxx

yyxxu

0,0 outinanan


Convection of a product cosine hill

LW-FE(Md) LW LW-FE LW-TG

LW-FE(Md) LW LW-FE: unstable LW-TG

Convection of a hill in a rotational field, ∆t=2π/200, after one complete rotation

Convection of a hill in a rotational field, ∆t=2π/120, after one complete rotation


Convection of a product cosine hill

Convection of a hill in a rotational field, ∆t=2π/200, after 3/4 and 1 complete rotation

LW-TG

Two step Taylor Galerkin Schemes


A third-order T-G scheme which can be effectively employed in non-linear multidimensional advection problems is obtained by considering the following two step procedure:

where the value of the parameter is left unspecified for the time being. In fact while the other coefficients in the two equations assume the value necessary for a third-order accuracy of the two discretisations combined, the parameter enters only the coefficient of fourth-order term in the overall series and therefore its value affects the amplification factor only within the fourth-order accuracy. The fully discrete counterpart of the equations above in the case of linear advection problem in one dimension is

The value of the parameter may be fixed in such a way to reproduce the excellent phase accuracy of the LW-TG scheme:

2

221

2

22

~)(

2

1

)(3

1~

t

ut

t

utuu

t

ut

t

utuu

nnnn

nnnn

nj

nj

nj

nj

nj

nj

nj

nj

Uc

UcUU

UcUc

UU

~26

11

3~

6

11

22

012

220

2

9

1



The modified equation is

The third order accuracy is preserved by the two-step procedure. Moreover, the new scheme has the same phase response. The amplification factor takes the form

5

542

4

4

4222

)451(180

)3

1(24 x

ucc

xa

x

uctxa

x

ua

t

u

2sin2sin

2sin)1(

3

211),(),( 22

122 cciccRcG realcR ),(

Property analysis

1c2

3cTTG=LW-TG2



Propagation of a cosine profile



Multidimensional discretisation property LW-TG

LW-TG2

Amplification factor of the LW schemesDomain of numerical stability of the LW schemes

),(),(

,),(),(

),

ykxk

hyxy

ta

x

ta

aa

yxyx

yxyx

yx

λ

ν

(a

Multiple stages algorithms


Another important family of time integration schemes that account for only two time levels are those namedmulti-stage algorithms (generalisation of the Runge-Kutta method).Let write the spatial discretised scheme as follows:

where is the residual of the partial differential equations

The general form of an explicit multi-stage algorithm with l levels may be written according to

where and for consistency.

)(uRdt

du

)(uR

)()1(

)1()0()(

)1(2

)0()2(

)0(1

)0()1(

)0(

ln

ll

l

n

uu

Rtuu

Rtuu

Rtuu

uu

)( )()( kk uRR 1l

4

1,3

1,2

1,1

3

1,2

1,1

2

1,1

1234

323

12

Second order:

Third order:

Fourth order:

Maximum stability condition:

1lc

Hybrid Multi stages algorithms


In the case of the solution of steady problems, the objective is to have an efficient integration method which allows to get the steady state as fast as possible while the time accuracy is unimportant.That leads to a class of hybrid schemes of the multi-stage tipe where the residual of a generic l+1 stage is evaluated according to

where and are the convective and dissipative terms, respectively. In order to satisfy the consistency property, the following relations have to be satisfied

l

k

klk

klk

l uDuFR0

)1( )()(

)( kuF )( kuD

100

l

klk

l

klk

Example:

)5()1(

)1()4(5

)0()5(

)1()3(4

)0()4(

)1()2(3

)0()3(

)1()1(2

)0()2(

)0()0(1

)0()1(

)0(

)(

)(

)(

)(

)(

uu

DFtuu

DFtuu

DFtuu

DFtuu

DFtuu

uu

n

n

1

2/1

8/3

6/1

4/1

5

4

3

2

1

1

2/1

324.0

228.0

1375.0

5

4

3

2

1