1

Numerical Schemes for Advection Reaction Equation

Ramaz Botchorishvili

Faculty of Exact and Natural Sciences

Tbilisi State University

GGSWBS,Tbilisi, July 07-11, 2014

2

Outline Equations Operator splitting Baricentric interpolation and derivative Simple high order schemes Divided differences Stable high order scheme Multischeme Discretization for ODEs

3

Equation

)()(),(),0( 00 BVLuxuxu

),()(

tufx

vu

t

u

v(t,x) - velocityf (u,t) – reaction smooth functions

4

Operator splitting

),(),(

),,(

),(),(

,0)(

2/12/1

2/11

12/11

1

2/1

2/1

2/12/1

xtuxtu

ttttuft

u

xtuxtu

tttx

vu

t

u

nn

nn

nnn

n

nn

nn

nn

nn

5

Operator splitting

),(),(

),,(

),(),(

,0)(

2/12/1

2/11

12/11

1

2/1

2/1

2/12/1

xtuxtu

ttttuft

u

xtuxtu

tttx

vu

t

u

nn

nn

nnn

n

nn

nn

nn

nn

LA -> ODE ->LA -> ODE-> … - first order accurate

6

Operator splitting

),(),(

),,(

),(),(

,0)(

2/12/1

2/11

12/11

1

2/1

2/1

2/12/1

xtuxtu

ttttuft

u

xtuxtu

tttx

vu

t

u

nn

nn

nnn

n

nn

nn

nn

nn

LA -> ODE ->LA -> ODE-> … - first order accurate

LA -> ODE ->ODE->LA -> LA->ODE-> … - second order accurate

7

Interpolation

bxxxa n 10

nyyy ,,, 10

)(,),(0 xx n

n

iii xax

0

)()(

niyx ii ,,1,0,)(

8

Lagrange interpolation

n

iniin xlyxL

0, )()(

nixx

xx

xxxxxxxx

xxxxxxxxxl

n

ijj

ji

n

ijj

j

niiiiii

niini ,0,

)(

)(

)())(()(

)())(()()(

0

0

110

110,

n

jkjkj

n

jkjnikn nkyyxlyxL

00, 0,)()(

9

Lagrange interpolation

n

iniin xlyxL

0, )()(

nixx

xx

xxxxxxxx

xxxxxxxxxl

n

ijj

ji

n

ijj

j

niiiiii

niini ,0,

)(

)(

)())(()(

)())(()()(

0

0

110

110,

n

jkjkj

n

jkjnikn nkyyxlyxL

00, 0,)()(

Pros & cons

10

Barycentric interpolation

n

i i

iinn xxyxxL

0

)()(

n

jjn xxx

0

)()(

n

ijj

ji

ii

in

n

ijj

ji

n

ijj

j

niiiiii

niini

xxni

xxx

xx

xx

xxxxxxxx

xxxxxxxxxl

0

0

0

110

110,

)(

1,,0,)(

)(

)(

)())(()(

)())(()()(

11

Barycentric interpolation

n

i i

iinn xxyxxL

0

)()(

n

jjn xxx

0

)()(

n

ijj

ji

i

xx0

)(

1

12

Barycentric interpolation

Advantages

Efficient in terms of

arithmetic operations

Low cost for

introducing or

excluding new nodal

points = variable

accuracy

n

i i

iinn xxyxxL

0

)()(

n

jjn xxx

0

)()(

n

ijj

ji

i

xx0

)(

1

13

Baricentric derivative

 

 

 

14

Baricentric derivative

 

 

 

Advantages:

Easy for

implementing

Arithmetic

operations

High order accuracy

15

High order scheme for LA

n

nii in

nnniii

nnn xx

uvuvxL

2

,02

)()(1)(

n

ijj

ji

i

xx2

0

)(

1

x

vu

)(

Approximation for

N=1, 2nd orderN=2, 4th orderN=4 , 8th order…

16

High order scheme for LA

0)()(1 2

,0

1

n

nii in

kn

knn

ki

kii

n

kn

kn

xx

uvuvuu

n

ijj

ji

i

xx2

0

)(

1

First order accurate in time, 2n order accurate in space

17

High order scheme for LA

0)()(1 2

,0

1

n

nii in

kn

knn

ki

kii

n

kn

kn

xx

uvuvuu

n

ijj

ji

i

xx2

0

)(

1

First order accurate in time, 2n order accurate in space

Possible Problems conservation & stability

18

Firs order upwind

0])()[(])()[( 1111

1

h

uvuvuvuvuu ki

ki

ki

ki

ki

ki

ki

ki

ki

ki

19

Firs order upwind

0])()[(])()[( 1111

1

h

uvuvuvuvuu ki

ki

ki

ki

ki

ki

ki

ki

ki

ki

Numerical flux functions

20

Firs order upwind

0])()[(])()[( 1111

1

h

uvuvuvuvuu ki

ki

ki

ki

ki

ki

ki

ki

ki

ki

Numerical flux functions

Properties1. Consistency2. Conditional stability (CFL)3. First order in space and in time4. conservative

21

Firs order upwind

0])()[(])()[( 1111

1

h

uvuvuvuvuu ki

ki

ki

ki

ki

ki

ki

ki

ki

ki

Numerical flux functions

Properties1. Consistency2. Conditional stability (CFL)3. First order in space and in time4. conservative

ki

ki

ki vvv )()(

1|))(||)(|max ki

ki vv

h

i

ki

i

ki uu 1

22

High order conservative discretisation

0])()[(])()[( 1111

1

h

uvuvuvuvuu ki

ki

ki

ki

ki

ki

ki

ki

ki

ki

02/12/11

h

FFuu iiki

ki

23

High order conservative discretisation

0])()[(])()[( 1111

1

h

uvuvuvuvuu ki

ki

ki

ki

ki

ki

ki

ki

ki

ki

02/12/11

h

FFuu iiki

ki

Harten, Enquist, Osher, Chakravarthy:

given function in values in nodal points, how to interpolate at cell interfaces in order to get higher then 2 accuracy

24

Special interpolation/reconstruction procedure

g(x)= 𝑓ሺ𝑦ሻ𝑑𝑦𝑥−∞

g(𝑥𝑖+1 2Τ )=σ 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑖+1 2Τ𝑥𝑖−1 2Τ𝑖𝑗=−∞ =σ 𝑓ҧ𝑗∆𝑥𝑗𝑖𝑗=−∞

𝑓ҧ𝑗 = 1∆𝑥𝑗න 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ

P(x)-interpolation polinomial using g(𝑥𝑖+1 2Τ )

p(x)=P’(x)-derivative

25

Special interpolation/reconstruction procedure

g(x)= 𝑓ሺ𝑦ሻ𝑑𝑦𝑥−∞

g(𝑥𝑖+1 2Τ )=σ 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑖+1 2Τ𝑥𝑖−1 2Τ𝑖𝑗=−∞ =σ 𝑓ҧ𝑗∆𝑥𝑗𝑖𝑗=−∞

𝑓ҧ𝑗 = 1∆𝑥𝑗න 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ

P(x)-interpolation polinomial using g(𝑥𝑖+1 2Τ )

p(x)=P’(x)-derivative

26

Special interpolation/reconstruction procedure

g(x)= 𝑓ሺ𝑦ሻ𝑑𝑦𝑥−∞

g(𝑥𝑖+1 2Τ )=σ 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑖+1 2Τ𝑥𝑖−1 2Τ𝑖𝑗=−∞ =σ 𝑓ҧ𝑗∆𝑥𝑗𝑖𝑗=−∞

𝑓ҧ𝑗 = 1∆𝑥𝑗න 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ

P(x)-interpolation polinomial using g(𝑥𝑖+1 2Τ )

p(x)=P’(x)-derivative

27

Special interpolation/reconstruction procedure

g(x)= 𝑓ሺ𝑦ሻ𝑑𝑦𝑥−∞

g(𝑥𝑖+1 2Τ )=σ 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑖+1 2Τ𝑥𝑖−1 2Τ𝑖𝑗=−∞ =σ 𝑓ҧ𝑗∆𝑥𝑗𝑖𝑗=−∞

𝑓ҧ𝑗 = 1∆𝑥𝑗න 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ

P(x)-interpolation polinomial using g(𝑥𝑖+1 2Τ )

p(x)=P’(x)-derivative

28

Special interpolation/reconstruction procedure

1∆𝑥𝑗 𝑝ሺ𝑥ሻ𝑑𝑥𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ = 1∆𝑥𝑗 (g(𝑥𝑖+1 2Τ )- g(𝑥𝑖−1 2Τ ))= 1∆𝑥𝑗 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ =𝑓ҧ𝑗

P’(x)=g(x)+o(∆𝑥𝑘) p(x)=f(x)+ o(∆𝑥𝑘)

P(x)=σ 𝑔(𝑥𝑖−𝑟+𝑚−1 2ൗ�𝑘𝑚=0 )𝑙𝑚(𝑥)

P(x)- 𝑔(𝑥𝑖−𝑟−1 2ൗ�)=σ ቈ𝑔ቆ𝑥𝑖−𝑟−𝑚−1 2ൗ�ቇ− 𝑔(𝑥𝑖−𝑟−1 2ൗ�)𝑘𝑚=0 𝑙𝑚(𝑥)

𝑔(𝑥𝑖−𝑟+𝑚−1 2ൗ�) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)=σ 𝑔ҧ𝑖−𝑟+𝑗𝑚−1𝑗=0 ∆𝑥𝑖−𝑟+𝑗

29

Special interpolation/reconstruction procedure

1∆𝑥𝑗 𝑝ሺ𝑥ሻ𝑑𝑥𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ = 1∆𝑥𝑗 (g(𝑥𝑖+1 2Τ )- g(𝑥𝑖−1 2Τ ))= 1∆𝑥𝑗 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ =𝑓ҧ𝑗

P’(x)=g(x)+o(∆𝑥𝑘) p(x)=f(x)+ o(∆𝑥𝑘)

P(x)=σ 𝑔(𝑥𝑖−𝑟+𝑚−1 2ൗ�𝑘𝑚=0 )𝑙𝑚(𝑥)

P(x)- 𝑔(𝑥𝑖−𝑟−1 2ൗ�)=σ ቈ𝑔ቆ𝑥𝑖−𝑟−𝑚−1 2ൗ�ቇ− 𝑔(𝑥𝑖−𝑟−1 2ൗ�)𝑘𝑚=0 𝑙𝑚(𝑥)

𝑔(𝑥𝑖−𝑟+𝑚−1 2ൗ�) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)=σ 𝑔ҧ𝑖−𝑟+𝑗𝑚−1𝑗=0 ∆𝑥𝑖−𝑟+𝑗

30

Special interpolation/reconstruction procedure

1∆𝑥𝑗 𝑝ሺ𝑥ሻ𝑑𝑥𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ = 1∆𝑥𝑗 (g(𝑥𝑖+1 2Τ )- g(𝑥𝑖−1 2Τ ))= 1∆𝑥𝑗 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ =𝑓ҧ𝑗

P’(x)=g(x)+o(∆𝑥𝑘) p(x)=f(x)+ o(∆𝑥𝑘)

P(x)=σ 𝑔(𝑥𝑖−𝑟+𝑚−1 2ൗ�𝑘𝑚=0 )𝑙𝑚(𝑥)

P(x)- 𝑔(𝑥𝑖−𝑟−1 2ൗ�)=σ ቈ𝑔ቆ𝑥𝑖−𝑟−𝑚−1 2ൗ�ቇ− 𝑔(𝑥𝑖−𝑟−1 2ൗ�)𝑘𝑚=0 𝑙𝑚(𝑥)

𝑔(𝑥𝑖−𝑟+𝑚−1 2ൗ�) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)=σ 𝑔ҧ𝑖−𝑟+𝑗𝑚−1𝑗=0 ∆𝑥𝑖−𝑟+𝑗

31

Special interpolation/reconstruction procedure

1∆𝑥𝑗 𝑝ሺ𝑥ሻ𝑑𝑥𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ = 1∆𝑥𝑗 (g(𝑥𝑖+1 2Τ )- g(𝑥𝑖−1 2Τ ))= 1∆𝑥𝑗 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ =𝑓ҧ𝑗

P’(x)=g(x)+o(∆𝑥𝑘) p(x)=f(x)+ o(∆𝑥𝑘)

P(x)=σ 𝑔(𝑥𝑖−𝑟+𝑚−1 2ൗ�𝑘𝑚=0 )𝑙𝑚(𝑥)

P(x)- 𝑔(𝑥𝑖−𝑟−1 2ൗ�)=σ ቈ𝑔ቆ𝑥𝑖−𝑟−𝑚−1 2ൗ�ቇ− 𝑔(𝑥𝑖−𝑟−1 2ൗ�)𝑘𝑚=0 𝑙𝑚(𝑥)

𝑔(𝑥𝑖−𝑟+𝑚−1 2ൗ�) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)=σ 𝑔ҧ𝑖−𝑟+𝑗𝑚−1𝑗=0 ∆𝑥𝑖−𝑟+𝑗

32

Special interpolation/reconstruction procedure

1∆𝑥𝑗 𝑝ሺ𝑥ሻ𝑑𝑥𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ = 1∆𝑥𝑗 (g(𝑥𝑖+1 2Τ )- g(𝑥𝑖−1 2Τ ))= 1∆𝑥𝑗 𝑓ሺ𝑦ሻ𝑑𝑦𝑥𝑗+1 2Τ𝑥𝑗−1 2Τ =𝑓ҧ𝑗

P’(x)=g(x)+o(∆𝑥𝑘) p(x)=f(x)+ o(∆𝑥𝑘)

P(x)=σ 𝑔(𝑥𝑖−𝑟+𝑚−1 2ൗ�𝑘𝑚=0 )𝑙𝑚(𝑥)

P(x)- 𝑔(𝑥𝑖−𝑟−1 2ൗ�)=σ ቈ𝑔ቆ𝑥𝑖−𝑟−𝑚−1 2ൗ�ቇ− 𝑔(𝑥𝑖−𝑟−1 2ൗ�)𝑘𝑚=0 𝑙𝑚(𝑥)

𝑔(𝑥𝑖−𝑟+𝑚−1 2ൗ�) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)=σ 𝑔ҧ𝑖−𝑟+𝑗𝑚−1𝑗=0 ∆𝑥𝑖−𝑟+𝑗

33

Special interpolation/reconstruction procedure

ቀP(x) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)ቁ′=p(x)

𝑓𝑗+1 2Τ = 𝑓ቀ𝑥𝑗+1 2ൗ�ቁ= 𝑝ቀ𝑥𝑗+1 2ൗ�ቁ= ൬𝑃ሺ𝑥ሻ− 𝑔ቀ𝑥𝑖−𝑟−1 2ൗ�ቁ൰ቚ𝑥=𝑥𝑗+1 2ൗ�′

Baricentric derivative?

1∆𝑥𝑖 𝜕𝑓𝜕𝑥 𝑑𝑥𝑥𝑖+1 2Τ𝑥𝑖−1 2Τ = 1∆𝑥𝑖 (𝑓𝑖+1 2Τ − 𝑓𝑖−1 2Τ )

34

Special interpolation/reconstruction procedure

ቀP(x) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)ቁ′=p(x)

𝑓𝑗+1 2Τ = 𝑓ቀ𝑥𝑗+1 2ൗ�ቁ= 𝑝ቀ𝑥𝑗+1 2ൗ�ቁ= ൬𝑃ሺ𝑥ሻ− 𝑔ቀ𝑥𝑖−𝑟−1 2ൗ�ቁ൰ቚ𝑥=𝑥𝑗+1 2ൗ�′

Baricentric derivative?

1∆𝑥𝑖 𝜕𝑓𝜕𝑥 𝑑𝑥𝑥𝑖+1 2Τ𝑥𝑖−1 2Τ = 1∆𝑥𝑖 (𝑓𝑖+1 2Τ − 𝑓𝑖−1 2Τ )

35

Special interpolation/reconstruction procedure

ቀP(x) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)ቁ′=p(x)

𝑓𝑗+1 2Τ = 𝑓ቀ𝑥𝑗+1 2ൗ�ቁ= 𝑝ቀ𝑥𝑗+1 2ൗ�ቁ= ൬𝑃ሺ𝑥ሻ− 𝑔ቀ𝑥𝑖−𝑟−1 2ൗ�ቁ൰ቚ𝑥=𝑥𝑗+1 2ൗ�′

Baricentric derivative?

1∆𝑥𝑖 𝜕𝑓𝜕𝑥 𝑑𝑥𝑥𝑖+1 2Τ𝑥𝑖−1 2Τ = 1∆𝑥𝑖 (𝑓𝑖+1 2Τ − 𝑓𝑖−1 2Τ )

36

Special interpolation/reconstruction procedure

ቀP(x) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)ቁ′=p(x)

𝑓𝑗+1 2Τ = 𝑓ቀ𝑥𝑗+1 2ൗ�ቁ= 𝑝ቀ𝑥𝑗+1 2ൗ�ቁ= ൬𝑃ሺ𝑥ሻ− 𝑔ቀ𝑥𝑖−𝑟−1 2ൗ�ቁ൰ቚ𝑥=𝑥𝑗+1 2ൗ�′

Baricentric derivative?

1∆𝑥𝑖 𝜕𝑓𝜕𝑥 𝑑𝑥𝑥𝑖+1 2Τ𝑥𝑖−1 2Τ = 1∆𝑥𝑖 (𝑓𝑖+1 2Τ − 𝑓𝑖−1 2Τ )

37

Special interpolation/reconstruction procedure

ቀP(x) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)ቁ′=p(x)

𝑓𝑗+1 2Τ = 𝑓ቀ𝑥𝑗+1 2ൗ�ቁ= 𝑝ቀ𝑥𝑗+1 2ൗ�ቁ= ൬𝑃ሺ𝑥ሻ− 𝑔ቀ𝑥𝑖−𝑟−1 2ൗ�ቁ൰ቚ𝑥=𝑥𝑗+1 2ൗ�′

Baricentric derivative?

1∆𝑥𝑖 𝜕𝑓𝜕𝑥 𝑑𝑥𝑥𝑖+1 2Τ𝑥𝑖−1 2Τ = 1∆𝑥𝑖 (𝑓𝑖+1 2Τ − 𝑓𝑖−1 2Τ )

02/12/11

h

FFuu iiki

ki

High order accurate approximation

38

Special interpolation/reconstruction procedure

ቀP(x) − 𝑔(𝑥𝑖−𝑟−1 2ൗ�)ቁ′=p(x)

𝑓𝑗+1 2Τ = 𝑓ቀ𝑥𝑗+1 2ൗ�ቁ= 𝑝ቀ𝑥𝑗+1 2ൗ�ቁ= ൬𝑃ሺ𝑥ሻ− 𝑔ቀ𝑥𝑖−𝑟−1 2ൗ�ቁ൰ቚ𝑥=𝑥𝑗+1 2ൗ�′

Baricentric derivative?

1∆𝑥𝑖 𝜕𝑓𝜕𝑥 𝑑𝑥𝑥𝑖+1 2Τ𝑥𝑖−1 2Τ = 1∆𝑥𝑖 (𝑓𝑖+1 2Τ − 𝑓𝑖−1 2Τ )

02/12/11

h

FFuu iiki

ki

High order accurate approximation

39

Components of high order scheme

Discretization of the divergence operator • baricentric interpolation • baricentric derivative• ENO type reconstruction procedure (Harten, Enquist, Osher, Chakravarthy): • Given fluxes in nodal points• Interpolate fluxes at cell interfaces in such a way

that central finite difference formula provides high order (higher then 2) approximation

• Use adaptive stencils to avoid oscillations

40

Adaptation of interpolation

Use adaptive stencils to avoid oscillations•interpolate with high order polynomial in all cell interfaces inside of the stencil of the polynomial•If local maximum principle is satisfied then value at this cell interface is found•If local maximum principle is not satisfied then change stencil and repeat interpolation procedure for such cell interfaces•If after above procedure local maximum principle is not respected then reduce order of polynomial and repeat procedure for those interfaces only

41

Convergence in one space dimension Algorithm ensures

Uniform bound of approximate solutions Uniform bound of total variation

ConclusionsApproximate solution converge a.e. to solution of the original problem

42

Extension to higher spatial dimension Cartesian meshes:

strightforward Hexagonal meshes:

Directional derivates => div needs three directional derivatives in 2D

Implementation with baricentric derivatives without adaptation procedure See poster ( Tako & Natalia)

Implementation with adaptation – not yet

43

Better ODE solvers

Different then polinomial basis fanctions, e.g approach B.Paternoster, R.D’Ambrosio

44

Thank you