Discrete-analytical approximations based on global and local - - PowerPoint PPT Presentation

Discrete-analytical approximations based on global and local adjoint problems for atmosphere, ocean and environment studies

Penenko V.V. ICMMG SB RAS, Novosibirsk

Goals

• To develop a methodology for construction of

mutually agreed methods of complex models implementation in direct and inverse modes which take into account the processes of divers time-space scales

• To diminish uncertainty of models owing to

improvement of discrete approximations quality due to including solutions of global and local adjoint problems as well as some analytical solutions

Review Flux Corrected Transport (FCT) Schemes/ Monotonicity algorithms

• The mesh refinement schemes
• The monotone interpolation routines
• Overlapping and moving grids
• Richardson extrapolation
• Romberg’s method
• “Mother” domain- “daughter” domain interactions
• Averager procedures
• “Smoother-dismoother”
• Non-linear renormalization
• “Lagrangian-type” monotonization

Review Flux Corrected Transport (FCT) Schemes

• Richardson (1910)
• Romberg (1955
• Godunov S.K. (1959)
• Gol’din V.Ya, Kalitkin, Shishova (1965)
• Van Leer (1974-79) self-limiting diffusion, Taylor’s series

expansion

• A.Harten et al (1978-87) TVD, ENO
• Tremback et al (1987) Non-linear renormalization
• A.Bott ( 1989) Non-linear renormalization
• Smolarkiewicz and Grell (1992)

, ) ( ~       ∂ ∂ Φ + ∂ ∂ − = − + ∂ ∂ x RT x H v l u L t u

S S S S S

π σ π π π π , ) ( ~       ∂ ∂ Φ + ∂ ∂ − = + + ∂ ∂ y RT y H u l v L t v

S S S S S

π σ π π π π ) ( = + ∂ ∂

S S

L t π π

1

=       ∂ ∂ + ∂ ∂ + ∂ ∂

∫

σ π π π d y v x u t

S S S

. T R H

S

Φ − = ∂ ∂ π σ

Model of atmospheric dynamics

,

, / ) (

T S S S T

p p p p − ≡ − = π π σ

T S

p + ≡ Φ σπ

( )

S S S S

u v L x y ∂π ϕ ∂π ϕ ∂π ϕσ π ϕ = + + ∂ ∂ ∂σ &

, ) ( ) ( ~

B H S S

F F L L

ϕ ϕ

ϕ π ϕ π + + =

+ boundary and initial conditions dependent on domain and physical essence of problem

Model of transport and transformation of pollutants

i

f is the source term

i

S ) ( ϕ r is the pollutant transformation operator, } ; { D x t t Dt ∈ ≤ ≤ = r

.

i

S ) ( ϕ r = loss – production + deposition

Boundary and initial conditions ) , ( , ) (

t i i

t x q R Ω ∈ = ϕ r r ) ( ) , (

0 x

x v r r v ϕ = ϕ .

) , ( ) ( ) grad ( div = − − + − + ∂ ∂ ≡

i i i i c i i

r t x f S c u c t c L

i

r r r r ϕ µ π π ϕ

} , 1 {

,

m i ci = = ϕ r

is the state function

Variational form: hydrodynamics & transport & transformation models

{ }

; div ) ( ) ( ) ( ) ( 1 ) ( ) grad grad ( ) ( ) , ( ) Y, , (

* 1 s * * * * * * * * * * * * * * 4 1 * * *

∫ ∫ ∫ ∑ ∫

Ω + = ∗

= Ω +       ′ + ∂ ∂ + + +             ∂ ∂ − + ∂ ∂ − + − + ∂ ∂ ∂ ∂ − + − + − + Λ ≡

t t t

dt d u H dSdt d u t T H dDdt y vT v x uT u m T t T H H H u H u vu uv f I

n S S n i D i i i

π σ π π π π χχ α π π σ σ σ η π σ σ π π ϕ ϕ α

χ ϕ

r & & ϕ ϕ

( )

1 2 ( , )

t t

i i i i i i i D i i i i D

u dDdt t t t

∗ ∗ ∗ ∗

∂ψ   Λψ ψ ≡ + ψ −µ ψ ψ =   ∂     ∂ψ ∂ψ  = ψ − ψ +      ∂ ∂   

∫ ∫

div grad

( )+

−

∗ ∗

U U

i i i i

r r ψ ψ ψ ψ div div 2 1 −       ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂

∗ ∗ ∗

z z y y x x

i i iz i i iy i i ix

ψ ψ µ ψ ψ µ ψ ψ µ

}

+ + ∫

∗ ∗

dD dDdt q

t D i i i i

2 1 ψ ψ ψ

ψ

dt d Q U

i i n i i

t

Ω +

∗ ∗ Ω

∫

) 2 1 ( ψ ψ ψ

ψ

Main form for advective-diffusive operators in the integral identity of the variational form of the model

) , , ( ) , ( ) (

= ∗

+ ∂ ∂ ≡ Γ ≡ Φ

α

ϕ δ α ϕ α δ ϕ δ

k h k h k

Y Y I Y r r r r r r r

      + ∂ ∂ ∂ ∂ = Γ

= ∗

) , , (

α

ϕ δ α ϕ α δ

k h k

Y Y I Y r r r r r r

The main sensitivity relations The algorithm for calculation

• f sensitivity functions

} { ki

k

Γ = Γ r

are the sensitivity functions

} {

i

Y Y δ δ = r

are the parameter variations

N i K k , 1 , , 1 = = N N N dt dY

k

≤ = Γ − =

α α α α α

α η , , 1 ,

The fead-back relations

∑ ∫

= ∗ ∗ ∗

+ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂

n i D ik i ik i y ik i x

t

c c y c y c x c x c

1

[ { σ σ δµ δµ δµ

σ

− Ω ∂ ∂ − −

∗ Ω ∗ ∗

∫

dt d c n c dDdt c f c S

ik i n ik i ik i

t

δµ δ δ ] ) ( ϕ

} ] ) ) ( ( [ dt d c q R c c u dD c c

ik i D i ik i n t ik i

t

Ω − + +

∗ Ω ∗ = ∗

∫ ∫

δ δ δ δ ϕ

K k , 1 =

δ defines variations of corresponding functions; coefficients at δ are sensitivity functions

) (ϕ

k

Φ

= Φ ≡ Φ Y) ), ( ( ) ( δ δ ϕ ϕ

h k Y h k

grad

Sensitivity relations

Where do analytical solutions be effective?

• Advective-diffusive operators
• Non-stationary problems with linearized

main part and slowly varied non-linear part:

– linear part of positive sign – linear part of negative sign – linear part of alternating sign

( , ) B G Y f t ϕ ϕ ∂ + = ∂ r r r r

Variational principle: splitting and decomposition

t t ≤ ≤ ( , , ) ( ( , ) , ) I Y B G Y f t ϕ ϕ ϕ ϕ ϕ

∗ ∗

∂ = + − ∂ r r r r r r r r ( , , ) ( , , )

J j j

I Y I Y ϕ ϕ ϕ ϕ

∗ ∗ =

=∑ r r r r r r

1

( , , ) ( , , )

n j j k k

I Y I Y ϕ ϕ ϕ ϕ

∗ ∗ =

= ∑ r r r r r r

1

( , ) ( , )

n k k

G Y G Y ϕ ϕ

=

=∑ r r r r

1 n k k

f f

=

=∑ r Integral identity functional is approximated by cubature formula on a set of 4D finite volumes

Associative property of integral identity for construction

• f discrete approximations and splitting schemes

Analytical solutions with the use of local adjoint problems

Basic construction in the set of finite volumes

1 1 1 1

( )

i i i i i i i i

x x x x x x x x

L f dx L dx u f dx x x ϕ ϕ ϕ ϕ ϕ ϕ µ ϕ µ ϕ ϕϕ ϕ

+ + + +

∗ ∗ ∗ ∗ ∗ ∗ ∗

− = +   ∂ ∂ − + + − =   ∂ ∂  

∫ ∫ ∫

Integral identity fragment

а) functions and fluxes

, x ∂ ∂ ϕ ϕ µ

b) conditions at the outer boundaries (Dirichlet, Neumann, of the 3d type)

Conditions for the state functions:

are continuous at the inner boundaries

• f the cells;

1 j j

t t t + ≤ ≤ x - one of the space variables,

1

,

i i

L u a x x x x x x

+

∂ ∂ ∂ = − + ≤ ≤ ∂ ∂ ∂ ϕ ϕ ϕ µ ϕ

(1) 1/2 1 * (2) 1/2 1

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

i i i i i i i

x x x x x i n x x x x

∗ + − ∗ + +

 ≤ ≤  = = −  ≤ ≤   ϕ ϕ ϕ

* * * 1 1

( ) 0; ( ) 1; ( ) 0;

i i i i i i

x x x

− +

= = = ϕ ϕ ϕ

Basic elements for construction of discrete approximations

Local adjoint problems

1

0, , 1, 1

i i

L x x x i n ϕ

∗ ∗ +

= ≤ ≤ = −

(1) 1/2

( ),

i

x

∗ +

ϕ

(2) 1/2

( ),

i

x

∗ +

ϕ

{ }

* 1

( ) 1, ( ) 0, ;

i i

x x

∗ +

= = ϕ ϕ Fundamental solutions

{ }

* 1

( ) 0, ( ) 1, ;

i i

x x

∗ +

= = ϕ ϕ

under conditions under conditions

General structure

• f discrete-analytical approximations

with respect to spatial variables

1 1

;

i i i i i i i

a b c F

+ −

− + − = ϕ ϕ ϕ

*(2) 1/2 1/2 1

( )

i i i i

x a x

+ + +

∂ = − ∂ ϕ µ

*(1) 1/ 2 1/ 2 1

( )

i i i i

x c x

− − −

∂ = ∂ ϕ µ

*(1) *(1) 1/ 2 1/ 2 1/ 2 1/2

( ) ( )

i i i i i i

x b u x x

− − − − − −

  ∂ = +   ∂   ϕ µ ϕ

*(2) *(2) 1/ 2 1/ 2 1/2 1/2

( ) ( )

i i i i i i

x b u x x

+ + + + + +

  ∂ = − +   ∂   ϕ µ ϕ

i i i

b b b

+ −

= +

1 1

*(2) *(1) 1/ 2 1/2

( ) ( ) ( ) ( )

i i i i

x x i i i x x

F f x x dx f x x dx

+ −

+ −

= +

∫ ∫

ϕ ϕ

• approximation,
• stability,
• monotonicity,
• transportivity,
• differentiability with respect to parameters and the state functions,
• absence of conditional operators ( flux-corrector procedures)

Properties of numerical schemes: Properties of coefficients

0, 0, , , при

i i i i i i i

a c b a b c

+ −

≥ ≥ ≥ ≥ ≥ µ Matrix of coefficients

• is indecomposable
• has strict diagonal predominance
• Every element of inverse matrix is positive

At

i →

µ the properties of coefficients and matrix are the same. Re /2 10 u x = ∆ ≤ µ is grid control parameter

Time approximation of quasi-linear equations for transformation operators

( ) ( ) ( , ) t t F t t ∂ϕ + α ϕ + ϕ = ∂ ( ) t α

• matrix,

( , ) F t ϕ

• nonlinear operator

1 j j

t t t + ≤ ≤

Space variables participate as parameters

1 ( )

( ) ( ) ( , ( ))

t t t j j

t e t F e d

∆ −α∆ −α ∆ −τ +

ϕ = ϕ + τ ϕ τ τ

∫

In dependence of the form of matrix ( ) t α explicit or implicit formulas of desired order of accuracy are constructed with the use of adjoint problems in time

Comparative accuracy of analytical and numerical solutions

( ) a t t ∂ϕ + ϕ = ∂ 1 10 1 ( ) ( )exp( ), , , ( ) t at t a ϕ = ϕ − ≤ ≤ ≤ ϕ = 1 ( ) M a t = − ∆

1

1 ( ) M a t − = + ∆

1) Explicit scheme 2) Implicit scheme 3) Cranck-Nikolson scheme

1 j j j

M M

+

ϕ = ϕ = ϕ

( ) ( )

1

1 2 1 2 / / M a t a t

−

= + ∆ − ∆

Numerical schemes 4) 4-order scheme

( ) ( )

1 2 2

1 2 12 1 2 12 / / / / M a t a t a t a t

−

= + ∆ + ∆ − ∆ + ∆ 0 1 . a t ∆ = 41% 60% 0.83% 0.0001%

• Rel. errors

Novosibirsk region

Академгородок

Conclusion

• The hybrid discrete-analytical method to construct of numerical

models is proposed. It is based on approximations of integral identity presenting mathematical models in variational form.

• The schemes developed possess very useful properties like

monotonicity, transportivity, differentiability,etc.

• Accuracy of global and local description of the processes with the use
• f analytical solutions of the local adjoint problems is higher then that
• f traditional approximations which demand artificial monotonization .
• Proposed set of algorithms gives new possibilities for analysis &

synthesis of the behavior of multi-dimensional dynamic systems ( atmosphere, ocean, environment).

Спасибо за внимание