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)

Model of atmospheric dynamics ∂ π ∂ σ ∂ π   u H RT ~ + π − π = − π + S S ( ) , L u l v   ∂ ∂ Φ ∂   S S S t x x ∂ π ∂ σ ∂ π   v H RT ~ + π + π = − π + S S L ( v ) l u ,   ∂ ∂ Φ ∂ S S S   t y y ∂ π + π = S L ( ) 0 ∂ S t ∂ π H R = − S T . ∂ σ Φ ∂ π ∂ π ∂ π   1 u v ∫ + + σ = S S S d 0   ∂ ∂ ∂   t x y 0

σ = − π π ≡ − ( p p ) / , p p T S S S T , Φ ≡ σπ + p S T ∂π ϕ ∂π ϕ ∂π ϕσ & u v π ϕ = + + S S S L ( ) ∂ ∂ ∂σ S x y ~ π ϕ = π ϕ + + H B L ( ) L ( ) F F , ϕ ϕ S S + boundary and initial conditions dependent on domain and physical essence of problem

Model of transport and transformation of pollutants ∂ π c r r r r ϕ ≡ + π − µ + ϕ − − = i L div ( c u grad c ) ( S ) f ( x , t ) r 0 ∂ i c i i i i t i r ϕ = = is the state function { c i i 1 , m } , f is the source term i r is the pollutant transformation operator, ( ϕ S ) i r = ≤ ≤ ∈ . D t { 0 t t ; x D } r = loss – production + deposition ( ϕ S ) i Boundary and initial conditions r r ϕ = ∈ Ω ( R ) q , ( x , t ) i i t r r v v 0 x ϕ = ϕ ( x , 0 ) ( ) .

Variational form: hydrodynamics & transport & transformation models + { 4 n ∑ ∫ ∗ ϕ ϕ ≡ α Λ ϕ ϕ + π − + * * * ( , Y, ) ( , ) ( ) I f uv vu ϕ i i i = i 1 D t ∂ σ ∂ π * T π − + − + * * * ( u grad H u grad H ) ( H H ) ∂ σ ∂ t    ∂ π ∂ π  1 η σ − σ + − + − + * * * * * * ( & & T ) m ( u uT ) ( v vT )     σ ∂ ∂     x y   ∂ π } 1 r ∫ ∫ ′ α χχ + + π + π π σ * * * *   dDdt ( H T ) div u d dSdt χ ∂ S s  t  S 0 t ∫ + π Ω = * H u d dt 0 ; n Ω t

Main form for advective-diffusive operators in the integral identity of the variational form of the model ∂ψ ( )   ∗ ∫ ∗ Λψ ψ ≡ + ψ −µ ψ ψ =  i  ( , ) div u grad dDdt ∂ i i i i i i   t D t ∗   ∂ψ ∂ψ  1 ∫ ∗ = ψ − ψ +    i i   ∂ ∂ i i   2 t t  D t ( r r ) + 1 ψ ψ − ψ ψ ∗ ∗ div U div U i i i i 2 ∂ ψ ∂ ψ ∂ ψ ∂ ψ ∂ ψ ∂ ψ  ∗ ∗ ∗  µ + µ + µ −   i i i i i i ∂ ∂ ∂ ∂ ∂ ∂ ix iy iz   x x y y z z + ∫ 1 } ψ ψ ψ + ∗ ∗ t q dDdt dD ψ i i i i 0 2 D 1 ∫ ψ ψ + ψ Ω ∗ ∗ ( U Q ) d dt ψ i i n i i 2 Ω t

The main sensitivity relations ∂ r r r r r r r ∗ δ Φ ϕ ≡ Γ δ ≡ ϕ + α δ ϕ h h ( ) ( , Y ) I ( , Y Y , ) α = ∂ α k k k 0 The algorithm for calculation of sensitivity functions ∂ ∂   r r r r r ∗ Γ = ϕ + α δ ϕ  h  r I ( , Y Y , ) α = ∂ δ ∂ α k  k 0  Y The fead-back relations dY α = − η Γ α = ≤ , 1 , N , N N α α α α k dt r Γ = Γ { ki } are the sensitivity functions k r δ = δ are the parameter variations Y { Y } i = = k 1 , K , i 1 , N

Sensitivity relations δ Φ ϕ ≡ Φ ϕ δ = h h ( ) ( ( ), Y) grad k Y k ∗ ∗ ∗ ∂ ∂ ∂ ∂ ∂ ∂ n ∑ ∫ c c c c c c δµ + δµ + δµ + i ik i ik i ik { [ σ ∂ ∂ ∂ ∂ ∂ σ ∂ σ x y x x y y = i 1 D t ∂ c ∫ ∗ ∗ ∗ δ ( ϕ − δ − δµ Ω − i ) ] S c f c dDdt c d dt ∂ i ik i ik n ik n Ω t ∫ ∫ ∗ ∗ ∗ δ + δ + δ ϕ − δ Ω 0 [ ( ( ) ) ] } c c dD u c c R q c d dt = i ik t 0 n i ik i i ik Ω D t = k 1 , K δ defines variations of corresponding functions; coefficients at δ are sensitivity functions Φ ( ϕ ) k

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

Variational principle: splitting and decomposition Integral identity functional is approximated by cubature formula on a set of 4D finite volumes r ∂ ϕ r r r + ϕ = ≤ ≤ B G ( , ) Y f 0 t t ∂ t r ∂ ϕ r r r r r r r ∗ ∗ ϕ ϕ = + ϕ − ϕ I ( , , Y ) ( B G ( , ) Y f , ) ∂ t r r J r r = ∑ r r ∗ ∗ ϕ ϕ ϕ ϕ j I ( , , Y ) I ( , , Y ) = j 0 r = ∑ r n r r r r ∗ ∗ ϕ ϕ ϕ ϕ j j I ( , , Y ) I ( , , Y ) k = k 1 r r n r = ∑ r ϕ ϕ G ( , ) Y G ( , ) Y k = k 1 r n = ∑ f f k = k 1

Analytical solutions with the use of local adjoint problems Associative property of integral identity for construction of discrete approximations and splitting schemes Basic construction in the set of finite volumes x x + + i 1 i 1 ∫ ∫ ∗ ∗ ∗ ϕ − ϕ = ϕ ϕ + Integral identity ( ) L f dx L dx fragment x x i i x   ∗ ∂ ϕ ∂ ϕ + x i 1 + i 1 ∫ ∗ ∗ ∗ − µ ϕ + µ ϕ + ϕϕ − ϕ =   u f dx 0 ∂ ∂   x x x x i i ≤ ≤ t t t + x - one of the space variables, j j 1 ∂ ϕ ∂ ∂ ϕ ϕ = − µ + ϕ ≤ ≤ , L u a x x x + ∂ ∂ ∂ i i 1 x x x Conditions for the state functions : ∂ ϕ ϕ µ , а ) functions and fluxes are continuous at the inner boundaries ∂ x of the cells; b) conditions at the outer boundaries (Dirichlet, Neumann, of the 3d type)

Basic elements for construction of discrete approximations Local adjoint problems ∗ ∗ ϕ = ≤ ≤ = − L 0, x x x , i 1, n 1 + i i 1 Fundamental solutions { } ∗ ∗ ϕ ϕ = ϕ = (1) * ( ), x ( x ) 0, ( x ) 1, ; under conditions + + 1/2 i i i 1 { } ∗ ϕ = ϕ = * ∗ ϕ ( x ) 1, ( x ) 0, ; (2) under conditions ( ), x + i i 1 + 1/2 i  ∗ ϕ ≤ ≤ (1)  ( ), x x x x + − ϕ = = − i 1/2 i 1 i *  ( ) x , i 2, n 1 i ∗ ϕ ≤ ≤  (2)  ( ), x x x x + + i 1/2 i i 1 ϕ = ϕ = ϕ = * * * ( x ) 0; ( x ) 1; ( x ) 0; − + 1 1 i i i i i i

General structure of discrete-analytical approximations with respect to spatial variables − ϕ + ϕ − ϕ = a b c F ; + − 1 1 i i i i i i i ∂ ϕ ∂ ϕ *(2) *(1) ( ) x ( ) x = − µ + = µ − i 1/2 i 1/ 2 a c + − ∂ ∂ i i 1/2 i i 1/ 2 x x + − i 1 i 1   ∂ ϕ *(2) ( ) x + = − µ + + ϕ *(2) i 1/ 2   b u ( ) x + + + ∂ i i 1/ 2 i 1/2 i 1/2   x + 0 i   ∂ ϕ *(1) ( ) x − = µ − + ϕ *(1) i 1/ 2   b u ( ) x − − − ∂ i i 1/ 2 i 1/ 2 i 1/2   x − 0 i + − = + b b b i i i x x + i 1 i ∫ ∫ = ϕ + ϕ *(2) *(1) F f x ( ) ( ) x dx f x ( ) ( ) x dx + − i i 1/ 2 i 1/2 x x − i i 1