Conserved variables W=掘t=oj(x)X Conservation of the collision operator 蝌WX=jU·/X=0 A property: for any linear combination of f and feq, i.e fe=(1-b)f+ bfeg The conservation variables can be calculated by w=oj()fax
( )fd 轾犏 = = X 犏犏犏犏臌 W u ò E r r j x r Conserved variables 1 2 2 1 ( ) 轾犏 = 犏犏犏犏臌 j x x x [ ] 1 0 eq 蝌j j W X = - X = fd f f fd t Conservation of the collision operator A property: for any linear combination of f and f eq , i.e., = X ( )f d¢ W ò j x (1 ) eq f f f ¢= - + b b The conservation variables can be calculated by
Formulation: A finite-volume scheme ?fx籽f=W? Point 1: Updating rule for cell-center distribution function 1. integrating in cell j j+1 +1-f+ Mid-point Trapezoidal 2, transformation Dt f+ w= 2I-Dt 2Dt dt 3. update rule Key: distribution function at cell interface 12
# Formulation: A finite-volume scheme 1 eq t f f f f t ? 籽 = W ? - 轾 x 臌 1/ 2 1/ 2 1 1/ 2 1 1 / 2 2 n n n n j j j j n n j j f f t t f f x + + + + + - - + - = W + W D D 轾犏 轾 D x 臌 臌 j j+1 j+1/2 1. integratingin cell j: Mid-point Trapezoidal 2. transformation: 2 t f f = - D % W 2 2 2 2 2 eq t t t f f t t f f + D - D D W= + + D D = + + % t % t t 1/ 2 1/ 2 1/ 2 1 , 1/ 2 n n j n n j j j f t f x f f + + + + - + = + - D 轾 D 犏臌 % % x 3. update rule: Key: distribution function at cell interface 12 Point 1: Updating rule for cell-center distribution function
Point 2: Evolution of the cell-interface distribution function How to determine +l/2 Again?fx籽∫=W?饑 1. integrating along the characteristic line Dt wo h j+1 l/2 +l2 2 mple exp 2, transformation h h ∫+W t th 2t t h f(x)+(x +1/2-x, xDt)s j ?0 n+l/2 5n(x1+12 f f"(x+1)+(x xDt)s i* Slope 3. original: 2t +1/2 h 1221++2++h j+l/2 W+12=0y()j+2xdx
1 eq t f f f f t ? 籽 = W ? - 轾 Again x 臌 j j+1 j+1/2 1. integratingalong the characteristic line 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 ( ) 2 n n j n n j j j f f x h h + + + + + - = W + W - 轾犏臌 x Implicit explicit 2. transformation: 2 f h = - f W 2 2 2 2 2 eq h h f f h f f h h + - W= + + + + = t t t , 1 1/ 2 2 1/ 1/ 2 , 1/ 2 , 1 1 1/ 2 2 1 1 2 ( ) ( ) , 0 ( ) ( ) ( ) , 0 n j j j j j n n n j j j j j f x x x t x h f f x x f x t x s x x x s x + + + + + + + + + + + + + ìï + - - D ? ï ï ï = - = í ï ï ï + - - D < ïî So 1/ 2 1/ 2 n j f + + 2 t h D = 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 2 ( ) 2 2 n n n eq j j j h f f f h h + + + + + + = + + + W t t t 3. original: 1/ 2 1/ 2 1/ 2 1/ 2 ( ) ( ) n n j j y x x x f d + + W + + = ò 1 eq t f f f f t ? 籽 = W ? - 轾 Again x 臌 Point 2: Evolution of the cell-interfacedistribution function How to determine j s + j j s f + + = ? Slope