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FINITE DIFFERENCES FINITE VOLUMES CONSERVATIONS LAWS Feb April

3 pages
FINITE DIFFERENCES/FINITE VOLUMES & CONSERVATIONS LAWS Feb-April 09 Conservations laws : Derivation Let us consider a subset depending on time D(t) ? R3. Initially, for t = 0, any material particle in D(0) is identified by its coordinate ? . We define by x(?, t) the position at the time t of the particle that was initially at ? . The transformation (?, t) 7? x(?, t) is invertible and sufficiently regular. The material velocity u and jacobian of the transformation are : u(x, t) = ∂x ∂t and J(?, t) = ??x(?, t) = ( ∂xi ∂?j ) 1≤i≤3,1≤j≤3 For any function f(x, t) : R3?R+ 7? R continuously differentiable (that could represent a physical property), we define its particular derivative and sum over a moving volume : df dt = df(x(?, t), t) dt and If (t) = ∫ D(t) f(x, t)dx The aim here is to estimate the integral over the volume D(t) as a function of the initial position, and its variation in time in order to establish some conservation properties : 1.

  • ∂ui ∂xk

  • ∂? ∂t

  • boundary integrals into

  • ∂xk ∂?j

  • finite volume

  • cross product

  • relation into

  • conservation laws

  • ∂f ∂xi

  • ?x ·


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FINITE DIFFERENCES/FINITE VOLUMES& CONSERVATIONS LAWS
Conservations laws : Derivation
Feb-April 09
3 Let us consider a subset depending on timeD(t)R. Initially, fort= 0, any material particle inD(0)is identified by its coordinateξ. We define byx(ξ, t)the position at the timetof the particle that was initially atξ. The transformation(ξ, t)7→x(ξ, t)is invertible and sufficiently regular. The material velocityuand jacobian of the transformation are :   x∂xi u(x, t) =andJ(ξ, t) =rξx(ξ, t) = ∂t ∂ξj 1i3,1j3 3 + For any functionf(x, t) :R×R7→Rcontinuously differentiable (that could represent a physical property), we define its particular derivative and sum over a moving volume : Z df df(x(ξ, t), t) =andIf(t) =f(x, t)dx dt dt D(t) The aim here is to estimate the integral over the volumeD(t)as a function of the initial position, and its variation in time in order to establish some conservation properties : 1. Verifythat Z df =tf+u∙ rxfandIf(t) =f(x(ξ, t), t) det(J)dξ dt D(0) Answer. By using the standard derivation formulas of composed functions we get : 3 3 X X df ∂f∂xi∂f =tf+ =tf+ui=tf+u∙ rxf dt ∂xi∂t ∂xi i=1i=1 As for the integral, we perform a variable change in the first integral (by writingxas a function ofξ), + x:D(0)×R7→ D(t). First we havedx= det(J)dξ, then the integration domain becomesD(0)and the conclusion follows. 2. Showthat  3 X ∂ ∂xi(ξ, t)∂ui∂xk = ∂t ∂ξj∂xk∂ξj k=1 Answer. Again by inverting the derivatives w.r.t time and space and then by applying the derivatives to the composed functions, we get :   3 X ∂ ∂xi(ξ, t)∂ ∂xi(ξ, t)∂ui∂ui∂xk = == ∂t ∂ξj∂ξj∂t ∂ξj∂xk∂ξj k=1
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