Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

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Depending on the sign of the wave propagation velocities D L and D D the values of the variables R , U and P will be determined. In case where D L and D D are of the same sign, R , U and P take the values ( ρ L , u L , p L ) from zone I if D L and D D are positive, and values ( ρ D , u D , p D ) from zone IV if D L and D D are negative. If the velocities D L and D D are of opposite signs, for P and U the values p c . d . and u c . d . are taken, respectively, while R = ρ L if u c . d . > 0, or R = ρ D in case u c . d . < 0. The problem analyzed in this chapter is known in the literature as the Riemann problem . Roe scheme – approximation of Riemann problem Unlike Godunov model, which represents an approximate solution of the exact physi cal problem, in this chapter will be presented procedure for solving a system of linear hyperbolic equations, which approximate the exact Euler system of equation [50], [40] ∂ q ∂ t + A ( q L , q D ) ∂ q ∂ x = 0 , q ( x , 0 ) = q L , for x < x m q ( x , 0 ) = q D , for x > x m (4.30) where q is a vector of variable flow quantities q = ( ρ , ρ u , ρ e t ) T , (4.31) while ρ , u and e t are density, flow velocity and specific total fluid energy, respectively. In the first equation of system (4.30) A represents the locally constant Jacobi matrix ∂ F / ∂ q , where F is the flux vector F = ( ρ u ρ u 2 + p ρ u ( e t + p / ρ ) ) , (4.32) bearing in mind that e t = e u + u 2 / 2. Jump of the flow variables vector q through each characteristic line , shown on Fig ure 4.5, corresponding to eigenvalue λ ( j ) of matrix A , is proportional to the right eigenvec tor r ( j ) associated to eigenvalue λ ( j ) , ie. q D − q L = 3 ∑ j = 1 α j r ( j ) , (4.33) where α j is “strength” of the j th wave, while the jump the flux vector F can be represented by relation F D − F L = 3 ∑ j = 1 ∆ F j = 3 ∑ j = 1 α j λ ( j ) r ( j ) . (4.34) To solve the hyperbolic system equation (4.30) the flux vectors of flow variables at each boundary cells have to be known. At the cell boundary x = x m (Figure 4.5) flow variables flux, denoted by F m , is calculated on the basis of expression

k 1 ∑ j = 1

( j ) r ( j ) ,

α j λ −

F m = F L +

(4.35)