Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

147

In this chapter the procedure of second-order accuracy numerical scheme [29] forma tion, without oscillations present, capable of very accurate determination of flow quantities changes in the shock wave vicinity and contact discontinuity, will be outlined. Require ments that the scheme should fulfill in the mentioned cases are based on Godunov concept of monotonicity . Constant value of quantities in numerical solution within each computational cell, which was applied in the original Roe scheme [40], described in chapter 4.2.3, implies the first order accuracy in space discretization. Linear distribution of the numerical solution in the computational cell increases accuracy of spatial discretization to the second order, while quadratic distribution leads to the spatial discretization of the third order accuracy. In numerical solution of Euler equations, based on finite volume method, mean values of flow variables are used at the observed time. In the general case distribution of flow variables in the computation cell “ i ” is given by the expression for x i − 1 / 2 < x < x i + 1 / 2 , where U i is mean value of quantity U in computational cell i , ie. U i = 1 ∆ x Z i + 1 / 2 i − 1 / 2 U ( x ) dx , (4.115) while δ i U and δ 2 i U are approximations of the first and second derivative of quantity U . The constant k in the expression (4.114) can affect the error of the chosen approximation. For the values x = x i ± ∆ x / 2 values of flow variables at the boundary of the calculation cell are obtained U L i + 1 / 2 = U i + ( 1 − k ) 4 ( U i − U i − 1 )+ ( 1 + k ) 4 ( U i + 1 − U i ) , (4.116) ie. U D i + 1 / 2 = U i + 1 − ( 1 + k ) 4 ( U i + 1 − U i )+ ( 1 − k ) 4 ( U i + 2 − U i + 1 ) . (4.117) Analogous to the expressions (4.116) and (4.117) extrapolation of the numerical flux can be performed at the boundary of the calculation cell, which leads to relations U ( x ) = U i + 1 ∆ x ( x − x i ) δ i U + 3 k 2 ∆ x 2 ( x − x i ) 2 − ∆ x 2 12 δ 2 i U , (4.114)

( 1 − k ) 4

( 1 + k ) 4

+ i +

+ i ) ,

f + b i + 1 / 2 = f

( f +

+ i − 1 )+

( f +

i − f

i + 1 − f

(4.118)

( 1 + k ) 4

( 1 − k ) 4

f − f i + 1 / 2 = f − i + 1 −

( f − i + 2 − f − i + 1 ) .

( f − i + 1 − f − i )+

(4.119)

Numerical flux of the second order accuracy, on the boundary of the computational cell, is determined by the expression f ( 2 ) i + 1 / 2 = f i + 1 / 2 + 1 − k 4 ( f i − f i − 1 / 2 )+ 1 + k 4 ( f i + 1 − f i + 1 / 2 ) + + 1 + k 4 ( f i − f i + 1 / 2 )+ 1 − k 4 ( f i + 1 − f i + 3 / 2 ) , (4.120)