Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

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where the matrix E is defined as follows: E =   

   .

− 1 0 0 0 0 0 − 1 0 0 0 0 0 − 1 0 0 0 0 0 1 0 0 0 0 0 − 1

(4.20.9)

The introduced correction eliminates the flow through the boundary surface of the body, and in the case of symmetric flow it is similarly possible to perform with cells in the plane of symmetry of the aircraft. At the outer boundaries of the physical domain Whitfield [5] proved the assumption ∆ q n = 0. With mentioned corrections, solving algebraic systems equation (4.20.5) is possible without the use of huge memory space, which in addition with reduced factorization error, due to the presence of only two passes, represents a great advantage of this method over classic ADI schemes. If the process of solving the system of equation (4.20.5) in the computational domain occurs in “diagonal” planes, defined by relation i + j + k = const, program code can be completely vectorized for efficient usage on supercomputers. Application of central differential scheme in calculation of the residual term R n in rela tion (4.19.1) requires the introduction of additional terms of artificial viscosity to obtain stationary solutions of differential equations [45]. Method of stabilization of the factorized equation (4.20), based on introduction of dissipative terms of the second and fourth order modifying a fourth-order accurate scheme to a scheme of second order accuracy in the immediate vicinity of discontinuous changes, identical is to the procedure described in chapter 4.2.1. By switching to second-order accuracy it is possible to precisely determine the abrupt changes in the small number of calculation cells, thus avoiding inaccuracy in the vicinity of shock waves. A combination of dissipative terms is described by relation R n 1 = ( D 2 ξ + D 2 η + D 2 ζ − D 4 ξ − D 4 η − D 4 ζ ) q n i , j , k . (4.21) In the expression (4.21) the term D 2 ξ q n i , j , k is determined by relation D 2 ξ q n i , j , k = d ( 2 ) i + 1 / 2 , j , k − d ( 2 ) i − 1 / 2 , j , k , (4.21.1) where is d ( 2 ) i + 1 / 2 , j , k = ε ( 2 ) i + 1 / 2 , j , k J i + 1 / 2 , j , k ∆ t δ + ξ q n i , j , k , (4.21.2) where the term D 4 ξ q n i , j , k is calculated by expression D 4 ξ q n i , j , k = d ( 4 ) i + 1 / 2 , j , k − d ( 4 ) i − 1 / 2 , j , k , (4.21.3) in which is d ( 4 ) i + 1 / 2 , j , k = ε ( 4 ) i + 1 / 2 , j , k J i + 1 / 2 , j , k ∆ t δ 3 + ξ q n i , j , k . (4.21.4) In relations (4.21.2) and (4.21.4) terms δ + ξ q n i , j , k and δ 3 + ξ q n i , j , k represent forward-oriented differ ence operators δ + ξ q n i , j , k = q n i + 1 , j , k − q n i , j , k (4.21.5) and δ 3 + ξ q n i , j , k = q n i + 2 , j , k − 3 q n i + 1 , j , k + 3 q n i , j , k − q n i − 1 , j , k . (4.21.6)