Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

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The quantities α and β , present in matrices T and T − 1 , are calculated from relations α = ρ / ( c √ 2 ) and β = 1 / ( ρ c √ 2 ) , in which c is the local speed of sound propagation. By grouping non-positive eigenvalues of matrix A in Λ − and positive ones in Λ + Jacobi matrix A can be decomposed into matrix A + = T Λ + T − 1 , (4.57) and matrix A − = T Λ − T − 1 , (4.58) where is A = A + + A − . (4.59) The equation (4.45) can now be written in the form of an implicit difference scheme AM m − 3 / 2 ∆ q m − 3 / 2 + AA m − 1 / 2 ∆ q m − 1 / 2 + AP m + 1 / 2 ∆ q m + 1 / 2 = DS m − 1 / 2 , (4.60) where are AM m − 3 / 2 = − ∆ t ∆ x A + m − 1 , AA m − 1 / 2 = [ S m − 1 / 2 I + ∆ t ∆ x ( S m A + m − S m − 1 A − m − 1 ) − B m − 1 / 2 ∆ t ] , AP m + 1 / 2 = ∆ t ∆ x A − m , DS m − 1 / 2 = − ∆ t ∆ x ( S m F m − S m − 1 F m − 1 ) + ∆ t H m − 1 / 2 , (4.61) and I is unit matrix. Euler system of equations, represented by difference implicit scheme, also supple mented by appropriate boundary conditions [13] and [15], is solved by applying a standard block-diagonal procedure based on the inversion of third order matrices in the case of one dimensional flows. The problem of flow in several dimensions comes down to application the so-called ADI scheme [35], [35], which, unfortunately, in case of the 3D flow field requires a large amount of CPU time. In order to increase the efficiency of the exposed procedure in analysis of 2D and 3D flows, ADI scheme with block-diagonal inversion will be replaced by the LU method of implicit factorization, presented in the chapter 4.2.2 and literature [20], [21] and [31]. Decomposition of flux matrices in a factorized scheme provides diagonal terms dominance and implicitly introduces artificial viscosity, necessary for stabilization of the central difference schemes. In the case of one-dimensional flow, matrices A + and A − previously defined by the relations (4.57) and (4.58), will be formed so that eigenvalues of matrix A + are non-negative and eigenvalues of matrix A − are negative, ie. A + = ( A + r A I ) and A − = (4.62)

1 2

1 2

( A − r A I ) ,