Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

131

where r A ≥ max ( | λ ( j ) | ) . This approach of flux matrix decomposition, based on the spectral radius ρ ( A ) = max ( | λ ( j ) | ) of matrix A , contribute to easy realization of program code. Quantities averaged by the expressions (4.39) are used in calculation of matrix A elements. Decomposition of the flux matrix by applying the relation (4.57) and (4.58) requires far more complex coding, but gives increased numerical stability of solutions with accelerated convergence. Numerical results, using this method of matrix decomposition, will be presented in detail in the next chapter. The process of solving the system of equations (4.60) using LU implicit factorization method consists of two one-way passes, which is explained in detail in the chapter 4.2.2. To approve the accuracy of the presented numerical procedures, a comparison of such obtained results with the exact one from shock wave theory based on Rankine-Hugoniot equations [22], [8], will be performed. The first example relates to the one-dimensional problem of ideal gas flow in a pipe of constant cross section, caused by the sudden removal of the diaphragm that separates the two areas gases of different thermodynamic quantities. The diagram (Figure 4.6) shows the pressure and density distribution along the axis of the pipe after a certain time interval ∆ t .

Riemann problem

1.2

1.1

pr ROE (BD) ess. dens. ROE (BD) pr ROE (LU) ess. dens. ROE (LU) pr GODUNOV ess. dens. GODUNOV

1

0.9

0.8

0.7

p/p 1 ρ ρ/ 1

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.25

0.5 x/l

0.75

1

Figure 4.6: Pressure and density distribution in the pipe.

The diagram in Figure 4.6 shows a remarkable match of pressure and density distribu tion obtained by Godunov method application with the theoretical, in the shock wave zone and in zones of expansion wave and contact discontinuity. Despite of certain deviations in the zones of shock wave and contact discontinuity the results of the implicit block-diagonal procedure ( BD ) applied to the system of equations (4.39), obtained by the Roe scheme, as well as the methods of implicit LU factorization are quite acceptable. The main disadvan tage of Godunov model is reflected in its complexity that is a major obstacle in the field of two-dimensional and three-dimensional flow field analysis. Explicit nature of this scheme prevents the use of larger integration steps, which greatly increases the computational time. Considering almost identical quality of the solution of the block-diagonal procedure and the procedure of implicit LU factorizations, taking into account given simplicity of LU factorization, this procedure is practically the best choice in the process of computer