Mathematical Physics - Volume II - Numerical Methods

4.2 Solution of Euler equations

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As in the case of a smooth solution in a supersonic nozzle flow without the presence of a shock wave, number of iterations, required for the convergence of the discontinuous solution, decreases with increasing Courant number CFL .

10 -2

10 -3

re idu s al term

10 -4

CFL=5 CFL=10 CFL=20 CFL=25

10 -5

0

200

400

600

number of iterations

Figure 4.15: Convergence rate – subsonic outflow.

For transonic flow with shock wave arising (Figure 4.13), number of the required iterations needed to obtain a stationary solution is greater than in the case of completely supersonic flow without the presence of discontinuous changes of flow quantities. Residual member representing the mass flux error in the continuity equation after 1000 iterations for each Courant number is significantly larger compared to the corresponding value at entirely supersonic flow, which is also shown on the Figure 4.14. Numerical scheme error especially increases in the immediate vicinity of shock wave. Numerical solution error reduction is achieved by increasing the Courant number, whereby it should be noted that large values of the number CFL can reduce the stability of the numerical scheme. Formation method of Jacobi matrices A + and A − in the relations (4.57) and (4.58) also has a significant influence on the convergence of the numerical solution of the system of equations (4.60). Jacobi matrices obtained by flux vector decomposition increase the convergence rate of the numerical solution, making it at the same time more robust than the solution based on matrix formation, used in the Roe scheme. The cause of this lies in the fact that Jacobi matrices are determined by derivatives of the positive and negative flux vectors and in the general differ from“decomposed” Jacobi matrices. Having in mind the equation (4.59), it can be written f = AU = A + U + A − U = f + + f − (4.106) and ∂ f ∂ U = A = ∂ f ∂ U + + ∂ f ∂ U − = A + + A − , (4.107)