preconditioners for linearized discrete compressible euler equations

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Bernhard Pollul 4 blocks with 10 × 10 cells each. Adaptation of the grid was performed each time the density residual has decreased two orders of magnitude. On the finest level the iteration is stopped when the density residual is reduced by a factor of 10 4 . Grid refinement and coarsening are controlled by a multiscale wavelet technique. For test case A this results in a sequence of 14 (locally refined) grids. In table 2 we show the number of cells in each of these grids. We note that on the finer grids a change (refinement/coarsening) to Grid 1 2 3 4 5 6 7 # cells 400 1,600 4,264 7,006 11,827 15,634 21,841 Grid 8 9 10 11 12 13 14 # cells 25,870 28,627 30,547 31,828 33,067 33,955 34,552 Table 2: Test case A, sequence of grids the next grid does not necessarily imply a smaller finest mesh size. It may happen that only certain coarse cells are refined to obtain a better shock resolution. For a discussion of this adaptivity issue we refer to [10]. In figure 4 the final grid (34,552 cells) and the corresponding Mach number distribution is shown. The flow pattern downstream of the trailing edge has a complex shock configuration. Two oblique shocks are formed at the trailing edge. The supersonic region behind these oblique shocks is closed by a further normal shock. On each grid an implicit time integration is applied (cf. section 2.3). Currently the choice of the timestep is based on an ad hoc strategy. Starting with a CFL number equal to 1 the timestep is increased based on the rule CFL new = 1.1 CFL old . A maximum value of 1000 is allowed. Per timestep one inexact Newton iteration is applied. The linear systems with the approximate Jacobians (cf. section 2.3) are solved by a preconditioned BiCGSTAB method until the relative residual is smaller than 1E-2. Because the PBSPAI(0) preconditioners have shown a very poor behavior already for the relatively simple problems in section 4.1 and section 4.2 we decided not to use the PBSPAI(0) preconditioner in this test problem. An interface between QUADFLOW and the PETSc library [3] has been implemented. This makes the BiCGSTAB method and PBILU(0) preconditioner available. Recently also a PBGS routine has become available. The PETSc library offers many other iterative solvers and preconditioners. Here, besides the PBGS and PBILU(0) preconditioners we also consider block variants of ILU that allow more fill-in, namely the PBILU(1) and PBILU(2) methods. The arithmetic work in the QUADFLOW solver is dominated by the linear solves on the finest grids. We present results only for the two finest grids. The corresponding number of preconditioned BiCGSTAB iterations for each timestep on these two grids is given in figure 5. Note that different preconditioners may lead to (slightly) different numbers of timesteps. On grid 13 we have about 110 timesteps and then a change to grid 14 takes place. In these 110 timesteps on grid 13 the iteration count shows a clear 12
Bernhard Pollul Figure 4: Test case A. Left figure: computational grid. Right figure: Mach distribution, M min = 0.0, M max = 1.45, ∆M = 0.05 increasing trend. This is due to the increase of the CFL number. After the change to grid 14 one starts with CFL = 1 and a similar behaviour occurs again. test case A B Grid 13 14 10 11 PBGS 15.3 20.2 2.85 32.0 PBILU(0) 6.92 8.82 1.33 11.2 PBILU(1) 4.49 4.65 1.04 6.87 PBILU(2) 3.52 3.81 1.04 5.78 Table 3: Test cases A and B, average iteration count 13
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preconditioners for linearized discrete compressible euler equations

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