Motivation. The RBF interpolation used for spatial adaption in the experiments above results
in a dense linear system that was solved with SOR, as mentioned. We note that in our first experi- ments, the same under-relaxation parameter was used as in the previous exercises with the inviscid vortex patch test problem. This resulted in errors that were larger than the cases using standard remeshing schemes (the Λ3 scheme was also used, resulting in only a slight improvement overM40).
Subsequently, several tests were performed with different relaxation, and finally a parameter%= 0.4 provided the best results; this value was used in the case shown in Figure 4.3, with a hard-coded maximum of 100 iterations and a maximum norm of the residual in the iterations set to 1E-5.
The choice of a relaxation parameter is known to be problematic when using SOR, as it can be problem-dependent. In addition, this solution scheme is not practicable with large numbers of unknowns, more than, say, a few thousand. In particular, it would not be a practical scheme for the solution of flow problems of real fluid dynamical interest. Furthermore, SOR does not lend itself to parallelization. Fortunately, there is considerable experience in the RBF community using iterative methods and preconditioning to solve the difficult linear systems that arise.
Implementation of Iterative Method. As discussed for example in [27], Krylov subspace meth-
ods have proved to perform very well in the fitting with global basis function methods. The two principal exponents of this class of methods have been applied with success: the conjugate gradient
(CG) method [87], and the generalized minimal residual (GMRES) method [179]. The first of these is applied when the system matrix is symmetric and positive definite, which is true in many RBF applications, but ceases to be the case when some preconditioners are used. The authors of [27] have made freely available their code for the fitting of geophysical data1 which uses the Kelley im-
plementation of GMRES [94]. We modified this code to use the Gaussian basis function, as defined in (1.3), withk = 2, and incorporated it to the vortex method to be used in the spatial adaption processes.
Numerical Experiments. Using the GMRES method both for initialization and spatial adap-
tion, the numerical experiment previously performed with SOR and shown in Figure 4.3 has been reproduced. The errors at the end of the calculation are of the same order of magnitude as in the experiment using SOR in the spatial adaption step, with a final L2-norm velocity error of 2.2E-8.
This is quite positive considering that no preconditioning was used in this experiment (the condition number was recorded on each spatial adaption process, being always of order 108; the maximum
number of iterations was set to 500), although the SOR calculations were not preconditioned either. Once again, the core sizes were restarted to their original size of 0.025 upon each RBF-based spatial adaption step, so that the maximum value ofσwas 0.0287.
Comparison with Standard Remeshing. Next, numerical experiments were performed at-
tempting to produce comparable accuracies with both the RBF-spatial adaption and the standard remeshing, and find the decrease in h that was necessary in the latter case to obtain this accu- racy. For these experiments, a slightly coarser resolution was used as the base case, withσ= 0.03,
h/σ = 0.8, h = 0.024. Population control was enforced at a value of Γi = 2E-13, resulting in
N = 859 at initialization for the same initial condition of the Lamb vortex as used above (Γo= 1.0,
to= 2.0,ν = 0.001). A triangular lattice was used, where the remeshing cases utilized the hexagonal
redistribution scheme of Chatelain. For the same value ofh, the errors obtained using the hexagonal redistribution are plotted in Figure 4.5(a), whereas Figure 4.5(b) shows the results when using the RBF spatial adaption with GMRES as the solution method. The finalL2-norm velocity errors are
2.5E-4 and 1.4E-6, respectively; note that there is a one order of magnitude jump in the velocity error upon the first remeshing event.
A remeshing case is performed next where the value ofhhas been halved, withσ= 0.015. This improves the final velocity error to a value of 8.85E-5, while once again there is an initial remesh error of one order of magnitude (Figure 4.6(a)). An experiment with RBF spatial adaption using
σ= 0.045 is shown in Figure 4.6(b), where the final velocity error is 1.4E-4, an increase of about 60% with respect to the previous case. In conclusion, it can be estimated that a value ofhbetween
1
Available under the terms of the GNU General Public Licence, version 2 or later, with c°2002 by S. Billings. See
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 time, t Errors velocity, L2−norm vorticity, L2−norm velocity, max. rel. vorticity, max. rel.
Figure 4.4: Evolution of the errors in the calculation of a Lamb vortex with Gaussian blobs, Γo= 1.0,
to= 2.0 andν = 0.001 and σ= 0.025,h/σ= 0.7 andN = 2177, ∆t= 0.01. Spatial adaption using
GMRES to solve the RBF linear system for particle strengths, performed every 10 steps.
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 time, t Errors velocity, L2−norm vorticity, L2−norm velocity, max. rel. vorticity, max. rel.
(a) Hexagonal redistribution scheme.
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 time, t Errors velocity, L2−norm vorticity, L2−norm velocity, max. rel. vorticity, max. rel.
(b) RBF spatial adaption (using GMRES).
Figure 4.5: Evolution of the errors in the calculation of a Lamb vortex with Gaussian blobs, Γo= 1.0,
to = 2.0 andν = 0.001 andσ= 0.03, h/σ= 0.8 andN(t= 0) = 859, ∆t= 0.02. Spatial adaption
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 time, t Errors velocity, L2−norm vorticity, L2−norm velocity, max. rel. vorticity, max. rel.
(a) Hexagonal redistribution scheme,h= 0.012.
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 time, t Errors velocity, L2−norm vorticity, L2−norm velocity, max. rel. vorticity, max. rel.
(b) RBF spatial adaption (GMRES),h= 0.036.
Figure 4.6: Evolution of the errors in the calculation of a Lamb vortex with Gaussian blobs, Γo= 1.0,
to= 2.0 andν = 0.001; ∆t= 0.02. Spatial adaption every 10 time steps.
two and three times smaller was necessary to obtain the same accuracy with remeshing compared to the RBF spatial adaption, in this test problem. This is, however, only indicative and a more detailed study at different resolutions would be required for a more conclusive assessment of the comparative accuracy. To do this exhaustively would entail considerable work, taking into account that different parameters affect the results simultaneously. For example, repeating the coarsest case, shown in Figure 4.6(b), but with more frequent spatial adaption —every 5 time steps—, results in a finalL2-norm velocity error of 4.9E-5, which is almost three times smaller than when the adaption
was performed every 10 steps, and almost two times smaller that the case with hexagonal remeshing shown in Figure 4.6(a).
Instead of pursuing further a detailed comparative study of the new mesh-less spatial adaption method with respect to the standard remeshing schemes used in vortex methods, we set about to perform a numerical study of the convergence properties of the new method. A first step in this direction is presented in the following section.
Summary of Algorithm. The basic implementation of the viscous vortex method, with core
spreading and RBF spatial adaption (as used in the Lamb-Oseen tests above) consists in the following algorithmic steps.
1. Distribute particles in a computational box chosen to be large enough to contain the initial vorticity distribution. These particles do not necessarily have to be on a regular arrangement, but in this section we have used square and triangular lattices for simplicity.
circulation strength of the initial particles using the standard initialization formula (local vorticity timesh2).
3. Eliminate particles whose approximate circulation strength is smaller than the chosen threshold for population control.
4. Solve an RBF interpolation system (using an iterative method) to obtain the strengths of the initial particles, after population control.
5. One basic time step — Calculate the particle velocities using the Biot-Savart law, and spread the cores to satisfy the diffusion term (note that there is no fractional step here: performing these steps in any order produces the same result!).
6. If spatial adaption is required (determined by a given adaption frequency, which is an input parameter), then: (i)distribute new particles as in the initialization step, but first determine the computational box with the maximum and minimum particle coordinates; (ii) calculate approximate circulation strengths using the standard initialization formula; (iii) eliminate particles whose approximate circulation strength is smaller than the chosen threshold for pop- ulation control; (iv) using the left-over particles, solve the RBF interpolation system for the new particle strengths, as during the initialization, but first reset the particle sizes to their original small value.