Special Scientific Computing Seminar
Abstract:
Following a suggestion of using a direct linear system solver for stability analysis of steady state flows [1] a possible
acceleration of 2D and 3D time-marching algorithms by implementation of a direct linear solver for the time-propagation
operator itself or for the inner iteration of a multigrid approach is discussed.
In the first case the solution is based on an LU factorization of the Stokes operator, defined on the whole computational
domain. The LU factorization is carried out by a direct multifrontal sparse solver (we use the MUMPS package) and is
performed only once at the beginning of the time-stepping procedure. The successive calculation of the velocity and
pressure fields is obtained by the backward substitution procedure also realized for sparse triangular L and U matrices.
Due to effective utilization of the matrix sparsity both LU- factorization and back substitution are relatively fast.
Another approach utilizes the block implicit multigrid solution based on a coupled line Gauss-Seidel smoother (CLGS) [2].
The line-wise smoother computes the pressure and the velocity corrections simultaneously over the entire row of finite
volumes. We derived an analytical solution of this assembled equations system. It was found that the dependence of memory
and CPU time consumption for both 2D and 3D configurations versus the total nodes number is almost linear [3], which is
typical for the multigrid method [4]. A standard V-cycle technique [4] is used for multigrid solution of the problem. The
developed approach can be effectively parallelized and demonstrates a significant speed up with increase of CPUs number.
A semi-implicit linearized time step implemented in the first approach serves as an effective preconditioner for solving
linear equation systems by Krylov-subspace-based iteration methods [5]. Combining this technique with BICGstab-based
Newton-Raphson and Arnoldi iterations for calculation of steady states and eigenvalues, respectively, allows us to compute
steady state solutions and bifurcation points.
Our stability results for 2D case are in good agreement with previous works [6,7]. The developed approach allows also for
the first incursion into stability analysis of 3D flows without any restriction on velocity boundary conditions.
Unfortunately, due to large memory demands the algorithm is restricted only to 503 grid resolution for a single 64-bit CPU
computer, which is insufficient for obtaining quantitatively reliable results [1]. This restriction is being removed now in
two different ways: by running the code on a massively parallel computer with a large distributed memory; or by a
replacement of the direct solver with the multigrid solver described in [3].
References
[1] Gelfgat A Yu. Inter. J. Numer. Meth. Fluids 2007; 53: 485-506.
[2] Paisley MF. Inter. J. Numer. Meth. Fluids 1999; 30: 441-459.
[3] Feldman Yu., Gelfgat A Yu. doi:10.1016/j.compstruc.2009.01.013.
[4] Trottenberg U, Oosterlee C, Schüller A. Multigrid, Ac. Press, London, 2001.
[5] Doedel E,Tuckerman L.S. Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, Springer,1999.
[6] Gelfgat A Yu, J. Comput. Phys. 2006; 211:513-530.
[7] Sahin M, Owens R.G. Inter. J. Numer. Meth. Fluids 2003; 42:79-88.