Alternating direction methods on multiprocessors
SIAM Journal on Scientific and Statistical Computing
A first course in the numerical analysis of differential equations
A first course in the numerical analysis of differential equations
Alternating Direction Methods for Parabolic Systems in m Space Variables
Journal of the ACM (JACM)
A multigrid tutorial: second edition
A multigrid tutorial: second edition
Multigrid
Nonlinear diffusion filtering on extended neighborhood
Applied Numerical Mathematics
Efficient and reliable schemes for nonlinear diffusion filtering
IEEE Transactions on Image Processing
A Fictitious Domain, parallel numerical method for rigid particulate flows
Journal of Computational Physics
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For the solution of elliptic problems, fractional step methods and in particular alternating directions (ADI) methods are iterative methods where fractional steps are sequential. Therefore, they only accept parallelization at low level. In [T. Lu, P. Neittaanmaki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, RAIRO Model. Math. Anal. Numer. 26 (6) (1992) 673-708], Lu et al. proposed a method where the fractional steps can be performed in parallel. We can thus speak of parallel fractional step (PFS) methods and, in particular, simultaneous directions (SDI) methods. In this paper, we perform a detailed analysis of the convergence and optimization of PFS and SDI methods, complementing what was done in [T. Lu, P. Neittaanmaki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, RAIRO Model. Math. Anal. Numer. 26 (6) (1992) 673-708]. We describe the behavior of the method and we specify the good choice of the parameters. We also study the efficiency of the parallelization. Some 2D, 3D and high-dimensional tests confirm our results.