Local fourier analysis of space-time relaxation and multigrid schemes.
Friedhoff, Stephanie.
MacLachlan, Scott P.
Börgers, Christoph.
2013.
- We consider numerical methods for generalized diffusion equations that are motivated by the transport problems arising in electron beam radiation therapy planning. While Monte Carlo methods are typically used for simulations of the forward-peaked scattering behavior of electron beams, rough calculations suggest that grid-based discretizations can provide more efficient simulations if the discretizations ... read morecan be made sufficiently accurate, and optimal solvers can be found for the resulting linear systems. The multigrid method for model two-dimensional transport problems presented in [C. Börgers and S. MacLachlan, J. Comput. Phys., 229 (2010), pp. 2914-2931] shows the necessary optimal scaling with some dependence on the choice of scattering kernel. In order to understand this behavior, local Fourier analysis can be applied to the two-grid cycle. Using this approach, expressions for the error-propagation operators of the coarse-grid correction and relaxation steps, projected onto the fine-grid harmonic spaces, can be found. In this paper, we consider easier problems of the form of generalized diffusion problems in space-time that are analogous to model two-dimensional transport problems. We present local Fourier analysis results for these space-time model problems and compare with convergence factors of Börgers and MacLachlan. Since one of our model problems is the diffusion equation itself, we also compare to convergence factors for the diffusion equation of [S. Vandewalle and G. Horton, Computing, 54 (1995), pp. 317-330]. The results presented here show that local Fourier analysis does not offer its usual predictivity of the convergence behavior of the diffusion equation and the generalized diffusion equations until we consider unrealistically long time intervals. © 2013 Society for Industrial and Applied Mathematics.read less
- S. Friedhoff, S. MacLachlan, and C. Börgers, "Local Fourier Analysis of Space-Time Relaxation and Multigrid Schemes," SIAM Journal on Scientific Computing, vol. 35, no. 5, pp. S250-S276, Jan. 2013. doi: 10.1137/120881361.
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