Comparison of efficiency between the Time Stepping and Waveform Relaxation methods for two-dimensional Fourier equation
Sebastião Romero Franco1; Marcio Augusto Villela Pinto2; Ana Paula da Silveira Vargas3
1 Federal University of Paraná, Graduate Program in Numerical Methods in Engineering. PR, Brazil; 2 Federal University of Paraná, Department of Mechanical Engineering. Curitiba - PR, Brazil; 3 Federal Technologic University of Paraná, Department of Mathematics. Apucarana - PR, Brazil
doi:10.20906/CPS/COB-2015-0231
Resumo
This paper presents a methodology and comparisons of parameters between Time Stepping and Waveform Relaxation methods, used to solve the two-dimensional linear time-dependent heat diffusion problem, governed by Fourier equation. The numerical model is obtained by employing finite-difference method, using central second order approximation, Cranck-Nicolson and one variant, designated herein as Hybrid Cranck-Nicolson method for discretization in space and time, respectively. In the solution of the system of equations that resulted from discretization, we used the geometric multigrid method with Correction Scheme, V-cycle, Gauss-Seidel solver, restriction by injection, half weithing and full weithing, prolongation by bilinear interpolation and standard coarsening ratio in the spatial coordinates directions. Tests were accomplished to optimize the relationship between time and spatial discretizations, as well as multigrid parameters. We concluded that, for finer grids (greater reduction in discretization error), Time Stepping method is more efficient than Waveform Relaxation method, and the Hybrid Cranck-Nicolson solver is more efficient than Cranck-Nicolson, in relation to the analyzed parameters.
Palavras-chave: Multigrid; Waveform Relaxation; Time Stepping; Fourier equation; CPU time