By Hector L Juarez
ISBN-10: 9535100521
ISBN-13: 9789535100522
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28) Mass–Consistent WindNumerical Field Techniques Models: Numerical Techniques by L2–Projection Methods Mass–Consistent Wind Field Models: by L2–Projection Methods 33 11 Then, from (27)–(28) we obtain A ( B q ) = A φq = −∇ · (S −1 ∇φq ) = q. (29) This shows that B can be used as an optimal preconditioner. Therefore, the additional cost of the preconditioned conjugate gradient algorithm is the solution of an elliptic problem at each iteration. However, this additional cost is offsetted by two nice properties: a) the preconditioning must reduce drastically the number of iterations (from about 1000 to less than 20, based on previous experience in CFD); b) there is a significant reduction of degrees of freedom in the discrete version of the elliptic problem associated to operator B.
Performance of the CG–algorithm for three different cases. 4 Preconditioned conjugate gradient method The CPU time to solve the problem with the CG–algorithm, at the level of accuracy shown in Table 2, is about twice the CPU time needed to solve the problem with the E1–algorithm. In order to make this algorithm more reliable we need to speed up the iterative algorithm to get at least a comparable computational efficiency. Fortunately, we have found a good preconditioner for the iterative algorithm.
2007; 2006), but we also include some additional ideas and recent results. The variational method proposed by Sasaki uses the continuity equation ∇ · u = 0, where u is the wind velocity vector field on a given domain Ω. The method is based on the minimization of the functional L defined by L (u, λ) = 1 2 Ω S u − uI · u − uI + λ [∇ · u ] dV , (1) where uI is an initial observed wind field, λ is a Lagrange multiplier and S is a diagonal matrix with weighting parameters αi > 0, i = 1, 2, 3, called Gaussian precision moduli, related to the scales of the respective components of the velocity field.
Fluid dynamics, computational modeling and applications by Hector L Juarez
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