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Application of Numerical Methods to Selected Model Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
There is another useful family of iterative methods that has been widely used in solving large systems of algebraic equations. These are loosely related to the conjugate gradient method, which, in its original form, was a direct method. The methods are also known in some circles as Krylov methods or Krylov subspace methods after the Russian scientist A. N. Krylov. The methods solve a system of the form [A]u = C by repeatedly performing matrix–vector multiplications involving [A]. The conjugate gradient method (Hestenes and Stiefel, 1952) is the “original” Krylov subspace iterative method. The conjugate gradient method requires that [A] be symmetric and positive definite, which limits its use for many equations arising in CFD. However, a modification known as the generalized minimal residual method (GMRES) (Saad and Schultz, 1986) is often used for the more general systems of equations arising in CFD. To provide a background for the development of GMRES, the conjugate gradient method will be outlined first.
A process-based hydrological model for continuous multi-year simulations of large-scale watersheds
Published in International Journal of River Basin Management, 2023
Marcela Politano, Antonio Arenas, Larry Weber
Partial differential conservation Equations (1), (8) and (9) are solved using a finite volume method. Divergence terms in the equations are converted to surface integrals, which can be evaluated as surface fluxes, using the divergence theorem. The resulting system of ordinary differential equations is solved using the library CVODE of SUNDIALS developed at the Lawrence Livermore National Laboratory (Hindmarsh & Serban, 2016). The Backward Differentiation Formulas (BDFs) with Newton iterations recommended for stiff problems are used. A scaled preconditioned GMRES (Generalized Minimal Residual method) solver is used for the solution of the linear system within the Newton corrections.
SENSMG: First-Order Sensitivities of Neutron Reaction Rates, Reaction-Rate Ratios, Leakage, keff, and α Using PARTISN
Published in Nuclear Science and Engineering, 2018
A more efficient method of solving Eq. (14) was recently implemented into Oak Ridge National Laboratory’s (ORNL’s) Denovo code.9 The method takes advantage of Denovo’s generalized minimal residual method (GMRES) and Krylov solvers.