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Flow and riverbed erosion-deposition simulation around submerged water intake
Published in Silke Wieprecht, Stefan Haun, Karolin Weber, Markus Noack, Kristina Terheiden, River Sedimentation, 2016
The pressure decomposition technique and θ semi-implicit method are used, with the solution procedure being split into two steps. First, with the implicit parts of non-hydrostatic pressures excluded, the provisional velocity field and free surface are obtained by solving a 2-D Poisson equation. Second, the theory of the differential operator is employed to derive the 3-D Poisson equation for non-hydrostatic pressures, which is solved to obtain the non-hydrostatic pressures and to update the provisional velocity field. When the non-orthogonal sigma-coordinate transformation is introduced, additional terms append, resulting in a 15-diagonal, diagonally dominant but unsymmetric linear system of the 3-D Poisson equation for non-hydrostatic pressure. The Biconjugate Gradient Stabilized (BiCGstab) method used to solve the resulting 3-D unsymmetric linear system instead of the conjugate gradient method, which only applies for symmetric, positive-definite linear systems. The detail of model equations and discretization are found in HU (2011, 2013).
A coupled multi–field dynamic model for anisotropic porous materials
Published in Alphose Zingoni, Current Perspectives and New Directions in Mechanics, Modelling and Design of Structural Systems, 2022
N. De Marchi, M. Ferronato, G. Xotta, V.A. Salomoni
A fully-coupled fully implicit FE model, for the solution of the equations representing the dynamics of fully saturated anisotropic porous media, was presented. For the efficiency and robustness of the inner linear solver, the Bi-CGStab algorithm was used, accelerated by an ad-hoc preconditioning technique. The developed numerical tool proved to be efficient and reliable for modeling the wave propagation in anisotropic soils.
A Synopsis of the Strategies and efficient Resolution Techniques Used for Modelling and Numerically Simulating the Drying Process
Published in Ian Turner, Arun S. Mujumdar, Mathematical Modeling and Numerical Techniques in Drying Technology, 1996
In an attempt to reduce the condition number of the sparse matrix A, the system of equations is premultiplied by a preconditioning matrix P, which is easily invertible and a good approximation to A-1. Thus, instead of solving the system Ax=b, the solution to the modified system (PA)x=Pb is sought. The choice of a good preconditioning matrix can often lead to a reduction in the number of iterations required to converge the chosen conjugate gradient scheme and hence, offers an efficient solution method which reduces substantially the overall computation time of the linear solver. In the search for the best preconditioning matrix it is necessary to find a good approximation to A-1 such that the memory overhead and computing cost per iteration are small. In fact, it is possible to use any of the classical iterative schemes as a preconditioner and in this instance, the complete linear solver for the inner iteration stage of the non-linear solver can be viewed as a two step procedure with an inner loop based on a simple classical iterative scheme acting as a preconditioner for an outer loop of the Bi-CGSTAB method. In another context, the conjugate gradient solver is simply being used to accelerate the convergence of the classical iterative scheme.
A Quasi–Monte Carlo Method With Krylov Linear Solvers for Multigroup Neutron Transport Simulations
Published in Nuclear Science and Engineering, 2023
Sam Pasmann, Ilham Variansyah, C. T. Kelley, Ryan McClarren
For each experiment presented in Sec. IV, two Krylov methods, GMRES (Ref. 27) and BiCGSTAB (Ref. 28), were used. GMRES is one of the most common Krylov methods. When solving , GMRES minimizes over the k’th Krylov subspace. For every iteration, GMRES stores an additional Krylov vector. For problems that require many iterations, this may lead to memory constraints. BiCGSTAB is a low-storage Krylov method that is memory bounded throughout the algorithm. However, the memory savings come from information that is thrown out with each iteration, and therefore, BiCGSTAB will generally require more iterations to converge than GMRES. Nonetheless, as we will observe in Sec. IV, both Krylov methods will require far fewer iterations than SI.
Flexible GMRES solver for interpolating MLPG analysis of heat conduction
Published in Numerical Heat Transfer, Part B: Fundamentals, 2022
Abhishek Kumar Singh, Krishna Mohan Singh
The BiCGSTAB method (a stabilized variant of BiCG method) is a Krylov subspace method. Algorithm of preconditioned BiCGSTAB method is presented below [24]:Set tolerance (tol_rel), initial guess Compute Set For k = 0, 1, 2, … until convergence DoIf (k==0) Else Solve (1st preconditioner call)Solve (2nd preconditioner call)Check for convergence. If converged, exit the iteration loop.End Do.
Scheduled relaxation Jacobi method as preconditioner of Krylov subspace techniques for large-scale Poisson problems
Published in Numerical Heat Transfer, Part B: Fundamentals, 2020
Ankita Maity, Krishna Mohan Singh
We have opted for two Krylov subspace solvers: (i) the preconditioned conjugate gradient (PCG) method for symmetric systems and (ii) the biconjugate gradient stabilized (BiCGSTAB) method for general systems. Both of these methods are very easy to program and parallelize and are very robust. We provide a brief outline of the algorithm of these methods below.