Iterative solvers

Solution of the algebraic equations. For creeping flows, the algebraic equations are hnear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. 

In order to exploit the sparseness of the matrix, iterative solvers can be applied. The iterative procedure is initialized with a guess for the solution vector O. In... 

In CED, a number of different iterative solvers for linear algebraic systems have been applied. Two of the most successful and most widely used methods are conjugate gradient and multigrid methods. The basic idea of the conjugate gradient method is to transform the linear equation system Eq. (38) into a minimization problem... 

In contrast to the conjugate gradient method, the multigrid method is rather a general framework for iterative solvers than a specific method. The multigrid method exploits the fact that the iteration error... 

Multigrid methods have proven to be powerful algorithms for the solution of linear algebraic equations. They are to be considered as a combination of different techniques allowing specific weaknesses of iterative solvers to be overcome. For this reason, most state-of-the-art commercial CFD solvers offer the multigrid capability. 

Vuik, C., Fast iterative solvers for the discretized incompressible Navier-Stokes equations , Delft University of Technology, TMI TR93-98 (1993). 

Major step Major iteration Objective function CPU time (s) NLP or MILP iterations Solver... 

A typical use for this model would be to solve for the number of moles of a gas, given its identity, pressure, volume, and temperature. The iterative solver is used for this purpose. You must decide which variable to choose for iteration and what a reasonable initial guess is. Real gases approach ideal behavior at low pressure and moderate temperatures. Since the compressibility factor z is 1 for an ideal gas, and since knowing z along with P, V, and T allows a calculation of n, we choose z as the iteration variable and 1.0 as the initial guess. 

A TKISolver model called RAINDROP.TK has beendeveloped to incorporate the full Charlson-Vong model of cloud water equilibrium Q2), including the temperature dependence of all equilibrium constants. The iterative solver makes it possible to compute the pH at charge neutrality without having to make plots of intermediate results. The Rule Sheet is shown in Figure 3. 

Using Jacobi s method to compute the inverse of the Laplacian is rather slow. Faster convergence may be achieved using successive over-relaxation (SOR) (Bronstein et al. 2001 Demmel 1996). The iterative solver can also be written in the Gauss-Seidel formulation where already computed results are reused. 

Horn (1974) / Blake (1985) Described in Section 7.2. Zeros were removed from the input by transforming each channel with data in the range [0, 1] according to y = (255,v + l)/256. The extensions of Blake (1985) are used, i.e. the threshold is applied after computing the first derivative. The threshold is set to 3. The inverse of the Laplacian is computed using an SOR iterative solver. (threshold=3)... 

Equations are solved sequentially using an iterative solver. The technique is inherently suited for solving non-linear problems, where non-linearity arises either from material behaviour, geometry or boundary conditions. 

A model reduction-based optimisation framework for large-scale simulators using iterative solvers... 

A Model Reduction-Based Optimisation Framework for Large-Scale Simulators Using Iterative Solvers... 

Compute the steady state using an iterative solver and evaluate x, f f, H,Z... 

In order to avoid the need for an iterative solver, we can extend the CQMOM algorithm described in Section 3.2.3 to include the additional moments needed to compute the parameters cr p. However, in order to apply the algorithm, we must assume that the conditional PDF has the same form for all values of the conditioning variable. As an example, consider the bivariate case with ( 1, 2)- Conditioning on 1, the bivariate NDF can be written as = /( i)/( 2l i). With extended CQMOM (ECQMOM) for 2 conditioned on... 

Finite-difference techniques were used to compute numerical solutions as column-breakthrough curves because of the nonlinear Freundlich isotherm in each transport model. Along the column, 100 nodes were used, and 10 nodes were used in the side-pore direction for the profile model. A predictor-corrector calculation was used at each time step to account for nonlinearity. An iterative solver was used for the profile model whereas, a direct solution was used for the mixed side-pore and the rate-controlled sorption models. 

Solve for dy(<)/dt using Simulink simulation and iterative solver (ode45). 

In our contribution, we address this aspect and describe numerical methods based on the use of efficient iterative solvers, which exploit the conjugate gradient (CG) method, its generalization and the space decomposition preconditioners. The efficiency of these solvers will be illustrated by the solution of elasticity and thermo-elasticity problems arising from the finite element analysis of selected benchmarks with computations performed on a PC cluster. The introduced ideas could be useful also for the solution of more complicated coupled problems. 

In this paper, we shall touch the development of such numerical methods intended for the solution of the coupled evolution problems as e.g. thermoelasticity, which is described in Section 2. Here we also discuss the discretization of the evolution problems. As the computational demands are concentrated mainly in the solution of the arising linear systems, we shall focus on the application of suitable, efficient and parallelizable iterative solvers for these linear systems. Section 3 deals with some general techniques enhancing the efficiency of the iterative solution of discrete evolution problems. Section 4 is devoted to a short discussion of the numerical results. In Section 5, we shall describe solvers, which exploit the domain decomposition and parallel computations. Here we also mention another division techniques as displacement decomposition or composite grid methods. 

The BMT3 problem was solved by means of a simple code implemented in MATLAB. In near future, the described ideas will be also implemented and tested within a scientific FEM system GEM (Institute of Geonics AS CR). This system aims at the solution of large 3D problems, for which the iterative solvers are crucial. In the next section, we try to give an idea about the expected efficiency of a large 3D modelling with parallel computations. 

In Section 3, we discussed an acceleration of iterative solvers, now we shall touch the question of the choice of a suitable iterative methods and show that efficient methods can be found in the class of space decomposition - subspace conection methods, see Blaheta et. al. (2003). Remember that we are interested in the solution of linear systems with symmetric positive definite matrices and... 

The aim of this paper was to show numerical techniques suitable for efficient modelling of large-scale coupled problems, which are typical for the assessment of repositories of the spent nuclear fuel. The use of iterative solvers and parallel computing is the essential part of these methods. 

---