Iterative linear solvers

G. H. Golub, H. A. van der Vorst. Closer to the solution Iterative linear solvers (in The State of the Art in Numerical Analysis). p. 63-92, Oxford, 1997. Clarendon Press. 

The direct linear solver PARDISO is used to solve the continuity and momentum equations and the iterative linear solver GMRES is applied for the species equations. 

The left-hand side of Figure 6.15 shows a similar process as described for Figure 6.14 but for the transport of diluted species physics, which includes species equations. The chosen solver is the iterative linear solver GMRFS with the specific parameters shown on the right-hand side of Figure 6.15. 

FIGURE 6.15 Solving species equations with iterative linear solver GMRES for the laminar static mixer. (Image made using COMSOL Multiphysics and is provided courtesy ofCOMSOL .)... 

The computational model is time independent and thus a stationary solver can be used. The continuity and momentum equations for the noncatalytic part and catalytic bed of the porous reactor with injection needle are solved simultaneously with the iterative linear solver GMRES. 

Applying Gershgorin s theorem to study the convergence of iterative linear solvers... 

Within the above scheme, we implemented the generalized minimal residual (GM-RES) method [52], which is a robust linear solver that ensures convergence of the iterative solution. 

The first term of Eq. 7 vanishes if feasible points (i.e. steady states) are computed at each iteration [6]. Clearly the calculation of the basis Z from Eq. 5 is expensive, as the large system Jacobians need to be constructed and inverted. Furthermore, in the case of input/output simulators Jacobians are not explicitly available to the optimisation procedure, or even to the solver itself as is the case of solvers using iterative linear algebra (e.g. Newton-Krylov solvers). In such cases the Jacobian can only be numerically approximated, with great computational expense in terms of CPU and memory requirements. For this purpose here we compute only reduced-order Jacobians... 

Linear Solvers Although general purpose solvers exist, major improvements in efficiency can be gained by exploiting structural features of the linear equations that are particular to reservoir simulation. The key parts of any linear solver are (i) a method of pre-conditioning, essentially an approximation to the system of equations that can be solved directly, but without storage or speed problems (ii) an iteration scheme. 

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. 

For the computation of compressible flow, the pressure-velocity coupling schemes previously described can be extended to pressure-velocity-density coupling schemes. Again, a solution of the linearized, compressible momentum equation obtained with the pressure and density values taken from a previous solver iteration in general does not satisfy the mass balance equation. In order to balance the mass fluxes into each volume element, a pressure, density and velocity correction on top of the old values is computed. Typically, the detailed algorithms for performing this task rely on the same approximations such as the SIMPLE or SIMPLEC schemes outlined in the previous paragraph. 

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... 

An iterative solution method for linear algebraic systems which damps the shortwave components of the iteration error very fast and, after a few iterations, leaves predominantly long-wave components. The Gauss-Seidel method [85] could be chosen as a suitable solver in this context. 

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. 

This problem is very small, however, with only two decision variables. As the number of decision variables increases, the number of iterations required by evolutionary solvers to achieve high accuracy increases rapidly. To illustrate this, consider the linear project selection problem shown in Table 10.9. The optimal solution is also shown there, found by the LP solver. This problem involves determining the optimal level of investment for each of eight projects, labeled A through H, for which fractional levels are allowed. Each project has an associated net present value (NPV) of its projected net profits over the next 5 years and a different cost in each of the 5 years, both of which scale proportionately to the fractional level of investment. Total costs in each year are limited by forecasted budgets (funds available in... 

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. 

These equations are linear and can be solved by a linear equation solver to get the next reference point (ah, A21). Iteration is continued until a solution of satisfectory precison is reached. Of course, a solution may not be reached, as illustrated in Fig. L,6c, or may not be reached because of round-off or truncation errors. If the Jacobian matrix [see Eq. (L.ll) below] is singular, the linearized equations may have no solution or a whole family of solutions, and Newton s method probably will fail to obtain a solution. It is quite common for the Jacobian matrix to become ill-conditioned because if ao is far from the solution or the nonlinear equations are badly scaled, the correct solution will not be obtained. 

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