Tridiagonal blocks

The system matrix of equation (6.108) contains the two diagonal blocks /3X and j3y and the two offdiagonal blocks Ax and Ay which are both banded tridiagonal. Rather than inverting a tridiagonal block as naively suggested earlier, it is much less costly to multiply the two sides of the block matrix equation (6.108) from the left by the nonsingular block matrix... 

It can be seen as four different submatrices An, A12, A21, A22, where An is a tridiagonal block with a block size 2x2, A12 is extremely sparse as A21, and the smallest matrix A22 is dense. 

The previous problem is solved in this example as it has a tridiagonal block structure with block sizes 2x2. The program is... 

Note that the object was initialized as follows and we only need to point out that the system is tridiagonal blocks with 2x2 block sizes ... 

An example using tridiagonal blocks structure is provided in Section 5.5. 

Ai i is a structured (i.e., diagonal, tridiagonal, tridiagonal blocks, staircase, etc.) submatrix. A necessary condition is that this matrix must be well conditioned. 

Figure 5.2 Ai structured (tridiagonal) block A and A2,i sparse unstructured blocks A2 dense block.

A distillation column model was chosen to test the various solvers. First-principle distillation models yield a tridiagonal block structure, each block... 

Adding two PI controllers to the set of equations increases the number of dynamic equations the DAE solver must integrate by 2 and inserts 14 unstructured elements into the well-structured tridiagonal block Jacobian, as shown in Figure 5.4. 

Figure 5.4 Partially structured tridiagonal block Jacobian. The points in row 941 and 942 and in column 941 and 942 represent the unstructured elements Specifically, the circles are elements of matrices A, 2 and A2,i, while the diamonds belong to the matrix A2,2.

The intrinsic structure of a distillation column, on the other hand, results in tridiagonal blocks of the correlated mathematical problem. A tridiagonal blocks structure of this kind can he exploited to efficiently and accurately solve the numerical problem. 

The basic algorithm is Newton s method, which exploits the structured form of the system (tridiagonal block). 

The unknowns are taken in a large vector in the order [Rq, Pq, Co,ifo,Ri,Pi,Ci, ifi,..., RN,PN,CN,ifNV but are lumped into the vector of four-point vectors U = [Ri, Pi, Ci, ifiY, i = 0... N, to prepare for the block-tridiagonal procedure for solving the system. The system of equations (13.37) is nonlinear, and the Newton method is used to solve it. At each index i we have three 4x4 blocks in the Jacobian matrix L the left-hand block for the elements at index / — 1 M,, the middle block for index /, and Q the right-hand block for index t -I-1, that symbol chosen here in order to avoid clashes with the concentration symbol R. They produce a tridiagonal block system. For this example, three-point BDF was used, started with one BI step. 

Hence, we can maintain the block tridiagonal structure of the Jacobian in (4.5) if we introduce the parameter a as a dependent variable at m of the m -h 1 grid points and if we specify a normalization condition at the remaining grid point that does not introduce nonzero Jacobian entries outside of the three block diagonals. The success of this procedure depends upon the choice of the normalization condition. 

In flame extinction studies the maximum temperature is used often as the ordinate in bifurcation curves. In the counterflowing premixed flames we consider here, the maximum temperature is attained at the symmetry plane y = 0. Hence, it is natural to introduce the temperature at the first grid point along with the reciprocal of the strain rate or the equivalence ratio as the dependent variables in the normalization condition. In this way the block tridiagonal structure of the Jacobian can be maintained. The flnal form of the governing equations we solve is given by (2.8)-(2.18), (4.6) and the normalization condition... 

The Jacobian of the system is a square matrix, but importantly, because the residuals at any mesh point depend only on variables at the next-nearest-neighbor mesh point, the Jacobian is banded in a block-tridiagonal form. Figure 16.10 illustrates the structure of the Jacobian in the form used by the linear-equation solution at a step of the Newton iteration,... 

Fig. 16.10 Illustration of the block-tridiagonal structure of the Jacobian matrix. The structure on the right would result if the mass flux were not defined as a variable at each mesh point.

The blocks of the block-tridiagonal structure correspond to the mesh, with each block being a square matrix with the dimension of the number of dependent variables at each mesh point (here the number of species, plus temperature, plus the mass flux). 

A better alternative approach is what will be called the Rudolph method [476], after the person who introduced it into electrochemical simulation. It was known before 1991 under various names, notably block-tridiagonal [280,412,470,471,528,570]. This comes from the fact that if one lumps the large matrix into a matrix of smaller matrices and vectors, the result is a tridiagonal system that is amenable to more efficient methods of solution. In the present context, we define some vectors... 

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