Block-tridiagonal solution

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,... 

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... 

Hindmarsh, A. C. Solution of Block-Tridiagonal Systems of Linear Algebraic Equations Lawrence Livermore National Laboratory Report UCID-30150, 1977. 

Because the Thomas algorithm can be applied to the block tridiagonal structure (15-67) of (15-70), submatrices of partial derivatives are computed only as needed. The solution of (15-65) follows the scheme in Section 15.3, given by 15-13) to (15-18) and represented in Fig. 15.4, where matrices and vectors A), Bj, C/,-Fy, and AX,- correspond to variables A By, Cy, Dy, and x, respectively. However, the simple multiplication and division operations in Section 15.3 are changed to matrix multiplication and inversion, respectively. The steps are as follows... 

In many applications, linear systems have block-band structure. In particular, the numerical solution of partial differential equations is frequently reduced to solving block tridiagonal systems (see Section II. A). For such systems, block triangular factorizations of A and block Gaussian elimination are effective. Such systems can be also solved in the same way as usual banded systems with scalar coefficients. This would save flops against dense systems of the same size, but the algorithms exploiting the block structure are usually far more effective. 

Because of the stiffness which is inherent to most Chemical Engineering problems, implicit integrators are commonly used in simulations. These integration methods require multiple solutions of the underlying linear equation system. The SFD discretization (1.1) leads to block-tridiagonal matrices that can be solved very efficiently. But, as the order of approximation is only two,... 

Hindmarsh AC (1977) Solution of block-tridiagonal systems of linear algebiaic equations. Technical Report Rept. UCID-30150, LLL... 

Marsh F, Potter DE (1981) Recurrence solution of a block tridiagonal matrix equation with Neumann, Dirichlet, mixed or periodic boundary conditions. Comput Phys Commun 24 185-190... 

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