Use of sparse matrix techniques in numerical solution of differential equations for renal counterflow systems

✍ Scribed by R.P. Tewarson; A. Kydes; John L. Stephenson; R. Mejia

• Publisher: Elsevier Science
• Year: 1976
• Tongue: English
• Weight: 791 KB
• Volume: 9
• Category: Article
• ISSN: 0010-4809
• DOI: 10.1016/0010-4809(76)90010-0

✦ Synopsis

Mammals concentrate urine by intricate counterflow systems in their kidneys. The mathematical models of such systems involve the numerical solution of a system of differential equations. (A preliminary report of this work was given in Notices of the American Mathematical Society 21, A498 (1974).) This leads to a system of nonlinear algebraic equations, that are solved by the Newton-Raphson method. The basic step in the Newton-Raphson method is the repeated solution of a system of linear equations with the Jacobian of the nonlinear equations as the coefficient matrix. It is shown that the zero-nonzero structure of the Jacobian can be predicted from the physical connectivity of the model. We exploit this zero-nonzero structure to permute the Jacobian to a bordered block triangular form, which is then used to compute the correction to the solution vector of the nonlinear equations. For this computation a modified form of partial pivoting is used. Results are given that show the high accuracy of the computed solutions, the optimum use of the internal computer storage, and the fast rate of computation.