Tridiagonal matrix algorithm

Abstract: For the first time, a numerical solution code, based on Levenberg–Marquardt method is presented for solving non-linear problem of inverse heat transfer in axisymmetric stagnation flow impinging on a cylinder rod to determine time-dependent wall temperature by temperature distribution at a specific point in the fluid region. Also, the effect of noisy data on the final result has been studied. For this purpose, the numerical solution of the dimensionless temperature and the convective heat transfer in a radial incompressible flow on a cylinder axis is carried out as a direct problem. In the direct problem, the free stream is steady with an initial flow strain rate of $$\overline{k}$$ . Using similarity variable and appropriate transformations, momentum and energy equations are converted into semi-similar equations. The new equation systems are then discretized using an implicit finite difference method and solved by applying the tridiagonal matrix algorithm (TDMA). The wall temperature is then estimated by applying the Levenberg–Marquardt parameter estimation approach. This technique is an iterative approach based on minimizing the least-square summation of the error values, the error being the difference between the estimated and measured temperatures. Results of the inverse analysis indicate that the Levenberg–Marquardt algorithm is an efficient and acceptably stable technique for estimating wall temperature in axisymmetric stagnation flow. The maximum value of the sensitivity coefficient is related to the estimation of polynomial wall temperature and its value is 0.1952 also the minimum value of the sensitivity coefficient is 8.62 × 10–6 which is related to the triangular wall temperature. The results show that the parameter estimation error in calculating the triangular and trapezoidal wall temperature is greater than the others because the maximum value of RMS error is obtained for these two cases, which are 0.451 and 0.479, respectively, the reason for the increase in error in estimating these functions is the existence of points where the first derivative of the function does not exist. This method also exhibits considerable stability for noisy input data.

Abstract: A parallel algorithm for computing the finite difference solution to the elliptic equations with non-separable variables is presented. The resultant matrix is symmetric positive definite, thus the preconditioning conjugate gradient or the chebyshev method can be applied. A differential analog to the Laplace operator is used as preconditioner. For inversion of the Laplace operator we implement a parallel version of the separation variable method, which includes the sequential FFT algorithm and the parallel solver for tridiagonal matrix equations (dichotomy algorithm). On an example of solving acoustic equations by the integral Laguerre transformation method, we show that the algorithm proposed is highly efficient for a large number of processors.

Abstract: In this paper the parallel algorithm for solving tridiagonal Toeplitz linear system is described. A rounding error analysis is made under assumption for strictly diagonally dominant matrices. Also some numerical experiments are given.