Tridiagonal matrix algorithm

Abstract: This paper is concerned with the numerical solution of the singularly perturbed differential-difference equations with small shifts called delay and advanced parameters. A fourth order finite difference method with a fitting factor is proposed for the solution of the singularly perturbed differential-difference equations with mixed shifts. The delay and advanced shifts are managed by Taylor series and an asymptotically equivalent singularly perturbed two-point boundary value problem is obtained. A fitting factor is introduced in the fourth order finite difference scheme for the problem which takes care of the small values of the perturbation parameter. This fitting factor is obtained from the asymptotic solution of singular perturbations. Thomas algorithm is used to solve the discrete system of the difference scheme. Convergence of the proposed method is analyzed. Maximum absolute errors in comparison with the several numerical experiments aretabulated to illustrate the proposed method.

Abstract: An algorithm is presented for solving a system of linear equations Bu = k where B is tridiagonal and of a special form. This form arises when discretizing the equation - d/dx (p(x) du/dx) = k(x) (with appropriate boundary conditions) using central differences. It is shown that this algorithm is almost twice as fast as the Gaussian elimination method usually suggested for solving such systems. In addition, explicit formulas for the inverse and determinant of the matrix B are given.

Abstract: Three-dimensional direct numerical simulations of effusion cooling are performed for a supersonic flat-plate boundary-layer flow at laminar and turbulent state. The cooling film is generated by blowing in wall-normal direction through one discrete spanwise slit. We consider both a modeled blowing approach, where the cooling-gas mass-flux and temperature distribution are prescribed at the wall, and a simulated blowing approach using an additional blowing channel attached to the main computational domain. Due to the increased mixing of the hot mean flow with the cooling gas a rapid dissolution of the cooling film is observed for the modeled blowing. As a consequence, the cooling effectiveness is significantly decreased compared to a laminar core flow. When including the channel, turbulent mixing occurs also near the channel outlet leading to a reduction of the cooling effectiveness compared to the modeled case. The applied numerical method is based on finite differences (FDs): Using compact FDs tridiagonal sets of equations have to be solved employing the pipelined Thomas algorithm in order to compute the spatial derivatives globally. The solution of these tridiagonal systems turned out to be a major bottleneck when porting the code from the previously used vector machines with few, powerful compute nodes to massively parallel systems. In order to avoid processor idling, fully explicit or sub-domain compact FDs are implemented and applied to the effusion-cooling problem. The numerical results and performance data are compared to the regular, globally compact FD scheme.