Tridiagonal matrix algorithm

Abstract: In this paper, we derive the general expression of the r-th power for some n-square complex tridiagonal matrices. Additionally, we obtain the complex factorizations of Fibonacci polynomials.

Abstract: An exponential compact higher order scheme is developed for stationary convection‐diffusion type of differential equations which inludes incompressible flow equations in stream function vorticity formulation. The scheme is, ingeneral, fourth order accurate however for one‐dimensional constant convection and diffusion coefficients, the scheme is O(h6) and produces a tri‐diagonal system of equations that can be solved efficiently using Thomas algorithm. For two‐dimensional problems, the scheme produces an O(h4+k4) accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit (ADI) procedure. The efficiency of the developed scheme is measured using wave number analysis. The analysis shows that the developed scheme has a much better spectral resolution than any of the existing higher order schemes.

Abstract: We have recently developed a vectorized Thomas solver for quasi-block tridiagonal linear algebraic equation systems using Streaming SIMD Extensions (SSE) and Advanced Vector Extensions (AVX) in operations on dense blocks [D. Barnaś and L. K. Bieniasz, Int. J. Comput. Meth., accepted]. The acceleration caused by vectorization was observed for large block sizes, but was less satisfactory for small blocks. In this communication we report on another version of the solver, optimized for small blocks of size up to four rows and/or columns.We have recently developed a vectorized Thomas solver for quasi-block tridiagonal linear algebraic equation systems using Streaming SIMD Extensions (SSE) and Advanced Vector Extensions (AVX) in operations on dense blocks [D. Barnaś and L. K. Bieniasz, Int. J. Comput. Meth., accepted]. The acceleration caused by vectorization was observed for large block sizes, but was less satisfactory for small blocks. In this communication we report on another version of the solver, optimized for small blocks of size up to four rows and/or columns.

Abstract: Implicit finite-difference schemes of approximate factorization and predictor-corrector schemes based on a special splitting of operators are proposed for the numerical solution of the Navier-Stokes equations governing a viscous compressible heat-conducting gas. The schemes are based on scalar tridiagonal Gaussian elimination and are unconditionally stable. The accuracy and efficiency of the algorithms are confirmed by computing two-dimensional flows of complex geometry.