Tridiagonal matrix algorithm

Abstract: First, the Cartesian and generalized coordinate systems and the coordinate transformation are introduced. We also discuss the method of virtual boundaries and the need to introduce a forcing term to represent the geometry. Next, we present the formulation for low Mach number, valid for most cases of reactive flows. Then the large-eddy simulations formulation is discussed as an improvement of Reynolds averaged Navier-Stokes (RANS) formulation, which is less expensive than direct numerical simulations (DNS) formulation. Subsequently, for a reactive flow model, the equations of momentum, energy, enthalpy, and chemical species are written as a general equation, which is approximated by methods of finite difference, finite volume, and finite element, to be integrated by Runge-Kutta methods. After that, approximations of order 3 and 4 are given, as well as some compact schemes of order of approximation 6. Then, we discuss some of the main methods used in the flow solution such as Gauss-Seidel, simplified Runge-Kutta, tridiagonal matrix algorithm (TDMA), Newton, strongly modified implicit procedure (MSI), and LU-SSOR, which is an LU decomposition with the introduction of dissipation. Then, we indicate some methods for solving stiff systems of equations, such as Newton’s method and Rosenbrock’s method, which can be seen as a combination of the methods of Newton and Runge-Kutta. After that, the principal boundary conditions, such as permeable and impermeable wall, symmetry and cut, far field and periodic are given, which are common in jet diffusion flames, and in reactive flows in porous media. Finally, some techniques for the acceleration of convergence as local time-stepping, residual smoothing, and the multigrid technique are introduced. Moreover, some numerical implementation details and the analysis of uncertainties for the solution of reactive flows is discussed.