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Review of Basic Laws and Equations
Published in Pradip Majumdar, Computational Methods for Heat and Mass Transfer, 2005
A tridiagonal matrix is a banded matrix with a bandwidth of three. For example A=a11a12a21a22a23a32a33a34a43⋱⋱⋱⋱an−1,nan,n−1ann
Freshwater Inflows to Estuaries
Published in Larry W. Mays, Optimal Control of Hydrosystems, 1997
The alternating direction implicit (ADI) method is used to solve the transport equation. Thus, theoretically, it is unconditionally stable for any size of time or spatial step. The linear system of equations result in a tridiagonal matrix which is efficiently solved using the Thomas algorithm. The ADI method is carried out in two steps. At time step t + 1, the x-derivatives are written in implicit form and y-derivatives in explicit form. At time step t + 2, the direction is switched so that the y-derivatives are written in implicit form and x-derivatives in explicit form. The resultant two sets of simultaneous equations are solved directly without iteration.
Linear Algebra
Published in James P. Howard, Computational Methods for Numerical Analysis with R, 2017
However, while the LU decomposition is the workhorse of numerical linear algebra, we should not forget the value in specialized solvers. The tridiagonal matrix solver is an especially efficient solution to solving tridiagonal banded matrix problems. Though tridiagonal banded matrices are not especially common, they have a specific application in interpolation for generating cubic splines, as we will see in the next chapter. Other application-specific solvers, such as the iterative methods, only apply to diagonally dominant matrices, but these commonly arise in polynomial optimization problems, making these somewhat constrained matrices appear more frequently than we might expect.
A numerical model for simulating soil moisture dynamics and root water uptake under saline irrigation
Published in ISH Journal of Hydraulic Engineering, 2023
Satendra Kumar, K S Hari Prasad, C S P Ojha
where, is a square matrix of coefficients of equation (25), is the vector of unknowns pressure heads at the corresponding nodes and, time steps and iterations ( and is the forcing vector. The resulting system of a tridiagonal matrix is solved by employing the Thomas algorithm. Since the variables , and are functions of dependent variable , hence, iterations are carried out to obtain the best possible solution, till the convergence criterion between the two iterative steps is met.
A study on MHD flow of SWCNT-Al2O3 /water hybrid nanofluid past a vertical permeable cone under the influence of thermal radiation and chemical reactions
Published in Numerical Heat Transfer, Part A: Applications, 2023
The Keller-box technique is used for numerical solution of Eqs. (9)–(11), together with boundary condition (12). We choose this scheme because of its flexibility and it is found to be very effective in solving the non-linear problem with an error of order Methodology considered by Cebeci and Bradshaw [51] has been implemented. Computational steps (as explained by Anwar et al. [52]) involved in this scheme to get a numerical solution are as follows: To reduce the obtained ordinary differential equations (ODE) into the system of first-order equations.To write the reduced equations in finite difference equations.To linearize the equations using Newton method and writing them in vector form.To solve the linear equations which provide the tridiagonal matrix.
Analytical and field verification of a 3D hydrodynamic and water quality numerical scheme based on the 2D formulation in CE-QUAL-W2
Published in Journal of Hydraulic Research, 2020
Hussein A. M. Al-Zubaidi, Scott A. Wells
There are several methods for solving a set of linear algebraic equations; some are more suitable than others, based on the accuracy and numerical efficiency. Therefore, any suitable method could be used here. The form of the system of linear algebraic equations plays a major role in choosing the solution method. The solution of the momentums and transport equation ends with a coefficients matrix in a tri-diagonal form where the main diagonal elements are positive and the two off-diagonal elements are negative. The most convenient direct method that is widely and easily used in solving a system of linear equations in which [A] is a tri-diagonal matrix is the Thomas algorithm (Patankar, 1980), also called the tri-diagonal matrix algorithm. We employed a time splitting technique in solving the equations of momentum and transport since these equations lead to a simple tri-diagonal matrix which is diagonally predominant and can be easily solved by Thomas algorithm.