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Numerical Algorithms
Published in George Qin, Computational Fluid Dynamics for Mechanical Engineering, 2021
The purpose of this chapter is to cover some iterative numerical algorithms commonly used in computational fluid dynamics (CFD). The finite difference method usually results in a large system of linear equations, and how to solve such equations efficiently is the main concern with regard to numerical algorithms. The numerical algorithms, or numerical solution methods, can be divided into two categories, namely the direct methods and iterative methods. A direct method gives the exact solution to the finite difference equations (note that it is not the exact solution to the original differential equation. They differ by the global error) in one go. For example the tridiagonal matrix algorithm (TDMA; also known as Thomas algorithm) is a direct method. On the other hand, an iterative method approaches the exact solution of finite difference equations by generating successive approximations to it. And it takes infinite iterations to really reach the exact solution. Of course we do not need to wait that long because typically we only want a sufficiently accurate estimate of the exact solution. The difference between the estimate solution given by an iterative method and the exact solution to the finite difference equations is called iterative error.
Numerical Methods for Conduction Heat Transfer
Published in Yogesh Jaluria, Kenneth E. Torrance, Computational Heat Transfer, 2017
Yogesh Jaluria, Kenneth E. Torrance
The set of algebraic equations obtained in the implicit formation is then solved by direct or iterative methods, as outlined earlier, to obtain the time-dependent temperature distribution. The Gaussian elimination method for the tridiagonal matrix that arises in this case is the most frequently used scheme for solving the linear algebraic equations and is known as the tridiagonal matrix algorithm (TDMA), as discussed earlier. For nonlinear equations, one must resort to iterative procedures. The Crank-Nicolson method is among the most popular methods for one-dimensional transient problems because of its higher accuracy in time. Appendix B.5 gives the computer program for this method. Roberts and Selim (1984) have compared several explicit and implicit schemes for solving the one-dimensional transient diffusion problem. They considered the accuracy, execution time, and programming effort for all these methods, providing guidelines for the selection of an appropriate scheme.
Review of Basic Laws and Equations
Published in Pradip Majumdar, Computational Methods for Heat and Mass Transfer, 2005
Block Tridiagonal Matrix Algorithm. The second approach, known as the block tridiagonal matrix algorithm as discussed in Chapter 2 , is also applicable to the system of Equation 7.58 by writing it in a N2×N2 coefficient matrix in a block tridiagonal form as γ0TN−IN−INTNTNTNTNTNTNTN−IN−INTN1
Avoiding under-relaxations in SIMPLE algorithm
Published in Numerical Heat Transfer, Part B: Fundamentals, 2020
Discretized equations are solved with a tridiagonal matrix algorithm (TDMA). Different steps of the 2-D solver are presented as follows:Guess pressure, velocity, and scalar fields.Solve momentum equations to obtain tentative velocity fields and using Eqs. (15) and (18) provided that the pressure-gradients in momentum equations are calculated from Eq. (11).Calculate using relations like Eq. (12) to construct the fictitious mass source and solve Eq. (34) (e.g., exclude the first term on the left-hand side when considering the SIMPLE algorithm) for the pressure-correction Update velocity and pressure fields.Solve scalar field using Eq. (15) with corrected velocity and pressure fields.Repeat steps 2-5 until convergence is achieved.
Channel Wall Cooling by Evaporative Falling Water–Ethanol and Water–Methanol Films
Published in Heat Transfer Engineering, 2020
Monssif Najim, M’barek Feddaoui, Abderrahman Nait Alla, Adil Charef
To solve the parabolic equations (Eqs. (1)–(8)), a fully implicit scheme is employed. The numerical solution is realized using finite difference method. Each finite-difference equation’s system forms a tridiagonal matrix that is solved using the tridiagonal matrix algorithm [25]. It is still indispensable to satisfy the global mass flow constraint. This is done by the correction of the pressure gradient and axial velocity profile at each axial step, according to Raithby and Schneider method [26]. Moreover, the generated grid is nonuniform to enhance numerical solutions stability and accuracy.