Iterative Method of Thomas Algorithm on The Case Study of Energy Equation

Implicit method is one of the finite difference method and is widely used for discretization some of ordinary or partial differential equations, such like: advection equation, heat transfer equation, burger equation, and many others. Implicit method is unconditionally stable and has been proved with the approximation of Von-Neumann stability criterion. Actually, implicit method is always identical to block matrices (tri-diagonal matrices or penta-diagonal matrices). These matrices can be solved numerically by Thomas algorithm including Gauss elimination using pivot or not, backward or forward substitution. Furthermore, it can be also solved using LU decomposition method with the elimination of lower triangle matrices first and then the elimination of upper triangle matrices. In this research, Thomas algorithm is used to solve numerically for the problem of convective flow on boundary layer, especially for energy equation with the variation of Prandtl number ( 𝑃 𝑟 ).


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INTRODUCTION
Differential equations are equations that involve an unknown function and derivatives. There will be times when solving the exact solution for the equation may be unavailable or the means to solve it will be unavailable. At these times and most of the time explicit and implicit methods will be used in place of exact solution (Bui, 2010), (Němec et al, 2017). Hasan et al (2013) explained that implicit method is one of the finite difference method and is widely used for discretization some of ordinary or partial differential equations, such like: advection equation, heat transfer equation, burger equation, and many others.
Actually, implicit method is always identical to block matrices, its means tridiagonal matrices or penta-diagonal matrices (Zhu, 1994). These matrices can be solved numerically by Thomas algorithm including Gauss elimination using pivot or not, backward or forward substitution. The Thomas algorithm requires two recursive loops, forward and backward (Hollig, et al 2001), (Chapra, 2010). Furthermore, the Gaussian Elimination is a simple, systematic algorithm to solve systems of linear equations. It is the workhorse of linear algebra, and, as such, of absolutely fundamental importance in applied mathematics (Peter, 2008).
In this research, Thomas algorithm is used to solve numerically for the problem of convective flow on boundary layer, especially for energy equation with the variation of Prandtl number Pr. Finally, the implicit method is unconditionally stable under the Von-Neumann stability criteria which gives also the same results of equation (8). This statement can be seen in Figure 3 which always give converge results for any input of Prandtl number (Pr). Thomas algorithm is very appropriate to make an easy of discretization iteratively (in the tri-diagonal matrices) based on the implicit method.

PRELIMINARIES
Implicit method is one of the finite difference method and is unconditionally stable. This can be shown by the following simple example of one-dimensional heat transfer where is thermal conductivity, initial condition: ( , 0) = ( ) and boundary condition: . Every element of in positive real number, then it is still in range of | | ≤ 1.
Furthermore, it can be concluded that Implicit method is unconditionally stable. After studying Von-Neumann stability for 1D unsteady Heat Transfer, then the next steps are to create tri-diagonal matrix by doing discretization first for the following 1D unsteady Heat where is thermal conductivity. Furthermore, defined an initial condition: ( , 0) = ( ) and boundary conditions: (0, ) = ( , ) = 0. By doing approximation for time with the forward difference and for space with the central difference, then obtained By rearranging the above discretization, then obtained Once again, by rearranging and grouping the same terms + 1 and , then obtained where parameter is defined as is started from index 1, then Equation (4) is iterated by = 2, 3, 4, … , − 1. In this case, the boundary conditions can be rewritten as: (1, ) = ( , ) = 0. To give some descriptions, the following implicit scheme for this case is given in Figure 1. Or rewritten as follows
Based on the tri-diagonal matrix that has been obtained from the discretization of energy equation on convective flow of boundary layer, then the tri-diagonal matrix must be written as shown this below Furthermore, this matrix is eliminated using Gauss elimination, then obtained Equations (3) and (4) can be written recursively as shown this below where = 2, 3, … , .
When the tri-diagonal matrix of Thomas algorithm in Equation (9) is applied into the tri-diagonal matrix in Equation (8), then obtained the following similarity Furthermore, Thomas algorithm is applied on tri-diagonal matrix of energy equation that is a derivation of convective flow on boundary layer, then obtained the following figure  Figure 3 shows that when the Prandtl number ( ) is increased then temperature distribution is decreased. In this case, it is related to heat transfer that is more increased, so