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Comparing the convergence rate (number of iterations) and transient (explicit and implicit) and justify the numerics involved.

Aim: To compare the convergence rate (number of iterations) and transient (explicit and implicit) and justify the numerics involved. Abstract Equation of two-dimensional heat conduction equation is $\frac{\partial T}{\partial t} = α * (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}})$ $frac{delT}{delt}=alpha**(frac{del^2T}{delx^2}+frac{del^2T}{dely^2})$ where $α$ $alpha$ is the thermal diffusion coefficient.…

Faizan Akhtar
updated on 22 Feb 2021

Aim: To compare the convergence rate (number of iterations) and transient (explicit and implicit) and justify the numerics involved.

Abstract

Equation of two-dimensional heat conduction equation is

$\frac{\partial T}{\partial t} = α * (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}})$ $frac{delT}{delt}=alpha**(frac{del^2T}{delx^2}+frac{del^2T}{dely^2})$

where $α$ $alpha$ is the thermal diffusion coefficient.

The equation is best described as a transient state heat conduction equation. If $\frac{\partial T}{\partial t} = 0$ $frac{delT}{delt}=0$ it is called steady state equation.

Applying numerical discretization for the above-mentioned equation.

Governing equation

Case-1

Transient state two-dimensional heat conduction equation

Nomenclature

Nodes representing "X" axis="i".

Nodes representing "Y" axis="j".

At any (i,j) temperature is represented by T(i,j)

The temperature at the top is represented by T(i,j+1)

The temperature at the bottom is represented by T(i,j-1)

The temperature at the right is represented by T(i+1,j)

The temperature at the left is represented by T(i-1,j)

The number of time steps is represented by "n".

Explicit approach

In an explicit approach forward differencing will be applied to the left-hand side of the equation whereas backward differencing is applied to the right-hand side of the equation. The right-hand side of the equation will contain terms from the previous time steps whereas the left-hand side of the equation will contain the term from the present time step.

The two-dimensional heat conduction equation under explicit approach is best described as under

$T {(i, j)}^{n + 1} = T {(i, j)}^{n} + \frac{α * d t (T {(i + 1, j)}^{n} + T {(i - 1, j)}^{n} - 2 T {(i, j)}^{n})}{{d x}^{2}} + \frac{α * d t (T {(i, j + 1)}^{n} + T {(i, j - 1)}^{n} - 2 T {(i, j)}^{n})}{{d y}^{2}}$ $T(i,j)^(n+1)=T(i,j)^n+frac{alpha**dt(T(i+1,j)^n+T(i-1,j)^n-2T(i,j)^n)}{dx^2}+frac{alpha**dt(T(i,j+1)^n+T(i,j-1)^n-2T(i,j)^n)}{dy^2}$

The Matlab output graph for transient state explicit approach is shown below.

Implicit approach

In an explicit approach forward differencing will be applied to the left-hand side of the equation whereas backward differencing is applied to the right-hand side of the equation. The right-hand side as well as left hand side of the equation will contain terms from the present time steps except T(i,j) of LHS will be of previous time steps.

As compared to explicit implicit requires iterative solvers to reach to a particular solution.

Iterative methods are as follows

Here $K 1 = \frac{α * d t}{{d x}^{2}}$ $K1=frac{alpha**dt}{dx^2}$

$K 2 = \frac{α * d t}{{d y}^{2}}$ $K2=frac{alpha**dt}{dy^2}$

Jacobi method: As the solution marches forward jacobi will use the value of previous time steps and compute the present value till convergence.

$T {(i, j)}^{n + 1} = \frac{T p r e v i o u s {(i, j)}^{n} + K 1 * (T o l d {(i + 1, j)}^{n + 1} + T o l d {(i - 1, j)}^{n + 1}) + K 2 * (T o l d {(i, j + 1)}^{n + 1} + T o l d {(i, j - 1)}^{n + 1})}{1 + 2 * K 1 + 2 * K 2}$ $T(i,j)^(n+1)=frac{Tprevious(i,j)^n+K1**(Told(i+1,j)^(n+1)+Told(i-1,j)^(n+1))+K2**(Told(i,j+1)^(n+1)+Told(i,j-1)^(n+1))}{1+2**K1+2**K2}$

The Matlab output graph for transient state implicit approach under jacobi method is shown below.

jac

Gauss-Seidel method: For calculating the value of a temperature at a node 2 it will use the updated value at node 1 and previous value of node 3.

$T {(i, j)}^{n + 1} = \frac{T p r e v i o u s {(i, j)}^{n} + K 1 * (T o l d {(i + 1, j)}^{n + 1} + T {(i - 1, j)}^{n + 1}) + K 2 * (T {(i, j + 1)}^{n + 1} + T {(i, j - 1)}^{n + 1})}{1 + 2 * K 1 + 2 * K 2}$ $T(i,j)^(n+1)=frac{Tprevious(i,j)^n+K1**(Told(i+1,j)^(n+1)+T(i-1,j)^(n+1))+K2**(T(i,j+1)^(n+1)+T(i,j-1)^(n+1))}{1+2**K1+2**K2}$

The Matlab output graph for transient state implicit approach under Gauss-Seidel method is shown below.

Successive over relaxation method:

It is a variant of "Gauss-Seidel method". It uses a relaxation factor $ω$ $omega$ . When $ω > 1$ $omegagt1$ it is over relaxation. When $ω < 1$ $omegalt1$ it is under relaxation.

$T {(i, j)}^{n + 1} = (1 - ω) * T o l d {(i, j)}^{n} + ω * T (g a u s s - s e i d e l) (i, j)$ $T(i,j)^(n+1)=(1-omega)**Told(i,j)^n+omega**T(gauss-seid el)(i,j)$

The Matlab output graph for transient state implicit approach under SOR method is shown below.

SOR

Case-2

Steady state two-dimensional heat conduction equation

Nomenclature

Nodes representing "X" axis="i".

Nodes representing "Y" axis="j".

At any (i,j) temperature is represented by T(i,j)

The temperature at the top is represented by T(i,j+1)

The temperature at the bottom is represented by T(i,j-1)

The temperature at the right is represented by T(i+1,j)

The temperature at the left is represented by T(i-1,j)

The number of time steps is represented by "n".

Implicit approach

In an explicit approach forward differencing will be applied to the left-hand side of the equation whereas backward differencing is applied to the right-hand side of the equation. The right-hand side as well as left hand side of the equation will contain terms from the present time steps except T(i,j) of LHS will be of previous time steps.

As compared to explicit implicit requires iterative solvers to reach to a particular solution.

Iterative methods are as follows

Here $K = \frac{2 * ({d x}^{2} + {d y}^{2})}{{d x}^{2} * {d y}^{2}}$ $K=frac{2**(dx^2+dy^2)}{dx^2**dy^2}$

Jacobi method: As the solution marches forward jacobi will use the value of previous time steps and compute the present value till convergence.

$T (i, j) = \frac{1}{K} * (T o l d (i + 1, j) + T o l d (i - 1, j) + T o l d (i, j - 1) + T o l d (i, j + 1))$ $T(i,j)=frac{1}{K}**(Told(i+1,j)+Told(i-1,j)+Told(i,j-1)+Told(i,j+1))$

The Matlab output graph for steady state implicit approach under jacobi method is shown below.

Gauss-Seidel method: For calculating the value of a temperature at a node 2 it will use the updated value at node 1 and previous value of node 3.

$T (i, j) = \frac{1}{K} * (T o l d (i + 1, j) + T (i - 1, j) + T (i, j - 1) + T o l d (i, j + 1))$ $T(i,j)=frac{1}{K}**(Told(i+1,j)+T(i-1,j)+T(i,j-1)+Told(i,j+1))$

The Matlab output graph for steady state implicit approach under Gauss-Seidel method is shown below.

Successive over relaxation method:

It is a variant of "Gauss-Seidel method". It uses a relaxation factor $ω$ $omega$ . When $ω > 1$ $omegagt1$ it is over relaxation. When $ω < 1$ $omegalt1$ it is under relaxation.

$T (i, j) = ω * \frac{1}{K} * (T o l d (i + 1, j) + T (i - 1, j) + T (i, j - 1) + T o l d (i, j + 1)) + (1 - ω) * T o l d (i, j)$ $T(i,j)=omega**frac{1}{K}**(Told(i+1,j)+T(i-1,j)+T(i,j-1)+Told(i,j+1))+(1-omega)**Told(i,j)$

sor

The results from the graph can be tabulated as under

S No	Methods	Number of iterations	Computation time
1	Transient state explicit	161	28.365
2	Transient state (jacobi method)	208	19.94
3	Transient state (Gauss-Seidel method)	112	17.94
4	Transient state (Successive over relaxation method)	107	17.92
5	Steady state (jacobi method)	213	50.21
6	Steady state (Gauss-Seidel method)	114	26.39
7	Steady state (Successive over relaxation method)	73	19.27

Conclusion

It can be concluded that the successive over relaxation method gives faster rate of convergence than the other two methods as the number of iteration is least for successive over relaxation method and maximum for jacobi method.

It is also an important point to consider that steady state does not contain time derivative and $α$ $alpha$ terms in their equation.

Explicit method are time sensitive, wrong time steps may result in unstabe graph.

Implicit method are less time sensitive.This tells us the reason that computation time for implicit is more as compared to explicit.

The correct value of number of grid points, $α$ $alpha$ , $ω$ $omega$ are priorities for an accurate and fast solution which will further determine whether the method is economically feasible or not.