Second Order Schemes for the Heat Eq (Part 3)

Second Order Schemes for the Heat Eq (Part 3)

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Higher Order Finite Difference Schemes for the Heat Eq (Part 2)

Higher Order Finite Difference Schemes for the Heat Eq (Part 1)

Finite Difference Approximations for Three Points

Second Order Schemes for the Heat Eq (Part 3)

In this post, we compute the three points finite difference approximation formulas as Taylor Series. In three points formulae, the first two consecutive points must be known to find the third one. Finite difference relations are first developed using dependent variable w depending only on single independent variable x. The formulas obtained are further used for the approximation of derivatives either with respect to time or space. By working initially with w=w(x), the notation is being simplified without affecting the generality of the result. We now compute the second-order finite difference approximation using three points.

Second Order Forward Difference

Considering the series of Taylor expansion w_{i+1} and w_{i+2}

(1)   \begin{equation*} w_{i+1}=w_i+\Delta x \dfrac{\partial w}{\partial x}\big|{x_i}+\dfrac{\Delta x^2}{2} \dfrac{\partial^2 w}{\partial x^2}\big|{x_i} +\dfrac{\Delta x^3}{3!}\dfrac{\partial^3 w}{\partial x^3}\big|{x_i}+\cdots \end{equation*}

(2)   \begin{equation*} w{i+2}=w_i+2\Delta x \dfrac{\partial w}{\partial x}\big|{x_i}+2\Delta x^2 \dfrac{\partial^2 w}{\partial x^2}\big|{x_i} +\dfrac{4\Delta x^3}{3} \dfrac{\partial^3 w}{\partial x^3}\big|{x_i}+\cdots \end{equation*}

Multiplying equation (1) by 4 and then subtracting equation (2), we get

(3)   \begin{equation*} 4w{i+1}-w_{i+2}=3w_i+2\Delta x \dfrac{\partial w}{\partial x}\big|{x_i}- \dfrac{2\Delta x^3}{3} \dfrac{\partial^3 w}{\partial x^3}\big|{x_i}+\cdots\end{equation*}


Solve for \dfrac{\partial w}{\partial x}\big|{x_i}

(4)   \begin{equation*} \dfrac{\partial w}{\partial x}\big|{x_i}=\dfrac{-w_{i+2}+4w_{i+1}-3w_i}{2\Delta x}+\dfrac{\Delta x^2}{3} \dfrac{\partial^3 w}{\partial x^3}\big|{x_i}+\cdots \end{equation*}


The derivatives of higher order can be replaced by using mean value theorem

    \[\dfrac{\Delta x^2}{3} \dfrac{\partial^3 w}{\partial x^3}\big|{x_i}+\cdots =\dfrac{\Delta x^2}{3} \dfrac{\partial^3 w}{\partial x^3}\big|\xi\]

.

Higher Order Finite Difference Schemes for the Heat Eq (Part 2)

Thus

(5)   \begin{equation*} \dfrac{\partial w}{\partial x}\big|{x_i}=\dfrac{-w{i+2}+4w_{i+1}-3w_i}{2\Delta x}+\dfrac{\Delta x^2}{3} \dfrac{\partial^3 w}{\partial x^3}\big|\xi \end{equation*}


or

(6)   \begin{equation*} \dfrac{\partial w}{\partial x}\big|{x_i}+\dfrac{w_{i+2}-4w_{i+1}+3w_i}{2\Delta x}= \dfrac{\Delta x^2}{3} \dfrac{\partial^3 w}{\partial x^3}\big|\xi \end{equation*}

The expression at the right side of equation (6) is said to be truncation error (T.E) of finite difference approximation. Whereas, the “big \mathcal{O}” notation expresses the dependence of truncation error on mesh spacing.

    \[\dfrac{\Delta x^2}{3} \dfrac{\partial^3 w}{\partial x^3}\big|\xi = \mathcal{O}(\Delta x^2)\]


By using the big \mathcal{O} notation, equation (6) is written as,

(7)   \begin{equation*} \dfrac{\partial w}{\partial x}\big|{x_i}=\dfrac{-w{i+2}+4w_{i+1}-3w_i}{2\Delta x}+\mathcal{O}(\Delta x^2)\end{equation*}


The approximation of forward difference has a truncation error of order two i.e. \mathcal{O}(\Delta x^2).

Second Order Backward Difference

The Taylor series of w_{i-1} and w_{i-2} are

(8)   \begin{equation*} w_{i-1}=w_i-\Delta x \dfrac{\partial w}{\partial x}\big|{x_i}+\dfrac{\Delta x^2}{2} \dfrac{\partial^2 w}{\partial x^2}\big|{x_i} -\dfrac{\Delta x^3}{3!} \dfrac{\partial^3 w}{\partial x^3}\big|{x_i}+\cdots \end{equation*}

(9)   \begin{equation*} w{i-2}=w_i-2 \Delta x \dfrac{\partial w}{\partial x}\big|{x_i}+2 \Delta x^2 \dfrac{\partial^2 w}{\partial x^2}\big|{x_i} -\dfrac{4\Delta x^3}{3}\dfrac{\partial^3 w}{\partial x^3}\big|_{x_i}+\cdots\end{equation*}

Higher Order Finite Difference Schemes for the Heat Eq (Part 1)

Multiplying equation (8) by 4 and then subtracting from equation (9), we get

(10)   \begin{equation*} w_{i-2}-4w_{i-1}=-3w_i+2\Delta x \dfrac{\partial w}{\partial x}\big|{x_i}- \dfrac{2\Delta x^3}{3} \dfrac{\partial^3 w}{\partial x^3}\big|{x_i}+\cdots\end{equation*}


Solve for \dfrac{\partial w}{\partial x}\big|{x_i}

(11)   \begin{equation*} \dfrac{\partial w}{\partial x}\big|{x_i}=\dfrac{3w_i-4w_{i-1}+w_{i-2}}{2 \Delta x}+\dfrac{\Delta x^2}{3} \dfrac{\partial^3 w}{\partial x^3}\big|{x_i}+\cdots \end{equation*}

By using the notation of big \mathcal{O} , equation (11) is written as,

(12)   \begin{equation*} \dfrac{\partial w}{\partial x}\big|{x_i}=\dfrac{3w_i-4w_{i-1}+w_{i-2}}{2 \Delta x}+\mathcal{O}(\Delta x^2)\end{equation*}

The approximation of backward difference has the same truncation error as that of forward
difference approximation.

Second Order Central Difference

The Taylor expansion of w_{i-1} and w_{i+1} is

(13)   \begin{equation*} w_{i-1}=w_i-\Delta x \dfrac{\partial w}{\partial x}\big|{x_i}+\dfrac{\Delta x^2}{2} \dfrac{\partial^2 w}{\partial x^2}\big|{x_i}+\cdots\end{equation*}

(14)   \begin{equation*} w{i+1}=w_i+\Delta x \dfrac{\partial w}{\partial x}\big|{x_i}+\dfrac{\Delta x^2}{2} \dfrac{\partial^2 w}{\partial x^2}\big|{x_i} +\cdots \end{equation*}

By adding equation (13) with equation (14), we get

(15)   \begin{equation*} w{i+1}+w_{i-1}=2w_i+\Delta x^2 \dfrac{\partial^2 w}{\partial x^2}\big|{x_i} + \dfrac{\Delta x^4}{12} \dfrac{\partial^4 w}{\partial x^4}\big|{x_i}+\cdots\end{equation*}


(16)   \begin{equation*} \dfrac{\partial^2 w}{\partial x^2}\big|{x_i} =\dfrac{w{i+1}-2w_i+w_{i-1}}{\Delta x^2} -\dfrac{\Delta x^2}{12} \dfrac{\partial^4 w}{\partial x^4}\big|{x_i}+\cdots \end{equation*}

By using \mathcal{O} notation,

(17)   \begin{equation*} \dfrac{\partial^2 w}{\partial x^2}\big|{x_i} =\dfrac{w_{i+1}-2w_i+w_{i-1}}{\Delta x^2}+\mathcal{O}(\Delta x^2)\end{equation*}


It is known as approximation of second order central difference. From the above given formulae we have seen that by increasing the number of points, we get higher order scheme. These finite difference approximations are used in Euler explicit and implicit methods, and Adams Bashforth for numerical solution of heat equation. In the following section we first derive these methods.

Euler Method

Euler method is the first order numerical method to solve a differential equation of the form

(18)   \begin{equation*}\dfrac{\partial w }{\partial t}=f(t,w)\end{equation*}


Expanding the Taylor series of w^{m+1} about w^m , we have

(19)   \begin{equation*} w^{m+1} = w^m+ \Delta t \frac{\partial w}{\partial t}\big|^m + \frac{\Delta t^2}{2} \frac{\partial w^2}{\partial t^2}\big|^m+ \frac{\Delta t^3}{6} \frac{\partial w^3}{\partial t^3}\big|^m+\cdots \end{equation*}


(20)   \begin{equation*} \dfrac{w^{m+1} -w^m}{\Delta t}-\frac{\partial w}{\partial t}\big|^m = \frac{\Delta t}{2}\frac{\partial w^2}{\partial t^2}\big|^m+ \frac{\Delta t^2}{6} \frac{\partial w^3}{\partial t^3}\big|^m+\cdots\end{equation*}


By truncating the terms on the right hand side of equation (20) we have

(21)   \begin{equation*} \dfrac{w^{m+1} -w^m}{\Delta t}-\frac{\partial w}{\partial t}\big|^m= \mathcal{O}(\Delta t) \end{equation*}


By using \dfrac{\partial w }{\partial t}=f(t,w)

(22)   \begin{equation*} \dfrac{w^{m+1} -w^m}{\Delta t}-f(t^m,w^m)= \mathcal{O}(\Delta t) \end{equation*}

\label{46.2}

(23)   \begin{equation*}w^{m+1} = w^m+ \Delta t f(t^m,w^m)+ \mathcal{O}(\Delta t)\end{equation*}


which is first order Euler method.

Adams-Bashforth

There are many techniques for the approximation of the differential equation that are being used for the current state g^m with the previous data g^{m-1} , g^{m-2} , \cdots in an explicit state of formulation. We referred these techniques as
multistep techniques and the Adams-Bashforth techniques are the multistep techniques.
Expanding the Taylor series of w^{m+1} about w^m , we have

(24)   \begin{equation*} w^{m+1} = w^m+ \Delta t \frac{\partial w}{\partial t}\big|^m + \frac{\Delta t^2}{2} \frac{\partial w^2}{\partial t^2}\big|^m+ \frac{\Delta t^3}{6} \frac{\partial w^3}{\partial t^3}\big|^m+\cdots \end{equation*}


(25)   \begin{equation*} =w^m+ \Delta t g^m + \frac{\Delta t^2}{2} \frac{\partial g}{\partial t}\big|^m+ \frac{\Delta t^3}{6} \frac{\partial g^2}{\partial t^2}\big|^m+\cdots \end{equation*}


where g^m=\frac{\partial w}{\partial t}\big|^m. For the first order Adams-Bashforth method we
truncate the series at the second term. For the second order Adams-Bashforth method we truncate the
series at the third term. By using

(26)   \begin{equation*} \frac{\partial g}{\partial t}\big|^m=\frac{g^m-g^{m-1}}{\Delta t} \end{equation*}


and we can write,

(27)   \begin{equation*} w^{m+1} = w^m+ \Delta t \frac{3g^m-g^{m-1}}{2} +\mathcal{O}(\Delta t^3) \end{equation*}


or

(28)   \begin{equation*} \frac{w^{m+1}-w^m}{\Delta t} = \frac{3g^m-g^{m-1}}{2} +\mathcal{O}(\Delta t^2)\end{equation*}


which is a second order Adams Bashforth method.

Numerical Implementation

In this post we discuss the implementation of various schemes for numerical solution of heat conduction equation (??). These schemes are based on the finite difference approximation for two and three points in time and space. The results obtained from Matlab codes are analysed and compared for all the schemes. Furthermore, the stability of scheme shown provides which ones is the better scheme for solving heat equation.
The approximation of finite difference developed in previous chapter, are used to approximate the time and space derivatives in heat equation (??). The space and time derivatives are interchanged with finite difference approximations.
In order to do so, we require both spatial locations and time specifications for the values of “w” in finite difference relations. We therefore, need superscript notation “m” to denote time step of discrete solution and subscript notation i to denote spatial location. We start the analysis with the two points finite difference schemes for time followed by three points in space.

Schemes using Two Points in Time and Three Points in Space

These schemes have been discussed thoroughly by Gerald W. Recktenwald, for numerical solution of heat equation (??),

(29)   \begin{equation*} w_i^{m+1}=rw_{i+1}^m+(1-2r)w_i^m+rw_{i-1}^m\end{equation*}

(30)   \begin{equation*}\dfrac{\alpha}{\Delta x^2}w_{i-1}^m + \big(\dfrac{1}{\Delta t}+\dfrac{2\alpha}{\Delta x^2}\big)w_i^m-\dfrac{\alpha}{\Delta x^2} w_{i+1}^m=\dfrac{1}{\Delta t}w_i^{m-1}\end{equation*}


and

(31)   \begin{equation*} \dfrac{\alpha}{2\Delta x^2}w_{i-1}^m + \big(\dfrac{1}{\Delta t}+\dfrac{\alpha}{\Delta x^2}\big)w_i^m-\dfrac{\alpha}{2\Delta x^2}w_{i+1}^m=\dfrac{\alpha}{2\Delta x^2}w_{i-1}^{m-1} + \big(\dfrac{1}{\Delta t}-\dfrac{\alpha}{\Delta x^2}\big)w_i^{m-1}+\dfrac{\alpha}{2\Delta x^2}w_{i+1}^{m-1}\end{equation*}


These schemes (29), (30) and (31) are two points in time and three points in space.

FTCS Second Order Schemes for the Heat Eq

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