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Introduction to Linear Algebra with Mathematica

Glossary

Preface

Finite Difference for Heat Equations

Since one dimensional heat equation
\[ \frac{\partial u}{\partial t} = \alpha\frac{\partial^2 u}{\partial x^2} \]
contains second derivative, we need we look for a finite difference approximation to the second derivative, \( f ′′(x_0 ). Using Taylor's series, we have
\[ \begin{split} f \left( x_0 + \Delta x \right) &= f \left( x_0 \right) + f' \left( x_0 \right) \Delta x + \frac{1}{2}\, f'' \left( x_0 \right) \left( \Delta x \right)^2 + R_2^{+} , \\ f \left( x_0 - \Delta x \right) &= f \left( x_0 \right) - f' \left( x_0 \right) \Delta x + \frac{1}{2}\, f'' \left( x_0 \right) \left( \Delta x \right)^2 + R_2^{-} , \end{split} \]
where \( R_2^{\pm} . \) denote appropriate remainder terms. If we add both equations, divide by (∆x)² and ignore the remainder terms, we obtain the following finite difference approximation,
\[ f'' \left( x_0 \right) = \frac{f \left( x_0 - \Delta x \right) -2 f \left( x_0 \right) + f \left( x_0 + \Delta x \right) }{\left( \Delta x \right)^2} . \]
Because it involves x00 along with the points x0 ± ∆x on either side of it, this approximation is a centered difference approximation. ,/P.

 

We are going to demonstrate application of finite difference method to solve sme simple heat transfer equation, Namely, we consider homogeneous heat equation with no sources,
\begin{equation} \label{EqHeat.1} \frac{\partial u}{\partial t} = \alpha\frac{\partial^2 u}{\partial x^2} , \qquad 0 < x < \ell . \end{equation}
For the moment, the boundary conditions and initial conditions will be ignored.

We first form a grid on the x-axis by setting \( \displaystyle \Delta x = \frac{\ell}{N} > 0, \) for N > 0, and defining the grid points,

\begin{equation} \label{EqHeat.2} x_n = n\,\Delta x, \qquad n=0,1,2,\ldots , N. \end{equation}
We’ll also need to discretize the time variable: Let ∆t > 0 and define
\begin{equation} \label{EqHeat.3} t_k = k\,\Delta t, \qquad k=0,1,2,\ldots . \end{equation}
The finite difference scheme will be concerned with the values of u(x, t) on these grid points, i.e., the values u(xn, tk): For each n ≥ 0, the N + 1 set of values u(xn, tk), 0 ≤ n ≤ N , form a “snapshot” of the temperature function u on the spatial grid points xn.

or notational convenience, we use double scripts, with superscript for time variable:

\[ u_n^{(k)} = u( x_n , t_k ) \]
For time derivative, we employ the forward difference:
\[ \frac{\partial u}{\partial t} \left( x_n, t_k \right) \approx \frac{u( x_n , t_{k+1}) - u(x_n , t_k )}{\Delta t} = \dfrac{u_n^{(k+1)} - u_n^{(k)}}{\Delta t} . \]
The second spatial derivative is approximated by the following centered difference,
\[ \frac{\partial^2 u}{\partial x^2} \left( x_n, t_k \right) \approx \frac{u( x_{n+1} , t_{k}) - 2\, u(x_n , t_k ) + u(x_{n-1}, t_k )}{(\Delta x)^2} = \dfrac{u_{n+1}^{(k)} - 2\,u_n^{(k)} + u_{n-1}^{(k)}}{(\Delta x)^2} . \]
Substitution of these differences into the heat equation, Eq.\eqref{EqHeat.1}, and rearrangement yields the following difference scheme
\begin{equation} \label{EqHeat.4} u_n^{(k+1)} = u_n^{(k)} + \frac{k\,\Delta t}{\left( \Delta x \right)^2} \left[ u_{n+1}^{(k)} -2\,u_n^{(k)} + u_{n-1}^{(k)} \right] . \end{equation}

Crank--Nicholson Method

An important scheme, invented by John Crank and Phyllis Nicholson, is based on numerical approximations for solutions of heat equation at the point (n, t + ½(δ t) that lies between between the rows in the grid. Specifically, the approximation used for time derivative is obtained from the central difference formula,
\[ u_t \left( x, t + \frac{1}{2} \left( \Delta t \right) \right) = \frac{u(x, t + \Delta t) - u(x,t)}{\Delta t} + O \left( (\Delta t)^2 \right) . \]
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