2.3 Numerical Solution of the Heat Conduction Equation

Even though analytical solutions of Eq. (2.3) can be readily found in simplified cases, in practice numerical methods are more often applicable. In this study, an algorithm of the Numerical Algorithms Group Fortran Library (NAG, 1990) is used to integrate Eq. (2.3) between two depths $z_1$ and $z_2$. The integration is forward in time from $t_a$ to $t_b$ subject to the boundary conditions

$$\parbox{3cm}{ }\parbox{7.3cm}{$T(z=z_1 \dots z_2,t_a) = T_{a}(z)$}$$ (2.15)

$$\parbox{3cm}{ }\parbox{5.2cm}{$T(z_m,t=t_a \dots t_b) = T_{m}(t)$}\parbox{2cm}{$m=1,2$.}$$ (2.16)

The algorithm approximates the parabolic partial differential equation by an ordinary differential equation in time, obtained by replacing the spatial derivatives by finite differences on a regular mesh. The discretisation interval in the time direction is chosen by the routine to maintain a local accuracy specified by the user.