diff --git a/web/content/docs/benchmarks/heatconduction/heatconduction-neumann.pandoc b/web/content/docs/benchmarks/heatconduction/heatconduction-neumann.pandoc index 8b7dd7aa2f862dd5bc0dad513275800ea7d02f9e..570d50b144cd291b214ef0aad962cd13352db822 100644 --- a/web/content/docs/benchmarks/heatconduction/heatconduction-neumann.pandoc +++ b/web/content/docs/benchmarks/heatconduction/heatconduction-neumann.pandoc @@ -25,32 +25,28 @@ $$ \eqalign{ T(x, t=0) = T_0,\cr T(x) = g_D(x) &\quad \text{on }\Gamma_D,\cr -k{\partial T(x) \over \partial n} = g_N(x) &\quad \text{on }\Gamma_N, +\lambda {\partial T(x) \over \partial n} = g_N(x) &\quad \text{on }\Gamma_N, } $$ where $T$ could be temperature, the subscripts $D$ and $N$ denote the Dirichlet- and Neumann-type boundary conditions, $n$ is the normal vector pointing outside of $\Omega$, and $\Gamma = \Gamma_D \cup \Gamma_N$ and $\Gamma_D \cap \Gamma_N = \emptyset$. ## Problem specification and analytical solution -We solve the Parabolic equation on a line domain $[60\times 1]$ with $k = 3.2$ and $\rho C_\textrm{p} = 2.5e10^6$ w.r.t. the specific boundary conditions: +We solve the parabolic equation on a line domain $[60\times 1]$ with $\lambda = 3.2$ and $\rho C_\textrm{p} = 2.5 \times 10^6$ w.r.t. the specific boundary conditions: $$ \eqalign{ -k {\partial T(x) \over \partial n} = 2 &\quad \text{on } (x=0) \subset \Gamma_N.\cr -k {\partial T(x) \over \partial n} = 0 &\quad \text{on } (x=60) \subset \Gamma_N. +\lambda {\partial T(x) \over \partial n} = 2 &\quad \text{on } (x=0) \subset \Gamma_N.\cr +\lambda {\partial T(x) \over \partial n} = 0 &\quad \text{on } (x=60) \subset \Gamma_N. } $$ The solution of this problem is - -TODO: is not rendered correct - $$ \begin{equation} -T(x,t) = \frac{2q}{\lambda}\left((\frac{\alpha t}{\pi})^{\frac{1}{2}} e^{-x^2/4\alpha t} - \frac{x}{2}\textrm{erfc}( \frac{x}{2\sqrt{\alpha t}})\right), +T(x,t) = \frac{2q}{\lambda}\left(\left(\frac{\alpha t}{\pi}\right)^{\frac{1}{2}} e^{-x^2/4\alpha t} - \frac{x}{2}\textrm{erfc}\left( \frac{x}{2\sqrt{\alpha t}}\right)\right), \end{equation} $$ - where $T_\textrm{b}$ is the boundary temperature, $\textrm{erfc}$ is the complementary error function and $\alpha = \lambda/(C_p \rho)$ is the thermal diffusivity. ## Input files