Non stationary Black-Scholes equation with logarithmic prices.
\begin{array}{rcl} \partial_tu-\frac{\sigma^2}{2} \,u_{xx}+(\beta-\frac{\sigma^2}{2})u_x+r u& =&0 \quad\mbox{on } (-L,L),\\ u(-L,t)=0,\;u(L,t) & =&e^L-K\,e^{-r\,t},\\ u(x,0)&=&(e^x-K)_{+} \end{array}
Different snapshots of the solution. We use P2 finite elements and a row of triangles sitting on the 1d interval to use the freefem tools
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download example : BlackScholes-1d.edp