The author offers a way of treating the small disturbance equation for transonic flow around a wing profile with attached shock wave, using spectral numerical methods. The problem is formulated for the velocity potential \(\phi\) as follows: \[ 2\phi_{tx}=(k\phi_ x-\frac{\gamma +1}{2}\phi^ 2_ x)_ x+4\phi_{yy},\quad \phi (-1,y,t)=0,\quad \frac{\partial \phi}{\partial x}(1,y,t)=0, \]
\[ \frac{\partial \phi}{\partial y}(x,\pm 1,t)=F\pm (x),\quad \phi (x,y,0)=\phi_ 0(x,y), \] with the boundary conditions (being \(u=\phi_ x):\) \(\quad y(x)=- 1+\tau F(x),\quad | x| <x_ 0,\quad x_ 0\ll 1,\) \[ \phi (- 1,y,t)=0,\quad u(1,y,t)=0,\quad \phi_ y(x,1,t)=0. \]
\[ \phi_ y(x,- 1,t)=\left\{ \begin{matrix} F'(x),\quad | x| <x_ 0\\ 0,\quad | x| >x_ 0\end{matrix}.\right. \] Then two schemes are presented. One is spectral in x and y and of second order in t, and the other is spectral in x and of second order in y and t. The problem is discussed of approximating discontinuous solutions using spectral methods, and a method is presented to extract a highly accurate solution by fitting the standard Fourier approximation to a sum of a saw-tooth function and a truncated Fourier series. A similar method is also developed for a nonperiodic problem. Several numerical results are given and discussed.