91Ó°ÊÓ

Show that the Lax-Friedrichs scheme $$ \frac{v_{\ell, m}^{n+1}-\frac{1}{4}\left(v_{\ell+1, m}^{n}+v_{\ell-1, m}^{n}+v_{\ell, m+1}^{n}+v_{\ell, m-1}^{n}\right)}{k}+a \delta_{0 x} v_{\ell, m}^{n}+b \delta_{0 y} v_{\ell, m}^{n}=0 $$ for the equation \(u_{t}+a u_{x}+b u_{y}=0\), with \(\Delta x=\Delta y=h\), is stable if and only if \(\left(|a|^{2}+|b|^{2}\right) \lambda^{2} \leq 1 / 2\)

Use the Peaceman-Rachford ADI method to solve $$ u_{t}=2 u_{x x}+u_{y y} $$ on the unit square for \(0 \leq t \leq 1\). The initial and boundary data should be taken from the exact solution $$ u=\exp (1.68 t) \sin [1.2(x-y)] \cosh (x+2 y) $$ Use \(\Delta x=\Delta y=\Delta t=1 / 10,1 / 20\), and 1/40. Demonstrate the second-order accuracy.