Improving the convergence rate to steady state of parabolic ADI methods (Q1085958)

For the equation \(u_ t=\nu (u_{xx}+u_{yy})\), \(0\leq x\leq a\), \(0\leq y\leq a\), \(0\leq t\) standard ADI algorithms are investigated for their convergence to steady state. The analysis is carried out in the \(L_ 2\)- norm of the residuals. A correction speeding up convergence is proposed, and it is shown that there is only weak dependence on the mesh-length ratio \(\lambda =\Delta t/h^ 2\) (where \(h=\Delta x=\Delta y)\) for \(\lambda\) larger than the standard scheme's optimal value. Results of numerical experiments for various situations are displayed: Dirichlet problems, mixed Dirichlet-Neumann problems, problems with stretched grids and/or with variable coefficients.