A coupled double splitting ADI scheme for the first biharmonic using collocation

A new method for solving the first biharmonic equation via a double splitting into coupled systems of ordinary differential equations is presented. The first splitting reduces this fourth order partial differential equation into the coupled system uxx + uyy = v and vxx + vyy = f, where u carries both Dirichlet and Neumann boundary data and v carries no boundary data. The pair is then iteratively solved using coupled alternating direction collocation on the resulting systems of ordinary differential equations. This can also be viewed as an alternating direction method of lines for a system of partial differential equations. Although there is no separation of variables underlying the splitting, the method yields a convergent sequence of iterates for a variety of examples under a restricted range of acceleration parameters, and possesses O(h4) accuracy. Desirable features of the algorithm are discussed together with the reduction of bandwidth of the associated collocation matrices under intersticing of u and v variables. Interesting open questions are also discussed.