Solving PDEs on Surfaces of Pipe Geometries Using New Coordinate Transformations and High-order Compact Finite Differences

We introduce suitable coordinate systems for pipes and their variants that allow us to transform partial differential equations (PDEs) on the pipe surfaces or in the solid pipes into computational domains with fixed limits/ranges. Such a notion is reminiscent of the polar and cylindrical coordinates for their advantageous geometries. The new curvilinear coordinates are non-orthogonal in two directions, so the Laplace--Beltrami operators involve mixed derivatives. To deal with the variable coefficients arising from coordinate transformations and diverse surface geometries, we develop efficient fourth-order compact finite difference methods adaptable to various scenarios. We then rigorously prove the convergence of the proposed method for some model problem, and apply the solver to several other types of PDEs. We further demonstrate the efficiency and accuracy of our approach with ample numerical results. Here, we only consider the compact finite differences for the transformed PDEs for simplicity, but one can employ the spectral-collocation methods to efficiently handle such variable coefficient problems.