An efficient numerical method is described for solving partial differential equations in problems where traditional eulerian and lagrangian techniques fail. The approach makes use of the geometrical concept of ’natural neighbours’, the properties of which make it suitable for solving problems involving large deformation and solid-fluid interactions on a deforming mesh, without the need for regridding. The approach can also be applied to high-order partial differential equations (such as the Navier-Stokes equation), even in cases where the evolving mesh is highly irregular.