High-order accurate finite difference discretisations on fully unstructured dual quadrilateral meshes

We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate given function, an auxiliary function with greater smoothness properties is defined which when differentiated provides the derivatives original function. The method generalises traditional finite difference methods to arbitrary topology in any number dimensions order derivative and accuracy. demonstrate accuracy scheme using dual refinement based subdivision surfaces. applied solution range partial differential equations, including both linear nonlinear, second fourth time-dependent first equation.