Date of Award


Publication Type

Doctoral Thesis

Degree Name



Mechanical, Automotive, and Materials Engineering


Boundart condition implementation, Convection-diffusion equation, Cut-stencil finite difference, Higher-order accurate approximation, Method of manufactured problem, Prandtl’s stress function & lid-driven cavity flow


Ronald M Barron


Ram Balachandar




A new finite difference formulation, referred to as the Cartesian cut-stencil finite difference method (FDM), for discretization of partial differential equations (PDEs) in any complex physical domain is proposed in this dissertation. The method employs unique localized 1-D quadratic transformation functions to map non-uniform (uncut or cut) physical stencils to a uniform computational stencil. The transformation functions are uniquely determined by the coordinates of the points on the physical stencil. In its basic formulation, 2nd-order central differencing is used to approximate derivatives in the transformed PDEs. The resulting finite difference equations can be solved by classical iterative methods. In the case of a boundary node with a Dirichlet boundary condition, the prescribed value can be used directly in the calculations on the corresponding stencil adjacent to the boundary. However, for Neumann boundary nodes, discretization of the normal derivative in the Neumann condition is accomplished using one-sided approximations, producing an approximate value for the solution variable at the boundary. Then, the cut-stencil method allows stencils adjacent to boundaries to be treated in the same way as interior stencils, thus enabling finite difference calculations on arbitrarily complex domains. This new formulation can be combined with the higher-order compact Padé-Hermitian method to produce higher-order cut-stencil schemes. Three different Cartesian cut-stencil formulations based on local 4th-order approximations are proposed and analyzed. It has been shown that global 4th-order accuracy can be achieved when the same order of accuracy is implemented at Neumann boundaries. Comparison of numerical results for some manufactured problems with the exact solution verifies the capability of the cut-stencil method to deal with PDEs in regular and irregular shaped domains. Cartesian cut-stencil FDM solutions are also obtained for some classical engineering benchmark problems, including Prandtl’s stress function, steady or unsteady heat conduction and flow in a lid-driven cavity. This dissertation demonstrates that the Cartesian cut-stencil finite difference method has many desirable features of a high-end numerical simulation code including simplicity in formulation, meshing and coding, higher-order accuracy, high-fidelity solutions, reliable error estimator, applicable in different science and engineering fields, and can solve complicated nonlinear PDEs in complex geometries.