The treatment of the Neumann boundary conditions for a new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes

Recently we have developed a new 3-D numerical approach for the time dependent wave and heat equations as well as for the time independent Poisson equation on irregular domains with the Dirichlet boundary conditions. Its extension to the Neumann boundary conditions that was a big issue due to the presence of normal derivatives along irregular boundaries is considered in this paper. Trivial Cartesian meshes and simple 27-point uniform and nonuniform stencil equations are used with the new approach for 3-D irregular domains. The Neumann boundary conditions are introduced as the known right-hand side into the stencil and do not change the width of the stencil. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. Very small distances (0.1h−10⁻⁹h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. At similar 27-point stencils, the accuracy of the new approach is much higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the fourth order) tetrahedral finite elements with much wider stencils. The wave and heat equations can be uniformly treated with the new approach. The order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The new approach can be directly applied to other partial differential equations.