Hybrid nodal integral-finite element method (NI-FEM) for 2D, time-dependent Burgers’ equation in arbitrary geometries

Nodal integral method (NIM) is an efficient coarse mesh method that relies on transverse integration procedure and the analytical solution of the simplified form of the resulting ODEs. This restricts its development to regular rectangular (2D) or cuboid (3D) elements, which results in the limitation that the spatial domains must have boundaries that are captured by regular elements. The hybrid nodal-integral/finite-element method (NI-FEM) has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate planes can be discretized using coarse rectangular (2D) or cuboid (3D) elements and the rest of the domain can be discretized using triangular (2D) or tetrahedral (3D) elements. The development of the hybrid scheme requires the development of appropriate conditions on the interfaces between regions solved using the NIM and regions solved using the FEM. This approach has been developed by Wang and Rizwan-uddin [1] to solve the steady-state convection diffusion equation in irregular domains. NI-FEM is highly efficient as bulk of the region is solved using the coarse mesh NIM and the remaining region by the FEM. While our ultimate goal is to develop this hybrid scheme for the Navier-Stokes equations, the present work as a first step is aimed at extending this method to solve the time-dependent, 2D, coupled Burgers’ equations. Numerical results are compared with the exact solutions and are found to be in good agreement. The hybrid method is efficient compared to standalone conventional numerical schemes like FEM. The hybrid scheme developed is second order accurate in both space and time.