Efficient cell-centered nodal integral method for multi-dimensional Burgers equations

An efficient coarse-mesh nodal integral method (NIM), based on cell-centered variables and referred to as cell-centered NIM (CCNIM), is developed and applied for solving multi-dimensional, time-dependent, Burgers equations. Unlike traditional NIM, which utilizes surface-averaged variables as discrete unknowns, this innovative approach formulates the final expression of the numerical scheme using discrete unknowns represented by cell-centered or node-averaged variables. By relying on these cell centroids, the proposed CCNIM approach presents several advantages compared to traditional NIM. These include a simplified implementation process in terms of local coordinate systems, enhanced flexibility regarding the higher order of accuracy in time, straightforward formulation for higher-degree temporal derivatives, and offering a viable option for coupling with other physics. The multidimensional time-dependent Burgers' problems with known analytical solutions are solved in order to validate the developed scheme. Furthermore, a detailed comparison between the proposed CCNIM approach and other traditional NIM schemes is conducted to showcase its effectiveness. The simplicity and robustness of the approach provide a strong foundation for its seamless extension to more complex fluid flow problems.