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

TL;DR

This paper introduces a novel cell-centered nodal integral method (CCNIM) for efficiently solving multi-dimensional nonlinear Burgers equations, demonstrating quadratic convergence and advantages over traditional NIM in implementation and accuracy.

Contribution

The paper develops and validates a new CCNIM approach that extends traditional NIM to nonlinear problems with improved accuracy, simplicity, and flexibility.

Findings

Quadratic convergence in space and time (O[(Δx)^2, (Δt)^2)].

Effective handling of nonlinear convection terms with comparable or better accuracy.

Simplified implementation and enhanced flexibility for higher-order temporal derivatives.

Abstract

An efficient coarse-mesh nodal integral method (NIM), based on cell-centered variables and termed the cell-centered NIM (CCNIM), is developed and applied to solve multi-dimensional, time-dependent, nonlinear Burgers equations, extending the applicability of CCNIM to nonlinear problems. To overcome the existing limitation of CCNIM to linear problems, the convective velocity in the nonlinear convection term is approximated using two different approaches, both demonstrating accuracy comparable to or better than traditional NIM for nonlinear Burgers problems. 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. Using 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 multi-dimensional time-dependent Burgers problems (propagating shock, propagation, and diffusion of an initial sinusoidal wave, shock-like formation) 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 demonstrate its effectiveness. The proposed approach has shown quadratic convergence in both space and time, i.e., O[$(\Delta x)^2, (\Delta t)^2$], for the considered test problems. The simplicity and robustness of the approach provide a strong foundation for its seamless extension to more complex fluid flow problems.