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<p>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[<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((\Delta x)^2, (\Delta t)^2\)</EquationSource> </InlineEquation>], 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年7月22日

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  • Nadeem Ahmed,
  • Ram Prakash Bharti,
  • Suneet Singh

摘要

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.