Energy-Dissipation-Rate-Preserving Numerical Approximations on Staggered Grids for Rayleigh-Bénard Convection and Its Fast Implementation

doi:10.1007/s42967-025-00481-7

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (3): 953-971.doi: 10.1007/s42967-025-00481-7

Energy-Dissipation-Rate-Preserving Numerical Approximations on Staggered Grids for Rayleigh-Bénard Convection and Its Fast Implementation

Shouwen Sun, Liangliang Lei

School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu, 476000, Henan, China

收稿日期:2024-05-24 修回日期:2024-12-08 出版日期:2026-06-20 发布日期:2026-05-29
通讯作者: Shouwen Sun, Email: ssw@sqnu.edu.cn E-mail:ssw@sqnu.edu.cn
作者简介:Liangliang Lei, Email: leiliangliang@sqnu.edu.cn
基金资助:
This work is supported by the National Natural Science Foundation of China (No.12101387), the International Scientific and Technological Cooperation Projects of Henan Province, China, and Henan International Joint Laboratory of Optical Information Transmission and Application, China.

Energy-Dissipation-Rate-Preserving Numerical Approximations on Staggered Grids for Rayleigh-Bénard Convection and Its Fast Implementation

Shouwen Sun, Liangliang Lei

School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu, 476000, Henan, China

Received:2024-05-24 Revised:2024-12-08 Online:2026-06-20 Published:2026-05-29
Contact: Shouwen Sun, Email: ssw@sqnu.edu.cn E-mail:ssw@sqnu.edu.cn
Supported by:
This work was supported by the 2024 School-Level Project of Guangxi Financial Vocational College (No. 607001), the National Natural Science Foundation of China (No. 12361078), the Sichuan Science and Technology Program (No. 2024NSFSC0721), the Postdoctoral Fellowship Program of CPSF (No. GZB20230092), and the China Postdoctoral Science Foundation (No. 2023M740383).

摘要/Abstract

摘要： We develop a set of numerical approximations to discretize the non-isothermal incompressible hydrodynamic model for Rayleigh-Bénard convection (RBC), which ensures the negative energy dissipation rate with respect to adiabatic boundary conditions. Using the Crank-Nicolson (CN) method, the second-order backward difference method combined with the pressure-correction method, we propose two kinds of decoupled, linear, both second-order in time energy-dissipation-rate-preserving semi-discrete projection numerical algorithms. Meanwhile, the second-order fully discrete numerical algorithms are obtained by the use of a finite difference method on staggered grids in space. These numerical approximations are proved to preserve the property of the energy dissipation rate at the fully discrete level. Moreover, the numerical approximations are also unconditionally energy stable. The fast Fourier algorithm is applied to numerical implementation to enhance experimental efficiency. Convergence rate tests are conducted to verify the accuracy of the algorithms. Our simulations showcase that both hydrodynamic and thermal effects in resolving RBC within the non-isothermal hydrodynamic model of incompressible viscous fluid flow. In general, our structure-preserving projection schemes and implementation methods accurately and efficiently simulate RBC in nature.

关键词: Non-isothermal flow of incompressible viscous fluid, Energy-dissipation-rate-preserving approximations, Pressure-correction method, Unconditional stability, Rayleigh-Bénard convection (RBC)

Abstract: We develop a set of numerical approximations to discretize the non-isothermal incompressible hydrodynamic model for Rayleigh-Bénard convection (RBC), which ensures the negative energy dissipation rate with respect to adiabatic boundary conditions. Using the Crank-Nicolson (CN) method, the second-order backward difference method combined with the pressure-correction method, we propose two kinds of decoupled, linear, both second-order in time energy-dissipation-rate-preserving semi-discrete projection numerical algorithms. Meanwhile, the second-order fully discrete numerical algorithms are obtained by the use of a finite difference method on staggered grids in space. These numerical approximations are proved to preserve the property of the energy dissipation rate at the fully discrete level. Moreover, the numerical approximations are also unconditionally energy stable. The fast Fourier algorithm is applied to numerical implementation to enhance experimental efficiency. Convergence rate tests are conducted to verify the accuracy of the algorithms. Our simulations showcase that both hydrodynamic and thermal effects in resolving RBC within the non-isothermal hydrodynamic model of incompressible viscous fluid flow. In general, our structure-preserving projection schemes and implementation methods accurately and efficiently simulate RBC in nature.

Key words: Non-isothermal flow of incompressible viscous fluid, Energy-dissipation-rate-preserving approximations, Pressure-correction method, Unconditional stability, Rayleigh-Bénard convection (RBC)

中图分类号:

Shouwen Sun, Liangliang Lei. Energy-Dissipation-Rate-Preserving Numerical Approximations on Staggered Grids for Rayleigh-Bénard Convection and Its Fast Implementation[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 953-971.

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Energy-Dissipation-Rate-Preserving Numerical Approximations on Staggered Grids for Rayleigh-Bénard Convection and Its Fast Implementation

Energy-Dissipation-Rate-Preserving Numerical Approximations on Staggered Grids for Rayleigh-Bénard Convection and Its Fast Implementation

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