This study investigates the dimensionless quasi-geostrophic potential vorticity(QG-PV)equation with external sources.Employing the Gardner-Morikawa transformation and weakly nonlinear perturbation expansion,we derive ...This study investigates the dimensionless quasi-geostrophic potential vorticity(QG-PV)equation with external sources.Employing the Gardner-Morikawa transformation and weakly nonlinear perturbation expansion,we derive the nonlinear Boussinesq equation with external sources.We demonstrate the existence of explicit zero-order and first-order Wronskian solutions for the model equation whenα_4=0.Furthermore,using a modified Jacobi elliptic function method,we obtain soliton-like solutions for bothα_4=0 andα_4≠0.Analysis of these solutions reveals that the generalizedβ-plane approximation and shear flow are significant factors in inducing nonlinear Rossby waves,and that external sources play a crucial role in influencing Rossby wave behavior.展开更多
Fang等基于Boussinesq型方程和GPU并行加速技术开发的海岸和近海波浪模拟系统(Coastal and Offshore Wave Simulator,COWS)具有稳定性强、计算效率高的特点。本研究对COWS模型进行扩展,通过在动量方程中添加移动压力源项来模拟船舶实际...Fang等基于Boussinesq型方程和GPU并行加速技术开发的海岸和近海波浪模拟系统(Coastal and Offshore Wave Simulator,COWS)具有稳定性强、计算效率高的特点。本研究对COWS模型进行扩展,通过在动量方程中添加移动压力源项来模拟船舶实际航行过程中船行波的生成过程。压力项的位置通过三次样条插值法确定,采用二阶中心差分格式近似。利用扩展后的COWS模型,计算了一维矩形横截面航道船行波、浅水区域船首孤立波、开阔水域以及弯曲航道船行波四种不同情形,并将计算结果与解析解、实验数据以及与其他模型计算结果进行了对比,讨论了计算结果的异同和模型的精度。展开更多
In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,t...In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,the original Boussinesq system is transformed into an equivalent one.Then we discretize it using the second-order backward di erentiation formula(BDF2)and Crank-Nicolson(CN)to obtain two second-order time-advanced schemes.In both numerical schemes,a pressure-correction method is employed to decouple the velocity and pressure.These two schemes possess the desired property that they can be fully decoupled with satisfying unconditional stability.We rigorously prove both the unconditional stability and unique solvability of the discrete schemes.Furthermore,we provide detailed implementations of the decoupling procedures.Finally,various 2D numerical simulations are performed to verify the accuracy and energy stability of the proposed schemes.展开更多
We study the two-dimensional(2D)Cauchy problem of nonhomogeneous Boussinesq system for magnetohydrodynamics convection without heat diffusion in the whole plane.Based on delicate weighted estimates,we derive the globa...We study the two-dimensional(2D)Cauchy problem of nonhomogeneous Boussinesq system for magnetohydrodynamics convection without heat diffusion in the whole plane.Based on delicate weighted estimates,we derive the global existence and uniqueness of strong solutions.In particular,the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support.展开更多
Deep learning combining the physics information is employed to solve the Boussinesq equation with second-order time derivative.High prediction accuracies are achieved by adding a new initial loss term in the physics-i...Deep learning combining the physics information is employed to solve the Boussinesq equation with second-order time derivative.High prediction accuracies are achieved by adding a new initial loss term in the physics-informed neural networks along with the adaptive activation function and loss-balanced coefficients.The numerical simulations are carried out with different initial and boundary conditions,in which the relative L2-norm errors are all around 10^(−4).The prediction accuracies have been improved by two orders of magnitude compared to the former results in certain simulations.The dynamic behavior of solitons and their interaction are studied in the colliding and chasing processes for the Boussinesq equation.More training time is needed for the solver of the Boussinesq equation when the width of the two-soliton solutions becomes narrower with other parameters fixed.展开更多
Both the thickness effect and surface effect should be important in nano-indentation behavior of coatings due to the finite thickness and small indentation size.As a basic solution,the two-dimensional Boussinesq probl...Both the thickness effect and surface effect should be important in nano-indentation behavior of coatings due to the finite thickness and small indentation size.As a basic solution,the two-dimensional Boussinesq problem of a finite elastic layer bonded to a rigid substrate is studied in this paper,employing the surface-energy-density-based elastic theory.The Airy stress function and Fourier integral transform methods are adopted to solve the problem.A nalytical solutions of both the stress and displacement fields are well achieved for a finite elastic layer under a concentrated force and a uniform pressure.Unlike the classical solutions,it is discovered that both the thickness effect and surface effect will show significant influences on the Boussinesq elastic behaviors.The surface effect would harden the finite elastic layer and induce a more uniformly distributing displacements and stresses.Only when the thickness is sufficiently large,the Boussinesq solution of an elastic half space may represent that of a finite elastic layer case.A generalized hardness is further defined to include the coupling effects of thickness and surface for the Boussinesq problem of a finite elastic layer.Such a study would assist in the design and property evaluation of coatings and micro-devices with layer-substrate structures.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.12362027)the Scientific Research Ability of Youth Teachers of Inner Mongolia Agricultural University(Grant No.BR230110)+3 种基金Inner Mongolia National Science Fund for Excellent Young Scholars(Grant No.2025YQ033)Foundation for Basic Science Research Initiation at Inner Mongolia Agricultural University(Grant No.JC2021001)The Natural Science Foundation of Inner Mongolia Autonomous Region(2025MS01020)Supported by the Basic and Applied Basic Research Science and Technology Program Projects of Hohhot(2025-rule-basic-60)。
文摘This study investigates the dimensionless quasi-geostrophic potential vorticity(QG-PV)equation with external sources.Employing the Gardner-Morikawa transformation and weakly nonlinear perturbation expansion,we derive the nonlinear Boussinesq equation with external sources.We demonstrate the existence of explicit zero-order and first-order Wronskian solutions for the model equation whenα_4=0.Furthermore,using a modified Jacobi elliptic function method,we obtain soliton-like solutions for bothα_4=0 andα_4≠0.Analysis of these solutions reveals that the generalizedβ-plane approximation and shear flow are significant factors in inducing nonlinear Rossby waves,and that external sources play a crucial role in influencing Rossby wave behavior.
文摘Fang等基于Boussinesq型方程和GPU并行加速技术开发的海岸和近海波浪模拟系统(Coastal and Offshore Wave Simulator,COWS)具有稳定性强、计算效率高的特点。本研究对COWS模型进行扩展,通过在动量方程中添加移动压力源项来模拟船舶实际航行过程中船行波的生成过程。压力项的位置通过三次样条插值法确定,采用二阶中心差分格式近似。利用扩展后的COWS模型,计算了一维矩形横截面航道船行波、浅水区域船首孤立波、开阔水域以及弯曲航道船行波四种不同情形,并将计算结果与解析解、实验数据以及与其他模型计算结果进行了对比,讨论了计算结果的异同和模型的精度。
基金Supported by Research Project Supported by Shanxi Scholarship Council of China(2021-029)International Cooperation Base and Platform Project of Shanxi Province(202104041101019)+2 种基金Basic Research Plan of Shanxi Province(202203021211129)Shanxi Province Natural Science Research(202203021212249)Special/Youth Foundation of Taiyuan University of Technology(2022QN101)。
文摘In this paper,we construct two fully decoupled,second-order semi-discrete numerical schemes for the Boussinesq equations based on the scalar auxiliary variable(SAV)approach.By introducing a scalar auxiliary variable,the original Boussinesq system is transformed into an equivalent one.Then we discretize it using the second-order backward di erentiation formula(BDF2)and Crank-Nicolson(CN)to obtain two second-order time-advanced schemes.In both numerical schemes,a pressure-correction method is employed to decouple the velocity and pressure.These two schemes possess the desired property that they can be fully decoupled with satisfying unconditional stability.We rigorously prove both the unconditional stability and unique solvability of the discrete schemes.Furthermore,we provide detailed implementations of the decoupling procedures.Finally,various 2D numerical simulations are performed to verify the accuracy and energy stability of the proposed schemes.
文摘We study the two-dimensional(2D)Cauchy problem of nonhomogeneous Boussinesq system for magnetohydrodynamics convection without heat diffusion in the whole plane.Based on delicate weighted estimates,we derive the global existence and uniqueness of strong solutions.In particular,the initial data can be arbitrarily large and the initial density may contain vacuum states and even have compact support.
基金supported by the National Natural Science Foundation of China under Grant No.12475204.
文摘Deep learning combining the physics information is employed to solve the Boussinesq equation with second-order time derivative.High prediction accuracies are achieved by adding a new initial loss term in the physics-informed neural networks along with the adaptive activation function and loss-balanced coefficients.The numerical simulations are carried out with different initial and boundary conditions,in which the relative L2-norm errors are all around 10^(−4).The prediction accuracies have been improved by two orders of magnitude compared to the former results in certain simulations.The dynamic behavior of solitons and their interaction are studied in the colliding and chasing processes for the Boussinesq equation.More training time is needed for the solver of the Boussinesq equation when the width of the two-soliton solutions becomes narrower with other parameters fixed.
基金supported by the National Natural Science Foundation of China(Grant Nos.12032004,12293000,12293002,and 12272043).
文摘Both the thickness effect and surface effect should be important in nano-indentation behavior of coatings due to the finite thickness and small indentation size.As a basic solution,the two-dimensional Boussinesq problem of a finite elastic layer bonded to a rigid substrate is studied in this paper,employing the surface-energy-density-based elastic theory.The Airy stress function and Fourier integral transform methods are adopted to solve the problem.A nalytical solutions of both the stress and displacement fields are well achieved for a finite elastic layer under a concentrated force and a uniform pressure.Unlike the classical solutions,it is discovered that both the thickness effect and surface effect will show significant influences on the Boussinesq elastic behaviors.The surface effect would harden the finite elastic layer and induce a more uniformly distributing displacements and stresses.Only when the thickness is sufficiently large,the Boussinesq solution of an elastic half space may represent that of a finite elastic layer case.A generalized hardness is further defined to include the coupling effects of thickness and surface for the Boussinesq problem of a finite elastic layer.Such a study would assist in the design and property evaluation of coatings and micro-devices with layer-substrate structures.
