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
为了求出对称正则长波(symmetric regularized long wave,SRLW)方程的数值解,构造了一种新的高效紧致有限差分格式.采用经典的Crank-Nicolson(C-N)格式和外推技术对时间方向一阶导数进行离散化,使用四阶Padé方法和逆紧致算子分别...为了求出对称正则长波(symmetric regularized long wave,SRLW)方程的数值解,构造了一种新的高效紧致有限差分格式.采用经典的Crank-Nicolson(C-N)格式和外推技术对时间方向一阶导数进行离散化,使用四阶Padé方法和逆紧致算子分别对空间方向一阶和二阶导数进行离散化,使得构造的格式具有线性、非耦合和紧致的特点,极大地提高了求解效率.此外,还对新格式进行了守恒律、先验估计、稳定性、收敛性分析,证明了其在时间上达到二阶、在空间上达到四阶收敛精度.最后,通过一个数值算例验证了理论的正确性和格式的高效性.展开更多
In this paper,a composite numerical scheme is proposed to solve the threedimensional Darcy-Forchheimer miscible displacement problem with positive semi-definite assumptions.A mixed finite element is used for the fow e...In this paper,a composite numerical scheme is proposed to solve the threedimensional Darcy-Forchheimer miscible displacement problem with positive semi-definite assumptions.A mixed finite element is used for the fow equation.The velocity and pressure are computed simultaneously.The accuracy of velocity is improved one order.The concentration equation is solved by using mixed finite element,multi-step difference and upwind approximation.A multi-step method is used to approximate time derivative for improving the accuracy.The upwind approximation and an expanded mixed finite element are adopted to solve the convection and diffusion,respectively.The composite method could compute the diffusion flux and its gradient.It possibly becomes an eficient tool for solving convection-dominated diffusion problems.Firstly,the conservation of mass holds.Secondly,the multi-step method has high accuracy.Thirdly,the upwind approximation could avoid numerical dispersion.Using numerical analysis of a priori estimates and special techniques of differential equations,we give an error estimates for a positive definite problem.Numerical experiments illustrate its computational efficiency and feasibility of application.展开更多
在这篇文章里,我们考虑源自于控制消振问题的双曲方程,由于它的边值条件是需求解的控制函数,导致它不同于通常的偏微分方程的定解问题。在没有关于初始位移和初始速度的任何假设下,给出了控制函数的解析形式,截取级数的前2(N + 1)项作...在这篇文章里,我们考虑源自于控制消振问题的双曲方程,由于它的边值条件是需求解的控制函数,导致它不同于通常的偏微分方程的定解问题。在没有关于初始位移和初始速度的任何假设下,给出了控制函数的解析形式,截取级数的前2(N + 1)项作为数值逼近解,获得了数值逼近的收敛速度,数值实验验证了理论结果。In this paper, we are concerned with the development of numerical method for a class of hyperbolic equations, which are originated from the problem of vibration control and elimination in practice. Since its boundary condition is the control function which is needed to be solved, it is different from the ordinary initial value and boundary value problems of partial differential equations. Without any assumptions on initial displacement and initial velocity, the analytical form of the control function is given, and a numerical approximation solution is obtained by intercepting the first 2(N + 1) terms of the series, and a convergence rate of numerical approximation is analyzed. Numerical experiments confirm the theoretical results.展开更多
Physics informed neural networks(PINNs)are a deep learning approach designed to solve partial differential equations(PDEs).Accurately learning the initial conditions is crucial when employing PINNs to solve PDEs.Howev...Physics informed neural networks(PINNs)are a deep learning approach designed to solve partial differential equations(PDEs).Accurately learning the initial conditions is crucial when employing PINNs to solve PDEs.However,simply adjusting weights and imposing hard constraints may not always lead to better learning of the initial conditions;sometimes it even makes it difficult for the neural networks to converge.To enhance the accuracy of PINNs in learning the initial conditions,this paper proposes a novel strategy named causally enhanced initial conditions(CEICs).This strategy works by embedding a new loss in the loss function:the loss is constructed by the derivative of the initial condition and the derivative of the neural network at the initial condition.Furthermore,to respect the causality in learning the derivative,a novel causality coefficient is introduced for the training when selecting multiple derivatives.Additionally,because CEICs can provide more accurate pseudo-labels in the first subdomain,they are compatible with the temporal-marching strategy.Experimental results demonstrate that CEICs outperform hard constraints and improve the overall accuracy of pre-training PINNs.For the 1D-Korteweg–de Vries,reaction and convection equations,the CEIC method proposed in this paper reduces the relative error by at least 60%compared to the previous methods.展开更多
本文构造了基于Lax-Wendroff时间离散的有限差分HWENO (Hermite加权本质非振荡)格式,用于求解非线性退化抛物方程。与传统的Runge-Kutta时间离散方法相比,Lax-Wendroff方法在提高计算效率的同时,能够在解的光滑区域实现时空一致的高阶...本文构造了基于Lax-Wendroff时间离散的有限差分HWENO (Hermite加权本质非振荡)格式,用于求解非线性退化抛物方程。与传统的Runge-Kutta时间离散方法相比,Lax-Wendroff方法在提高计算效率的同时,能够在解的光滑区域实现时空一致的高阶精度。通过数值算例,验证了该方法的有效性。This paper constructs a finite difference HWENO (Hermite Weighted Essentially Non-Oscillatory) scheme based on Lax-Wendroff time discretization for solving nonlinear degenerate parabolic equations. Compared to traditional Runge-Kutta time discretization methods, the Lax-Wendroff method improves computational efficiency while achieving high-order accuracy in both space and time in smooth regions of the solution. The effectiveness of the method is validated through numerical examples.展开更多
在计算流体力学等领域中,双曲守恒方程及其对流占优问题可以通过加权基本无振荡(WENO)方法进行高精度的数值求解。本文旨在通过建立一个修正高阶有限差分WENO格式来求解双曲–椭圆混合型方程。通过引入特殊的通量分裂方法将通量分解为...在计算流体力学等领域中,双曲守恒方程及其对流占优问题可以通过加权基本无振荡(WENO)方法进行高精度的数值求解。本文旨在通过建立一个修正高阶有限差分WENO格式来求解双曲–椭圆混合型方程。通过引入特殊的通量分裂方法将通量分解为两部分,在这两个分量中分别应用双曲WENO算子,并在守恒律方程数值通量中加入高阶修正项,获得了一种可求解混合型守恒律方程的高精度有限差分WENO格式。该离散格式主要用于求解viscosity-capillarity容许性条件下的双曲–椭圆型范德华方程。数值测试验证了该算法的高精度和有效性,结果表明,该格式不仅能在强间断区域保持无振荡,在解的光滑部分保持高阶数值精度,还可以有效描述复杂波结构。Hyperbolic conservation law equations and convection-dominant situations in computational fluid dynamics and other areas can be solved numerically with great precision using the weighted essentially non-oscillatory (WENO) techniques. In this paper, we attempt to address the hyperbolic-elliptic mixed equations by developing a corrected high-order finite difference WENO scheme. By introducing a special flux splitting method, the flux is decomposed into two parts. We then apply the hyperbolic WENO operator to these two components, and add the higher order correction term to the numerical flux of the conservation laws, and finally a high-precision finite-difference WENO scheme is obtained. The discretization scheme is mainly used to solve the hyperbolic-elliptic van der Waals equations under the viscosity-capillarity admissibility criterion. It can be shown by numerical examples that the scheme not only can preserve the necessary no oscillation in the discontinuous region, but also retain high order numerical accuracy in the smooth part of the solution, and can effectively describe the complex wave structure.展开更多
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.
文摘为了求出对称正则长波(symmetric regularized long wave,SRLW)方程的数值解,构造了一种新的高效紧致有限差分格式.采用经典的Crank-Nicolson(C-N)格式和外推技术对时间方向一阶导数进行离散化,使用四阶Padé方法和逆紧致算子分别对空间方向一阶和二阶导数进行离散化,使得构造的格式具有线性、非耦合和紧致的特点,极大地提高了求解效率.此外,还对新格式进行了守恒律、先验估计、稳定性、收敛性分析,证明了其在时间上达到二阶、在空间上达到四阶收敛精度.最后,通过一个数值算例验证了理论的正确性和格式的高效性.
基金supported by the Natural Science Foundation of Shandong Province(ZR2021MA019)the National Natural Science Foundation of China(11871312)。
文摘In this paper,a composite numerical scheme is proposed to solve the threedimensional Darcy-Forchheimer miscible displacement problem with positive semi-definite assumptions.A mixed finite element is used for the fow equation.The velocity and pressure are computed simultaneously.The accuracy of velocity is improved one order.The concentration equation is solved by using mixed finite element,multi-step difference and upwind approximation.A multi-step method is used to approximate time derivative for improving the accuracy.The upwind approximation and an expanded mixed finite element are adopted to solve the convection and diffusion,respectively.The composite method could compute the diffusion flux and its gradient.It possibly becomes an eficient tool for solving convection-dominated diffusion problems.Firstly,the conservation of mass holds.Secondly,the multi-step method has high accuracy.Thirdly,the upwind approximation could avoid numerical dispersion.Using numerical analysis of a priori estimates and special techniques of differential equations,we give an error estimates for a positive definite problem.Numerical experiments illustrate its computational efficiency and feasibility of application.
文摘在这篇文章里,我们考虑源自于控制消振问题的双曲方程,由于它的边值条件是需求解的控制函数,导致它不同于通常的偏微分方程的定解问题。在没有关于初始位移和初始速度的任何假设下,给出了控制函数的解析形式,截取级数的前2(N + 1)项作为数值逼近解,获得了数值逼近的收敛速度,数值实验验证了理论结果。In this paper, we are concerned with the development of numerical method for a class of hyperbolic equations, which are originated from the problem of vibration control and elimination in practice. Since its boundary condition is the control function which is needed to be solved, it is different from the ordinary initial value and boundary value problems of partial differential equations. Without any assumptions on initial displacement and initial velocity, the analytical form of the control function is given, and a numerical approximation solution is obtained by intercepting the first 2(N + 1) terms of the series, and a convergence rate of numerical approximation is analyzed. Numerical experiments confirm the theoretical results.
基金supported by the National Natural Science Foundation of China(Grant Nos.1217211 and 12372244).
文摘Physics informed neural networks(PINNs)are a deep learning approach designed to solve partial differential equations(PDEs).Accurately learning the initial conditions is crucial when employing PINNs to solve PDEs.However,simply adjusting weights and imposing hard constraints may not always lead to better learning of the initial conditions;sometimes it even makes it difficult for the neural networks to converge.To enhance the accuracy of PINNs in learning the initial conditions,this paper proposes a novel strategy named causally enhanced initial conditions(CEICs).This strategy works by embedding a new loss in the loss function:the loss is constructed by the derivative of the initial condition and the derivative of the neural network at the initial condition.Furthermore,to respect the causality in learning the derivative,a novel causality coefficient is introduced for the training when selecting multiple derivatives.Additionally,because CEICs can provide more accurate pseudo-labels in the first subdomain,they are compatible with the temporal-marching strategy.Experimental results demonstrate that CEICs outperform hard constraints and improve the overall accuracy of pre-training PINNs.For the 1D-Korteweg–de Vries,reaction and convection equations,the CEIC method proposed in this paper reduces the relative error by at least 60%compared to the previous methods.
文摘本文构造了基于Lax-Wendroff时间离散的有限差分HWENO (Hermite加权本质非振荡)格式,用于求解非线性退化抛物方程。与传统的Runge-Kutta时间离散方法相比,Lax-Wendroff方法在提高计算效率的同时,能够在解的光滑区域实现时空一致的高阶精度。通过数值算例,验证了该方法的有效性。This paper constructs a finite difference HWENO (Hermite Weighted Essentially Non-Oscillatory) scheme based on Lax-Wendroff time discretization for solving nonlinear degenerate parabolic equations. Compared to traditional Runge-Kutta time discretization methods, the Lax-Wendroff method improves computational efficiency while achieving high-order accuracy in both space and time in smooth regions of the solution. The effectiveness of the method is validated through numerical examples.
文摘在计算流体力学等领域中,双曲守恒方程及其对流占优问题可以通过加权基本无振荡(WENO)方法进行高精度的数值求解。本文旨在通过建立一个修正高阶有限差分WENO格式来求解双曲–椭圆混合型方程。通过引入特殊的通量分裂方法将通量分解为两部分,在这两个分量中分别应用双曲WENO算子,并在守恒律方程数值通量中加入高阶修正项,获得了一种可求解混合型守恒律方程的高精度有限差分WENO格式。该离散格式主要用于求解viscosity-capillarity容许性条件下的双曲–椭圆型范德华方程。数值测试验证了该算法的高精度和有效性,结果表明,该格式不仅能在强间断区域保持无振荡,在解的光滑部分保持高阶数值精度,还可以有效描述复杂波结构。Hyperbolic conservation law equations and convection-dominant situations in computational fluid dynamics and other areas can be solved numerically with great precision using the weighted essentially non-oscillatory (WENO) techniques. In this paper, we attempt to address the hyperbolic-elliptic mixed equations by developing a corrected high-order finite difference WENO scheme. By introducing a special flux splitting method, the flux is decomposed into two parts. We then apply the hyperbolic WENO operator to these two components, and add the higher order correction term to the numerical flux of the conservation laws, and finally a high-precision finite-difference WENO scheme is obtained. The discretization scheme is mainly used to solve the hyperbolic-elliptic van der Waals equations under the viscosity-capillarity admissibility criterion. It can be shown by numerical examples that the scheme not only can preserve the necessary no oscillation in the discontinuous region, but also retain high order numerical accuracy in the smooth part of the solution, and can effectively describe the complex wave structure.
