本文基于MUSCL-Hancock数据重构方法提出了一种求解相对论流体力学方程的高分辨率熵相容格式(EC-MHM格式)。首先将一个修正的斜率限制器应用到MUSCL-Hancock方法的数据重构中,使之与熵相容格式结合,从而得到高分辨率的熵相容通量;在时间...本文基于MUSCL-Hancock数据重构方法提出了一种求解相对论流体力学方程的高分辨率熵相容格式(EC-MHM格式)。首先将一个修正的斜率限制器应用到MUSCL-Hancock方法的数据重构中,使之与熵相容格式结合,从而得到高分辨率的熵相容通量;在时间上,主要利用双曲守恒律方程的守恒型差分形式来更新下一时间层,从而提高了格式的计算效率。文中还证明了熵相容格式的收敛性。新格式在解的光滑区域具有高精度的特性,然而在间断区域,EC-MHM格式可以有效抑制非物理现象的发生;最后通过一系列数值算例验证了新格式具有无振荡、高分辨率等良好性能。This paper presents a high-resolution entropy-consistent scheme, termed the EC-MHM (Entropy-consistent, MUSCL-type High-resolution Method), for solving relativistic hydrodynamics equations, based on the MUSCL-Hancock data reconstruction methodology. Firstly, a modified slope limiter is applied to the data reconstruction of MUSCL-Hancock method, and combine it with the entropy consistent scheme, so as to obtain the high resolution entropy consistent flux. For the discretization of time derivative, the conservative finite difference scheme of hyperbolic conservation laws is adopted to update the solution at the next time level. The convergence of the entropy consistent scheme is also proved. In regions where the solution is smooth, the EC-MHM scheme exhibits high precision characteristics. Conversely, in discontinuous zones, the EC-MHM can effectively prevent the occurrence of non-physical phenomena. Finally, a series of numerical examples are simulated, and the new scheme is proved to have good properties such as no oscillation and high resolution.展开更多
本文研究了求解三维不可压缩矢量势磁流体力学方程组的一阶投影时间离散算法。该方程组是将原磁流体力学方程组中的磁场B写成旋度形式,即引入B = curlA。通过构造矢量势磁流体力学方程组的数值算法,使得磁场的数值解在全离散层面满足无...本文研究了求解三维不可压缩矢量势磁流体力学方程组的一阶投影时间离散算法。该方程组是将原磁流体力学方程组中的磁场B写成旋度形式,即引入B = curlA。通过构造矢量势磁流体力学方程组的数值算法,使得磁场的数值解在全离散层面满足无散度条件。本文主要通过构造一阶投影格式,使得速度场的数值解也满足无散度条件,且所构造的投影格式对于任意时间步长都是无条件稳定的。在合理的正则性假设下,我们得到了速度和磁矢量势的一阶时间收敛阶。最后,通过数值算例验证了收敛性结果。In this paper, we consider a first-order projection finite element scheme for the three-dimensional incompressible magnetohydrodynamic system. This system of equations is to write the magnetic field B in the original magnetohydrodynamic equations in the curl form, which introduces B = curlA. By constructing the numerical algorithm of the system, the numerical solution of the magnetic field satisfies the divergence-free condition in fully discrete level. In this paper, by constructing the first-order projection scheme so that the numerical solution of the velocity field satisfies the divergence-free condition, and the constructed projection scheme is unconditionally stable for any time step. Under a reasonable regularity assumption, we derive the first-order temporal convergence order of the velocity and magnetic vector potential. Finally, the convergence results are verified by numerical examples.展开更多
文摘本文基于MUSCL-Hancock数据重构方法提出了一种求解相对论流体力学方程的高分辨率熵相容格式(EC-MHM格式)。首先将一个修正的斜率限制器应用到MUSCL-Hancock方法的数据重构中,使之与熵相容格式结合,从而得到高分辨率的熵相容通量;在时间上,主要利用双曲守恒律方程的守恒型差分形式来更新下一时间层,从而提高了格式的计算效率。文中还证明了熵相容格式的收敛性。新格式在解的光滑区域具有高精度的特性,然而在间断区域,EC-MHM格式可以有效抑制非物理现象的发生;最后通过一系列数值算例验证了新格式具有无振荡、高分辨率等良好性能。This paper presents a high-resolution entropy-consistent scheme, termed the EC-MHM (Entropy-consistent, MUSCL-type High-resolution Method), for solving relativistic hydrodynamics equations, based on the MUSCL-Hancock data reconstruction methodology. Firstly, a modified slope limiter is applied to the data reconstruction of MUSCL-Hancock method, and combine it with the entropy consistent scheme, so as to obtain the high resolution entropy consistent flux. For the discretization of time derivative, the conservative finite difference scheme of hyperbolic conservation laws is adopted to update the solution at the next time level. The convergence of the entropy consistent scheme is also proved. In regions where the solution is smooth, the EC-MHM scheme exhibits high precision characteristics. Conversely, in discontinuous zones, the EC-MHM can effectively prevent the occurrence of non-physical phenomena. Finally, a series of numerical examples are simulated, and the new scheme is proved to have good properties such as no oscillation and high resolution.
文摘本文研究了求解三维不可压缩矢量势磁流体力学方程组的一阶投影时间离散算法。该方程组是将原磁流体力学方程组中的磁场B写成旋度形式,即引入B = curlA。通过构造矢量势磁流体力学方程组的数值算法,使得磁场的数值解在全离散层面满足无散度条件。本文主要通过构造一阶投影格式,使得速度场的数值解也满足无散度条件,且所构造的投影格式对于任意时间步长都是无条件稳定的。在合理的正则性假设下,我们得到了速度和磁矢量势的一阶时间收敛阶。最后,通过数值算例验证了收敛性结果。In this paper, we consider a first-order projection finite element scheme for the three-dimensional incompressible magnetohydrodynamic system. This system of equations is to write the magnetic field B in the original magnetohydrodynamic equations in the curl form, which introduces B = curlA. By constructing the numerical algorithm of the system, the numerical solution of the magnetic field satisfies the divergence-free condition in fully discrete level. In this paper, by constructing the first-order projection scheme so that the numerical solution of the velocity field satisfies the divergence-free condition, and the constructed projection scheme is unconditionally stable for any time step. Under a reasonable regularity assumption, we derive the first-order temporal convergence order of the velocity and magnetic vector potential. Finally, the convergence results are verified by numerical examples.
