This paper summarizes a Riemann-solver-free spacetime discontinuous Galerkin method developed for general conservation laws. The method integrates the best features of the spacetime Conservation Element/Solution Eleme...This paper summarizes a Riemann-solver-free spacetime discontinuous Galerkin method developed for general conservation laws. The method integrates the best features of the spacetime Conservation Element/Solution Element (CE/SE) method and the discontinuous Galerkin (DG) method. The core idea is to construct a staggered spacetime mesh through alternate cell-centered CEs and vertex-centered CEs within each time step. Inside each SE, the solution is approximated using high-order spacetime DG basis polynomials. The spacetime flux conservation is enforced inside each CE using the DG concept. The unknowns are stored at both vertices and cell centroids of the spatial mesh. However, the solutions at vertices and cell centroids are updated at different time levels within each time step in an alternate fashion. Thanks to the staggered spacetime formulation, there are no left and right states for the solution at the spacetime interface. Instead, the solution available to evaluate the flux is continuous across the interface. Therefore, no (approximate) Riemann solvers are needed to provide a unique numerical flux. The current method can be used to solve arbitrary conservation laws including the compressible Euler equations, shallow water equations and magnetohydrodynamics (MHD) equations without the need of any form of Riemann solvers. A set of benchmark problems of various conservation laws are presented to demonstrate the accuracy of the method.展开更多
研究舰艇附近水下爆炸问题对船体结构设计、爆炸冲击损害预测及人员安全保障至关重要。为此,提出改进扩散界面法中的六方程可压多相流模型以解决冲击波条件下热力学状态预测偏差,并为相关抗冲击机理研究与数值方法优化提供支撑。通过引...研究舰艇附近水下爆炸问题对船体结构设计、爆炸冲击损害预测及人员安全保障至关重要。为此,提出改进扩散界面法中的六方程可压多相流模型以解决冲击波条件下热力学状态预测偏差,并为相关抗冲击机理研究与数值方法优化提供支撑。通过引入混合能量校正方程及更精确的气体状态方程改进模型,在非结构网格系统构建数值算法程序,采用基于最小二乘重建和Barth-Jespersen限制器的二阶守恒定律的单调上游中心方案(Monotonic Upstream-centered Scheme for Conservation Laws,MUSCL)-Hancock格式、两相流带接触的Harten-Lax-van Leer(Harten-Lax-van Leer Contact,HLLC)黎曼求解器求解齐次双曲型方程,以Newton-Raphson迭代法求解瞬时压力松弛方程。研究结果表明:混合能量方程校正后,模型模拟流体冲击波速度和界面的结果与欧拉方程精确解高度吻合,解决界面附近数值振荡问题;相较于实验数据,改进型模型相对误差1.13%,准确度提升0.33%,且通过拟合冲击Hugoniot曲线获得更精确的刚性气体状态方程(Stiffened Gas Equation of State,SG-EOS)参数,同时可清晰呈现水下爆炸的冲击波传播、气泡胀缩及坍塌水射流现象,但在气泡界面清晰度、射流精细度上存在缺陷,主要受数值格式极端梯度下耗散特性限制。综上,改进型六方程可压多相流模型有效提升了舰艇附近水下爆炸模拟准确性,为深入研究舰艇抗冲击机理提供重要支撑,也为后续相关数值方法的优化奠定了坚实基础。展开更多
文摘This paper summarizes a Riemann-solver-free spacetime discontinuous Galerkin method developed for general conservation laws. The method integrates the best features of the spacetime Conservation Element/Solution Element (CE/SE) method and the discontinuous Galerkin (DG) method. The core idea is to construct a staggered spacetime mesh through alternate cell-centered CEs and vertex-centered CEs within each time step. Inside each SE, the solution is approximated using high-order spacetime DG basis polynomials. The spacetime flux conservation is enforced inside each CE using the DG concept. The unknowns are stored at both vertices and cell centroids of the spatial mesh. However, the solutions at vertices and cell centroids are updated at different time levels within each time step in an alternate fashion. Thanks to the staggered spacetime formulation, there are no left and right states for the solution at the spacetime interface. Instead, the solution available to evaluate the flux is continuous across the interface. Therefore, no (approximate) Riemann solvers are needed to provide a unique numerical flux. The current method can be used to solve arbitrary conservation laws including the compressible Euler equations, shallow water equations and magnetohydrodynamics (MHD) equations without the need of any form of Riemann solvers. A set of benchmark problems of various conservation laws are presented to demonstrate the accuracy of the method.
文摘研究舰艇附近水下爆炸问题对船体结构设计、爆炸冲击损害预测及人员安全保障至关重要。为此,提出改进扩散界面法中的六方程可压多相流模型以解决冲击波条件下热力学状态预测偏差,并为相关抗冲击机理研究与数值方法优化提供支撑。通过引入混合能量校正方程及更精确的气体状态方程改进模型,在非结构网格系统构建数值算法程序,采用基于最小二乘重建和Barth-Jespersen限制器的二阶守恒定律的单调上游中心方案(Monotonic Upstream-centered Scheme for Conservation Laws,MUSCL)-Hancock格式、两相流带接触的Harten-Lax-van Leer(Harten-Lax-van Leer Contact,HLLC)黎曼求解器求解齐次双曲型方程,以Newton-Raphson迭代法求解瞬时压力松弛方程。研究结果表明:混合能量方程校正后,模型模拟流体冲击波速度和界面的结果与欧拉方程精确解高度吻合,解决界面附近数值振荡问题;相较于实验数据,改进型模型相对误差1.13%,准确度提升0.33%,且通过拟合冲击Hugoniot曲线获得更精确的刚性气体状态方程(Stiffened Gas Equation of State,SG-EOS)参数,同时可清晰呈现水下爆炸的冲击波传播、气泡胀缩及坍塌水射流现象,但在气泡界面清晰度、射流精细度上存在缺陷,主要受数值格式极端梯度下耗散特性限制。综上,改进型六方程可压多相流模型有效提升了舰艇附近水下爆炸模拟准确性,为深入研究舰艇抗冲击机理提供重要支撑,也为后续相关数值方法的优化奠定了坚实基础。
