非稳态Hamilton-Jacobi方程的7阶加权紧致非线性格式

Hamilton-Jacobi (HJ) 方程是一类重要的非线性偏微分方程, 在物理学、流体力学、图像处理、微分几何、金融数学、最优化控制理论等方面有着广泛的应用. 由于HJ方程的弱解存在但不唯一, 且解的导数可能出现间断, 导致其数值求解具有一定的难度. 本文提出了非稳态HJ方程的7阶精度加权紧致非线性格式 (WCNS). 该格式结合了Hamilton函数的Lax-Friedrichs型通量分裂方法和一阶空间导数左、右极限值的高阶精度混合节点和半节点型中心差分格式. 基于7点全局模板和4个4点子模板推导了半节点函数值的高阶线性逼近和4个低阶线性逼近, 以及全局模板和子模板的光滑度量指标. 为避免间断附近数值解产生非物理振荡以及提高格式稳定性, 采用WENO型非线性插值方法计算半节点函数值. 时间离散采用3阶TVD型Runge-Kutta方法. 通过理论分析验证了WCNS格式对于光滑解具有最佳的7阶精度. 为方便比较, 经典的7阶WENO格式也被推广用于求解HJ方程. 数值结果表明, 本文提出的WCNS格式能够很好地模拟HJ方程的精确解, 且在光滑区域能够达到7阶精度; 与经典的同阶WENO格式相比, WCNS格式在精度、收敛性和分辨率方面更优, 计算效率略高. 

HIGH ORDER WEIGHT COMPACR NONLINEAR SCHEME FOR TIME-DEPENDENT HAMILTON-JACOBI EQUATIONS

Hamilton-Jacobi (HJ) equations are an important class of nonlinear partial differential equations. They are often used in various applications, such as physics, fluid mechanics, image processing, differential geometry, financial mathematics, optimal control theory, and so on. Because the weak solutions of the HJ equations exist but are not unique, and the spatial derivatives of the solutions may be discontinuous, numerical difficulties arise in numerical solutions of these equations. This paper presents a seventh-order weighted compact nonlinear scheme (WCNS) for the time-dependent HJ equations. This scheme is composed of the monotone Lax-Friedrichs flux splitting method for the Hamilton functions and the high-order hybrid cell-node and cell-edge central differencing for the left and right limits of first-order spatial derivatives in the numerical Hamilton functions. A high-order linear approximation scheme and four low-order linear approximation schemes for the unknowns at half nodes are derived based on a seven-point global stencil and four four-point sub-stencils, respectively. The smoothness indicators of the global stencil and four sub-stencils are also derived. In order to avoid non-physical oscillations of numerical solutions near the discontinuities and improve the numerical stability of the designed scheme, the WENO-type nonlinear interpolation technique is adopted to compute the unknowns at half nodes. The third-order TVD Runge-Kutta method is used for time discretization. The presented WCNS scheme is verified to have the optimal seventh order of accuracy for smooth solutions by theoretical analysis. For the sake of comparison, the classical seventh-order WENO scheme for solving hyperbolic conservation laws is also extended to solve the HJ equations. Numerical results show that the presented WCNS scheme can well simulate the exact solutions and can achieve seventh-order accuracy in smooth regions. Compared with the classical WENO scheme of the same order, the presented WCNS scheme has better accuracy, convergence and resolution, and its computational efficiency is slightly higher.