HIGH ORDER WEIGHT COMPACR NONLINEAR SCHEME FOR TIME-DEPENDENT HAMILTON-JACOBI EQUATIONS
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摘要: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.
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关键词:
- Hamilton-Jacobi方程/
- 7阶WCNS/
- WENO插值/
- 高分辨率/
- 间断捕捉能力
Abstract: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. -
图 1两种格式的计算误差与CPU时间关系曲线图
Figure 1.CPU times against numerical computing errors obtained with two schemes
图 2基于初值条件(33)在
$t = 11$ 时刻所得数值解比较Figure 2.Comparisons of numerical solutions at time
$t = 11$ based on the initial condition (33)
图 3基于初值条件(34)在不同时刻所得数值解比较
Figure 3.Comparisons of numerical solutions at different times based on the initial condition (34)
图 4一维(a)凸和(b)非凸Hamilton问题的数值解
Figure 4.Numerical solutions of 1D (a) convex and (b) nonconvex Hamilton problems
图 5二维凸Hamilton问题的数值解(a)曲面图和(b)截面图
Figure 5.(a) Surface and (b) cross-sectional diagram of the numerical solution of the 2D convex Hamilton problem
图 6二维非凸Hamilton问题的数值解(a)曲面图和(b)截面图
Figure 6.(a) Surface and (b) cross-sectional diagram of the numerical solution of the 2D nonconvex Hamilton problem
图 8二维最优控制问题的数值解(a), (b)曲面图和(c), (d)截面图
Figure 8.(a), (b) Surfaces and (c), (d) cross-sectional diagrams of the numerical solutions of the optimal control problem
8二维最优控制问题的数值解(a), (b)曲面图和(c), (d)截面图 (续)
8.(a), (b)Surfaces and (c), (d) cross-sectional diagrams of the numerical solutions of the optimal control problem (continued)
7二维完全问题的数值解(a), (b) 曲面图和(c), (d)截面图
7.(a), (b) Surfaces and (c), (d) cross-sectional diagrams of numerical solutions of the fully problem
图 9二维传播面问题的
$\varepsilon = 0$ 时数值解Figure 9.Numerical solutions of the propagating surface problem with
$\varepsilon = $ 0
图 10二维传播面问题的
$\varepsilon = 0.1$ 时数值解Figure 10.Numerical solutions of the propagating surface problem with
$\varepsilon = $ 0.1表 1一维情形时两种格式的数值误差、精度阶和CPU运行时间
Table 1.Numerical errors, convergence rates and CPU time obtained with two schemes for the 1D case
Method $N$ ${L_1}$error Order ${L_\infty }$error Order CPU time WCNS 60 2.00 × 10−8 — 3.72 × 10−8 — 0.078125 80 2.67 × 10−9 7.00 4.90 × 10−9 7.04 0.218750 100 5.60 × 10−10 7.00 1.02 × 10−9 7.05 0.453125 120 1.56 × 10−10 7.00 2.81 × 10−10 7.05 0.796875 140 5.33 × 10−11 6.98 9.52 × 10−11 7.03 1.312500 WENO 60 7.73 × 10−8 — 5.94 × 10−7 — 0.078125 80 1.33 × 10−8 6.13 1.31 × 10−7 5.24 0.203125 100 3.36 × 10−9 6.15 4.09 × 10−8 5.23 0.453125 120 1.14 × 10−9 5.94 1.57 × 10−8 5.24 0.828125 140 4.56 × 10−10 5.93 6.97× 10−9 5.27 1.093750 
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表 2二维情形时两种格式的数值误差、精度阶和CPU运行时间
Table 2.Numerical errors, convergence rates and CPU time obtained with two schemes for the 2D case
Method $M \times N$ ${L_1}$error Order ${L_\infty }$error Order CPU time WCNS 60 × 60 2.95× 10−8 — 4.80× 10−8 — 1.578125 80 × 80 3.95× 10−9 6.99 6.42× 10−9 6.99 4.843750 100 × 100 8.28 × 10−10 7.00 1.34× 10−9 7.01 12.65625 120 × 120 2.31 × 10−10 7.00 3.74 × 10−10 7.02 27.85937 140 × 140 7.86 × 10−11 7.00 1.27 × 10−10 7.01 54.92187 WENO 60 × 60 2.96× 10−8 — 1.88 × 10−7 — 1.140625 80 × 80 4.48× 10−9 6.57 3.84× 10−8 5.52 3.765625 100 × 100 1.07× 10−9 6.41 1.11× 10−8 5.55 9.984375 120 × 120 3.33 × 10−10 6.41 3.94× 10−9 5.70 22.09375 140 × 140 1.24 × 10−10 6.41 1.80× 10−9 5.09 43.14062 下载:
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