边界停止正立方体格式用于Hamilton--Jacobi--Bellman方程的误差界
Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton--Jacobi--Bellman Equations
- Soochow University（苏州大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对退化抛物型HJB方程，提出边界停止正立方体格式，推导误差界，在网格条件满足时获得最大范数收敛阶，并适用于秩亏扩散。
AI中文摘要：
我们推导了有界域上带Dirichlet数据的退化抛物型Hamilton--Jacobi--Bellman方程的边界停止正立方体格式的误差界。该格式在首次边界接触时平衡对映分支，并使用依赖于控制的有效时间。其停止屏障余项为\\(O(\tau\Delta t^{\gamma/2})\\)，插值分支的总质量至多为\\(2\tau/\Delta t\\)，其中\\(\tau\\)为有效时间。除以有效时间后，这些估计给出一个当\\(\tau\to0\\)时一致的相容性界。在常见的严格边界屏障和所述网格条件下，内部相容性、系数抖动、切换和局部比较给出\\[ \\|(u-u_h)^+\\|_\infty\le C(\Delta t^{1/4}+h\Delta t^{-1/2}), \qquad \\|(u_h-u)^+\\|_\infty\le C(\Delta t^{1/10}+h^{1/2}\Delta t^{-1/4}). \\] 因此，\\(\Delta t\asymp h^{10/7}\\)给出\\(O(h^{1/7})\\)的最大范数界。该分析适用于任何固定的中心对称正二次立方体格式，并允许秩亏扩散。数值例子检验了向冷壁的平行热损失以及依赖于控制的秩亏扩散。
英文摘要：
We derive error bounds for a boundary-stopped positive cubature scheme for degenerate parabolic Hamilton--Jacobi--Bellman equations with Dirichlet data on bounded domains. The scheme balances antipodal branches at their first boundary contacts and uses a control-dependent effective time. Its stopped-barrier remainder is \(O(τΔt^{γ/2})\), and the total mass of interpolated branches is at most \(2τ/Δt\), where \(τ\) is the effective time. After division by the effective time, these estimates give a consistency bound that is uniform as \(τ\to0\). Under a common strict boundary barrier and the stated mesh conditions, interior consistency, coefficient shaking, switching, and localized comparison yield \[ \|(u-u_h)^+\|_\infty\le C(Δt^{1/4}+hΔt^{-1/2}), \qquad \|(u_h-u)^+\|_\infty\le C(Δt^{1/10}+h^{1/2}Δt^{-1/4}). \] Thus \(Δt\asymp h^{10/7}\) gives an \(O(h^{1/7})\) maximum-norm bound. The analysis applies to any fixed centrally symmetric positive degree-two cubature and permits rank-deficient diffusion. Numerical examples examine parallel heat loss to a cold wall and control-dependent, rank-deficient diffusion.
