A CWENO large time-step scheme for Hamilton--Jacobi equations

TL;DR

The paper introduces a high‑order semi‑Lagrangian method for time‑dependent Hamilton–Jacobi–Bellman equations by coupling a semi‑Lagrangian discretization with Central Weighted ENO (CWENO) reconstructions to control spurious oscillations. It proves convergence in the state/time‑independent Hamiltonian setting and demonstrates through extensive 1D/2D tests that CWENO/ CWENOZ offer comparable or improved accuracy with substantial CPU‑time savings relative to traditional WENO approaches. The approach avoids dimensional splitting in 2D, reuses global reconstruction polynomials, and supports high‑order accuracy via RK time integration and CWENOZ smoothing. These results indicate a practically efficient and robust framework for solving HJB equations in control problems and front propagation scenarios with obstacles and complex geometries.

Abstract

We propose a high order numerical scheme for time-dependent first order Hamilton--Jacobi--Bellman equations. In particular we propose to combine a semi-Lagrangian scheme with a Central Weighted Non-Oscillatory reconstruction. We prove a convergence result in the case of state- and time-independent Hamiltonians. Numerical simulations are presented in space dimensions one and two, also for more general state- and time-dependent Hamiltonians, demonstrating superior performance in terms of CPU time gain compared with a semi-Lagrangian scheme coupled with Weighted Non-Oscillatory reconstructions.

Figures (5)

• Figure 1: Stencils of the two-dimensional $\mathsf{CWENO}$ and $\mathsf{CWENOZ}$ reconstructions. The red hatched region represents the cell $\Omega_{i,j}$ in which we compute the reconstruction. The vertices of $\mathcal{S}_{\text{opt}}$ are enclosed in the grey shaded region, while the stencils for the low degree polynomials are enclosed in the coloured squares: blue, green, orange and purple, respectively for the north-east, north-west, south-west and south-east polynomial.
• Figure 4: Left: the exact solution of Test 2 in red and the numerical one in blue computed with a third order $\mathsf{CWENOZ}$ reconstruction and $N=81$. Right: reconstructions count in the last time step.
• Figure 5: Left: the exact solution of Test 3 in red and the numerical one in blue computed with a third order $\mathsf{CWENOZ}$ scheme and $N=63$. Right: reconstructions count in the last time step.
• Figure 7: Test 4. Comparison of overshoots and undershoots of the numerical solutions computed with the traditional $\mathsf{WENO}$ scheme and with the central $\mathsf{WENO}$ ones on a $161\times161$ grid. In the left panels $\mathsf{WENO}$ and $\mathsf{CWENO}$ are compared, while in the right panels similar plots are shown for $\mathsf{WENO}$ and $\mathsf{CWENOZ}$. The exact solution is always represented with a thick black line. First row: scatter plots of all data points with $r\in[0,0.2]$ corresponding to the spike of the solution. Second row: scatter plots of all data points with $r\in[0.4,0.6]$ corresponding to regular region of the solution. Third row: scatter plots of all data points with $r\in[0.9,1.1]$ corresponding to the singular region of the initial data.
• Figure 8: Reachable sets \ref{['def:Rset']} computed at each time step for Test 5 on a grid $101\times 81$ and $T=3$, using $\mathsf{WENO}$ (left) and $\mathsf{CWENO}$ (right) reconstructions. In both panel the black line represents a reference solution computed on a grid $1001\times 801$, the red circle represents the target and the red rectangles represent the obstacles.

Theorems & Definitions (2)

• Definition 3.1
• Theorem 4.1