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Numerical Hopf-Lax formulae for Hamilton-Jacobi equations on unstructured geometries

Simone Cacace, Roberto Ferretti, Giulia Tatafiore

TL;DR

The paper addresses solving Hamilton-Jacobi equations on unstructured geometries by introducing a node-based Hopf-Lax discretization that omits interpolation, yielding a monotone Semi-Lagrangian scheme. A quadratic refinement is proposed to boost accuracy, and the authors extend the approach to both time-dependent and stationary problems, with Dirichlet boundary conditions handled via exit-time formulations and fast policy-iteration solvers. Theoretical guarantees are provided within the Barles–Souganidis framework, including error estimates under an inverse CFL condition, and the method is validated through extensive numerical experiments showing convergence, improved accuracy, and substantial GPU-driven speedups on large, complex meshes. Collectively, the work delivers an efficient, scalable toolkit for HJ equations on unstructured grids, enabling application to complex domains with rigorous convergence properties and practical performance gains.

Abstract

We consider a scheme of Semi-Lagrangian (SL) type for the numerical solution of Hamilton-Jacobi (HJ) equation on unstructured triangular grids. As it is well known, SL schemes are not well suited for unstructured grids, due to the cost of the point location phase; this drawback is augmented by the need for repeated minimization. In this work, we propose a scheme that works only on the basis of node values and connectivity of the grid. In a first version, we obtain a monotone scheme; then, applying a quadratic refinement to the numerical solution, we improve accuracy at the price of some extra computational cost. The scheme can be applied to both time-dependent and stationary HJ equations; in the latter case, we also study the construction of a fast policy iteration solver. We perform a theoretical analysis of the two versions, and validate them with an extensive set of examples, both in the time-dependent and in the stationary case.

Numerical Hopf-Lax formulae for Hamilton-Jacobi equations on unstructured geometries

TL;DR

The paper addresses solving Hamilton-Jacobi equations on unstructured geometries by introducing a node-based Hopf-Lax discretization that omits interpolation, yielding a monotone Semi-Lagrangian scheme. A quadratic refinement is proposed to boost accuracy, and the authors extend the approach to both time-dependent and stationary problems, with Dirichlet boundary conditions handled via exit-time formulations and fast policy-iteration solvers. Theoretical guarantees are provided within the Barles–Souganidis framework, including error estimates under an inverse CFL condition, and the method is validated through extensive numerical experiments showing convergence, improved accuracy, and substantial GPU-driven speedups on large, complex meshes. Collectively, the work delivers an efficient, scalable toolkit for HJ equations on unstructured grids, enabling application to complex domains with rigorous convergence properties and practical performance gains.

Abstract

We consider a scheme of Semi-Lagrangian (SL) type for the numerical solution of Hamilton-Jacobi (HJ) equation on unstructured triangular grids. As it is well known, SL schemes are not well suited for unstructured grids, due to the cost of the point location phase; this drawback is augmented by the need for repeated minimization. In this work, we propose a scheme that works only on the basis of node values and connectivity of the grid. In a first version, we obtain a monotone scheme; then, applying a quadratic refinement to the numerical solution, we improve accuracy at the price of some extra computational cost. The scheme can be applied to both time-dependent and stationary HJ equations; in the latter case, we also study the construction of a fast policy iteration solver. We perform a theoretical analysis of the two versions, and validate them with an extensive set of examples, both in the time-dependent and in the stationary case.

Paper Structure

This paper contains 19 sections, 3 theorems, 84 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Construction of the scheme, time-dependent case
  3. Restricting the Hopf--Lax formula to nodes
  4. Error estimates
  5. Boundary conditions
  6. Construction of the scheme, stationary case
  7. Boundary conditions
  8. Value iteration
  9. Exact policy iteration
  10. Modified policy iteration
  11. Unstructured implementation
  12. Quadratic refinement of the minimum search
  13. Numerical examples
  14. Evolutive case
  15. Quadratic refinement
  16. ...and 4 more sections

Key Result

Theorem 1

Let the basic assumptions hold, and assume, in addition, that $\Delta x=o(\Delta t)$. Then, the numerical solution $V^n$ defined by eq:hl_discr converges to the solution $u(x,t_n)$ of eq:hj locally uniformly on $\mathbb{R}^d\times [0,T]$ as $\Delta t,\Delta x\to 0$ for any $T>0$.

Figures (7)

  • Figure 1: Node achieving the minimum in \ref{['eq:hl_discr']} (left) and related stencil for quadratic refinement (right).
  • Figure 2: Test 1, numerical solution at initial time $t=0$ and final time $T=2$, for $\Delta x= 0.025$ and $\Delta t= 0.5 \sqrt{\Delta x}\approx 0.08$.
  • Figure 3: Test 2, numerical solution at initial time $t=0$ and final time $T=2$, for $\Delta x= 0.025$ and $\Delta t= 0.5 \sqrt{\Delta x}\approx 0.08$.
  • Figure 4: Test $1$ (left) and test $2$ (right) relative errors evolution, with respect to CPU times. Obtained applying scheme \ref{['eq:hl_discr']} with and without quadratic refinement, respectively for $\Delta x= 0.1, 0.05, 0.025, 0.0125$.
  • Figure 5: Sections of the numerical solutions of test $1$ at final time $T=2$. The numerical solutions are indicated as $u$QR and $u$ and are obtained applying scheme \ref{['eq:hl_discr']} with and without quadratic refinement, for $\Delta x= 0.0125$, $\Delta t= 0.5\sqrt{\Delta x}\approx 0.05$.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3: BMZ09
  • Remark 2
  • Remark 3
  • Remark 4