Asymptotic analysis for the eikonal equation with the dynamical boundary conditions

Abstract

We study the dynamical boundary value problem for Hamilton‐Jacobi equations of the eikonal type with a small parameter. We establish two results concerning the asymptotic behavior of solutions of the Hamilton‐Jacobi equations: one concerns with the convergence of solutions as the parameter goes to zero and the other with the large‐time asymptotics of solutions of the limit equation.

The large diffusion limit for the heat equation with a dynamical boundary condition

“…In particular, convergence of this type means that the influence of the initial function ϕ is lost in the limit, and we shall describe this phenomenon in more detail. A result in a similar spirit was established in [1] for the eikonal equation with the same dynamical boundary condition as in (1.1). More precisely, the following problem was considered in [1]:…”

“…This convergence does not look unexpected, see [1], but we are not aware of any previous result which would support this natural conjecture. In particular, convergence of this type means that the influence of the initial function ϕ is lost in the limit, and we shall describe this phenomenon in more detail.…”

“…We consider the problem 1) where N ≥ 2, R N + := R N −1 × R + , ∆ is the N -dimensional Laplacian (in x), ∂ t := ∂/∂t, ∂ ν := −∂/∂x N , ε ∈ (0, 1) and ϕ and ϕ b are measurable functions in R N + and R N −1 , respectively.…”

A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations

“…Number of steps [4,7] Theoretical convergence-order [9,18] Number of function evaluations per iteration [4,7] Solutions of system of linear equations per iteration [11,20] Number of Jacobian evaluations per iteration [2,2] Number of Jacobian LU-factorizations per iteration [1,1] Number of matrix-vector multiplications per iteration [7,13] Number of vector-vector multiplications per iteration [11,20] Steps ( Table 7: 2-D nonlinear HJ equation (5.5): absolute error in infinity norm of residue F(φ φ φ), t f = 0.8/π 2 , initial guess φ φ φ = 0, n t = 10, n x = 20, n y = 20.…”

“…The further extensions of these methods for unstructured grid can be found in [1]. Al-Aidarous et al [4,5] presented results concerning convergence result for the ergodic problem for Hamilton-Jacobi equations with Neumann-type boundary conditions and asymptotic analysis for the eikonal equation with the dynamical boundary conditions. The HJ equations contain partial derivatives in time and space.…”