Adaptive Filtered Schemes for First Order Hamilton-Jacobi Equations

Abstract

We consider a class of "filtered" schemes for first order time dependent Hamilton-Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter ε = ε(∆t, ∆x) > 0 which goes to 0 as (∆t, ∆x) is going to 0 and does not depend on the time t n . The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of ε = ε n (∆t, ∆x) at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold ε n . A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods.

Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

“…Let us start by giving a multidimensional extension of the smoothness indicators introduced in [20] and analyzed in [15] for the 1D case, which are necessary for the definition of the AF scheme that will be introduced in Sect. 3. We consider a uniform grid in space (x j , y i ) = (j∆x, i∆y), j, i ∈ Z, and a function f : R 2 → R. Before proceeding with the construction, let us recall some important features about multivariate interpolation.…”

“…For the reader convenience, in A we recalled the 1D counterpart, namely Prop. A.1, which has been proved in [15]. To simplify the computations, we will consider the case of a uniform grid with square stencils, that is ∆x = ∆y = ∆ and n = m = r, with r > 1.…”

“…with s ≥ 0 (see Lemma A.1, p. 37 of [15]). Consequently, since the two-dimensional undivided differences are obtained by successive one-dimensional computations, we can use (17) to rewrite the various terms in (14) in order to focus only on second order undivided differences.…”

“…This fact has caused some confusion in the related literature, since the scaling factor used in the definition of the indicators, when working with non-differentiable functions, is usually mistaken. In order to fill this gap, in [15,28] the authors proposed a rather detailed analysis of the considered indicators and then applied the constructions, in a slightly different setting, to define new Adaptive Filtered (AF) schemes, renewing the definition of [7]. In our approach the smoothness indicators are necessary in order to detect the regions in which it is safe to use the values of the numerical solution to compute the switching parameter ε n , which is automatically computed and does not depend on the problem, differently from the scheme in [7].…”

“…In this work we introduce a new definition of multidimensional smoothness indicators devised for functions which may present some discontinuity in the gradient, we analyze their properties and we extend our class of AF schemes [15] to more spatial dimensions. In particular, we propose a natural extension of the definition in [20] to structured two-dimensional grids.…”