Equivalence of Semi-Lagrangian and Lagrange-Galerkin Schemes Under Constant Advection Speed

Abstract

We compare in this paper two major implementations of large time-step schemes for advection equations, i.e., Semi-Lagrangian and Lagrange-Galerkin techniques. We show that SL schemes are equivalent to exact LG schemes via a suitable definition of the basis functions. In this paper, this equivalence will be proved assuming some simplifying hypoteses, mainly constant advection speed, uniform space grid, symmetry and translation invariance of the cardinal basis functions for interpolation. As a byproduct of this equivalence, we obtain a simpler proof of stability for SL schemes in the constant-coefficient case. I IMathematics subject classification: 65N12, 65M10, 49L25.

Field-Aligned Interpolation for Semi-Lagrangian Gyrokinetic Simulations

“…Remark 1 This result of positivity of the DFT has direct connection with results of Ferretti [9,10] stating equivalence between semi-Lagrangian and Lagrange-Galerkin methods under some assumptions, one of it being the positivity of the (continuous) Fourier transform. Such link may be further studied.…”

A Technique for High-order Treatment of Diffusion Terms in Semi-Lagrangian Schemes

“…In examining the convergence of the fully discrete scheme (3.10), we should take into account that convergence theory for SL schemes applied to first-order equations is itself somewhat incomplete. More precisely (see [1,8,11]), the unconditional stability of SL schemes, which is a widely observed fact in practice, has only been theoretically proved for the case of constant coefficient equation (although an extension to more general linear equations is in progress [12]). In what follows, the convergence analysis will be performed in a normalized Hölder norm, so that…”

Flux form Semi-Lagrangian methods for parabolic problems

“…First, we note that a well understood case is the construction of the nonconservative scheme with Ip in the form of a symmetric Lagrange interpolation, for odd values of p. In the pure advection case, this construction is known to be stable (see [32]). …”

“…It can be observed that, in the linear case, the errors obtained by the SL and FFSL methods are in general of the same order of magnitude. Indeed, as a result of the analysis in [32], the SL and FFSL methods should give exactly identical results in the constant coefficient case, provided that the centered Ip interpolation and the R p−1 reconstructions are employed, respectively. The slight differences in Tables 1 and 2 are due to the fact that, for simplicity of the present implementation, for the SL method the default MATLAB cubic interpolator was used, which implements a shape preserving cubic interpolator that is equivalent in accuracy, but not exactly identical to the centered I 3 interpolator.…”