Error estimates of fully semi-Lagrangian schemes for diffusive conservation laws

TL;DR

This paper provides error estimates and convergence rates for fully semi-Lagrangian schemes with high-order interpolation applied to one-dimensional nonlinear diffusive conservation laws, including Burgers equations.

Contribution

It establishes new error bounds and convergence rates for semi-Lagrangian schemes using spline and Hermite interpolations for diffusive conservation laws.

Findings

Error estimates in L2 and Hs norms are derived.

Convergence rates depend on interpolation degree and discretization parameters.

Numerical results confirm theoretical predictions.

Abstract

We present error estimates of the fully semi-Lagrangian scheme with high-order interpolation operators, solving the initial value problems for the one-dimensional nonlinear diffusive conservation laws, including the Burgers equations. We impose certain assumptions on the interpolation operator, which are satisfied by both spline and Hermite interpolations. We establish the convergence rates of $ O(\Delta t + h^{2 s} / \Delta t) $ in the $ L^2 $-norm and $ O(\Delta t + h^{s} / (\Delta t)^{1/2} + h^{2s} / \Delta t) $ in the $ H^s $-norm for the spatial mesh size $ h $ and the temporal step size $ \Delta t $, where the spline or Hermite interpolation operator of degree $ (2s - 1) $ is employed. The numerical results are in agreement with the theoretical analysis.