Nonlinear parametrization solver for fractional Burgers equations

TL;DR

This paper introduces a nonlinear parametrization method for solving fractional Burgers equations, effectively handling nonlocality and shocks with improved stability and accuracy over traditional spectral methods.

Contribution

The paper presents a novel sequential-in-time nonlinear parametrization approach that ensures stable, oscillation-free solutions for fractional Burgers equations, outperforming existing spectral schemes.

Findings

Achieves oscillation-free shock resolution

Accurately captures long-time dynamics

Requires fewer degrees of freedom

Abstract

Fractional Burgers equations pose substantial challenges for classical numerical methods due to the combined effects of nonlocality and shock-forming nonlinear dynamics. In particular, linear approximation frameworks-such as spectral, finite-difference, or discontinuous Galerkin methods-often suffer from Gibbs-type oscillations or require carefully tuned stabilization mechanisms, whose effectiveness degrades in transport-dominated and long-time integration regimes. In this work, we introduce a sequential-in-time nonlinear parametrization (STNP) for solving fractional Burgers equations, including models with a fractional Laplacian or with nonlocal nonlinear fluxes. The solution is represented by a nonlinear parametric ansatz, and the parameter evolution is obtained by projecting the governing dynamics onto the tangent space of the parameter manifold through a regularized least-squares formulation at each time step. This yields a well-posed and stable time-marching scheme that preserves causality and avoids global-in-time optimization. We provide a theoretical analysis of the resulting projected dynamics, including a stability estimate and an a posteriori error bound that explicitly decomposes the total error into contributions from initial condition fitting, projection residuals, and discretization of fractional operators. Our analysis clarifies the stabilizing role of regularization and quantifies its interaction with the nonlocal discretization error. Numerical experiments for both fractional Burgers models demonstrate that STNP achieves oscillation-free shock resolution and accurately captures long-time dynamics. The method consistently outperforms high-order spectral schemes augmented with spectral vanishing viscosity, while requiring significantly fewer degrees of freedom and avoiding ad hoc stabilization.