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Nonlinear parametrization solver for fractional Burgers equations

Haojun Qin, Zhiwei Gao, Jinye Shen, George Karniadakis

TL;DR

The paper tackles numerical solution of fractional Burgers equations that combine nonlocal diffusion with shock formation, which are challenging for linear discretizations. It introduces a sequential-in-time nonlinear parametrization (STNP) that represents the solution on a nonlinear parametric manifold via a surrogate Φ and evolves the parameters q(t) by projecting the PDE dynamics onto the tangent space using a regularized least-squares problem at each time step. The authors establish stability and a posteriori error estimates that decompose the total error into initial-condition fitting, projection residuals, and discretization of fractional operators; they also demonstrate energy-dissipation-consistent behavior under regularization. Numerical experiments on FBEFL and FBENN show that STNP delivers oscillation-free shock resolution and accurate long-time dynamics while using fewer degrees of freedom than high-order linear methods, outperforming SVV-augmented spectral methods and standard finite-difference approaches in challenging regimes.

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.

Nonlinear parametrization solver for fractional Burgers equations

TL;DR

The paper tackles numerical solution of fractional Burgers equations that combine nonlocal diffusion with shock formation, which are challenging for linear discretizations. It introduces a sequential-in-time nonlinear parametrization (STNP) that represents the solution on a nonlinear parametric manifold via a surrogate Φ and evolves the parameters q(t) by projecting the PDE dynamics onto the tangent space using a regularized least-squares problem at each time step. The authors establish stability and a posteriori error estimates that decompose the total error into initial-condition fitting, projection residuals, and discretization of fractional operators; they also demonstrate energy-dissipation-consistent behavior under regularization. Numerical experiments on FBEFL and FBENN show that STNP delivers oscillation-free shock resolution and accurate long-time dynamics while using fewer degrees of freedom than high-order linear methods, outperforming SVV-augmented spectral methods and standard finite-difference approaches in challenging regimes.

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.
Paper Structure (19 sections, 5 theorems, 76 equations, 4 figures, 1 algorithm)

This paper contains 19 sections, 5 theorems, 76 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Fractional calculus
  4. Discretization of the fractional derivatives
  5. Fractional Burgers equations
  6. FBEFL
  7. FBENN
  8. Methodology
  9. Nonlinear parametric approximation
  10. Sequential-in-time training
  11. Algorithm
  12. Theoretical analysis
  13. Regularization
  14. Stability
  15. Error estimate
  16. ...and 4 more sections

Key Result

Theorem 1

kiselev2008blow Assume that $1< \alpha \leq 2$, and the initial data $u_0(x) \in H^s(\mathbb{S}^1)$, for $s > \max(\frac{3}{2} - 2\alpha,0)$. Then there exists a unique global solution $u(x,t)$ of (FBEFL2) such that $u$ belongs to $C([0,\infty),H^s(\mathbb{S}^1))$. Moreover, $u(x,t)$ is real analyti

Figures (4)

  • Figure 1: Comparison of numerical solutions computed by STNP, WENO--Roe, and central-difference schemes. Both STNP and WENO--Roe suppress spurious oscillations, whereas the central-difference scheme exhibits pronounced oscillations near the shock.
  • Figure 2: Relative $L^2$ error versus the number of collocation points $N$ using first-order (left), second-order (middle), and third-order (right) finite-difference approximations of the fractional Laplacian, for networks of fixed width $10$ and depths 3, 5, and 7.
  • Figure 3: Comparison of spectral-SVV, STNP, and exact solutions. The first column shows $t=0.5$, the second column shows $t=1.0$; the top row corresponds to $\beta=0.6$ and the bottom row to $\beta=0.8$. Both methods suppress spurious oscillations, but STNP achieves higher accuracy near the shock.
  • Figure 4: Left: temporal evolution of the relative $L^2$ error for STNP and spectral-SVV; STNP remains more accurate over the entire time horizon. Middle: relative $L^2$ error of RK3 versus RK45, showing improved robustness with adaptive time stepping. Right: cumulative time reached at each integration step; RK45 automatically reduces the step size around $t\approx 0.3$.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 1: $L_{2}$ stability estimate
  • Proof 1
  • Remark 5.1
  • Theorem 2: Exact $L^2$ dissipation in the unregularized limit
  • Proof 2
  • Theorem 3: Error estimate
  • Proof 3