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Discontinuous Galerkin method for fractional convection-diffusion equations
Q. Xu, J. S. Hesthaven
TL;DR
The paper addresses the limited development of accurate and robust numerical methods for fractional convection-diffusion equations with fractional Laplacians. It rewrites the fractional operator as a low-order system and derives a local discontinuous Galerkin scheme. The scheme is stable, achieves the reported optimal convergence orders, and is supported by numerical examples.
Problem
Accurate and robust numerical methods for conservation laws with fractional Laplacian operators remain limited.
Method
The fractional Laplacian is rewritten as first-order derivatives and fractional integrals, yielding a low-order system treated with a local discontinuous Galerkin method.
Results
The semi-discrete scheme is stable, with ||u_h(x,T)|| ≤ ||u_0(x)||, and numerical examples show optimal O(h^k+1) convergence across 1 < α < 2.
Takeaways & Limitations
The reformulated LDG approach provides stability and optimal convergence for the considered fractional convection-diffusion problems.
Takeaways & Limitations
The paper does not provide theoretical analysis for the mixed boundary-condition scheme, despite computational evidence of excellent behavior and optimal convergence.
Abstract
from arXiv · showhide
We propose a discontinuous Galerkin method for convection-subdiffusion equations with a fractional operator of order $α(1<α<2)$ defined through the fractional Laplacian. The fractional operator of order $α$ is expressed as a composite of first order derivatives and fractional integrals of order $2-α$, and the fractional convection-diffusion problem is expressed as a system of low order differential/integral equations and a local discontinuous Galerkin method scheme is derived for the equations. We prove stability and optimal order of convergence O($h^{k+1}$) for subdiffusion, and an order of convergence of ${\cal O}(h^{k+1/2})$ is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.
1. Introduction.
The paper targets fractional convection-diffusion equations with fractional Laplacians, where accurate and robust numerical methods remain limited. It rewrites the fractional operator into a low-order system so a local discontinuous Galerkin method can be applied.
- Motivation: Fractional conservation laws generalize classical convection-diffusion equations and model applications including geomorphology, detonations, signal processing, and anomalous diffusion.For α < 1, solutions may lack smoothness and develop shocks; for α > 1, the nonlocal term smooths discontinuities.
- Related work: Accurate and robust numerical methods for conservation laws with fractional Laplacians remain limited despite extensive numerical work on nonlocal operators.Prior work includes difference, finite-difference, splitting, and discontinuous Galerkin methods, but reported numerical results did not confirm one analysis.
- Contribution: The local discontinuous Galerkin method is applied to the resulting system, with interface fluxes designed to support stability and local solvability of auxiliary variables.Direct application to higher-order spatial derivatives can be inconsistent, motivating the first-order-system formulation.
- Contribution: For 1 < α < 2, the paper rewrites the fractional operator as first-order derivatives composed with a fractional integral and converts the equation into low-order equations.This reformulation is intended to obtain a consistent, high-accuracy method.
2. Background and definitions.
The paper introduces fractional-calculus representations and fractional spaces needed to analyze the fractional Laplacian and its numerical approximation. It also states the bounded-domain setting and approximation tools used later.
- Fractional calculus: The analysis uses Riemann-Liouville fractional derivatives and introduces fractional integrals, derivatives, and their basic properties.The left and right fractional derivatives are used as the main definition for the developments and analysis.
- Fractional calculus: For 1 < α < 2, the fractional Laplacian is related to fractional derivatives under decay assumptions on derivatives of u at ±∞.The stated equivalence relies on derivatives through the relevant order disappearing as x approaches ±∞.
- Fractional spaces: The paper defines left, right, and symmetric fractional spaces with associated seminorms for the subsequent analysis.These definitions support the fractional-space results used in the numerical analysis.
- Domain and spaces: For numerical treatment, the problem is restricted from R to a bounded domain Ω = [a, b].The paper develops corresponding fractional-space results on the bounded domain and notes that L2(Ω) is embedded into the relevant spaces.
- Approximation: The approximation analysis uses piecewise polynomials of degree k on a mesh of width h and establishes bounds with constants independent of h.The lemmas cover fractional-integral and derivative approximation cases.
3. LDG scheme for the fractional convection-diffusion equation.
The LDG scheme first converts the fractional derivative into auxiliary variables and a low-order system, then discretizes that system with piecewise polynomial functions and numerical fluxes.
- System reformulation: The fractional derivative is rewritten as a composite of first-order derivatives and a fractional integral to avoid the lower-order accuracy of direct high-order differentiation.The resulting formulation is a low-order system suitable for a high-order discontinuous Galerkin scheme.
- System reformulation: Auxiliary variables p and q are introduced so the fractional convection-diffusion problem can be rewritten as a low-order system.The approximation seeks (u_h, p_h, q_h) in the discrete space.
- Spatial discretization: The discrete solution uses a piecewise polynomial space V_h on a partition of Ω = [a, b].The space is embedded in the fractional space required by the analysis.
- Numerical fluxes: Alternating-direction fluxes are used for the high-order derivative component, while any monotone flux may be used for the nonlinear convection component.The scheme is obtained by integration by parts and replacing interface fluxes with numerical fluxes.
- Boundary treatment: The bounded-domain formulation imposes homogeneous Dirichlet conditions outside Ω and uses a specified boundary flux with positive parameter β.These choices complete the boundary treatment of the LDG scheme.
4. Stability and error estimates.
The LDG scheme is analyzed through stability identities and projection-based error estimates for fractional diffusion and convection-diffusion problems. The analysis establishes stability, diffusion error bounds, and a reduced convergence order for the general convection-diffusion case, with upwind fluxes offering improvement.
- Stability: The semi-discrete scheme is stable and satisfies ∥u_h(x,T)∥ ≤ ∥u_0(x)∥ for any T > 0.
- Fractional diffusion: The fractional-diffusion error analysis applies projection estimates, fractional inequalities, and a bilinear-form bound to derive error estimates for sufficiently smooth solutions.The constant in the estimate is independent of h.
- Fractional convection-diffusion: The general fractional convection-diffusion analysis combines the fractional-diffusion estimate with a monotone numerical-flux bound to obtain an error estimate for piecewise polynomials of degree k ≥ 1.The result assumes a sufficiently smooth exact solution and f ∈ C3, with h sufficiently small.
- Fractional convection-diffusion: The general convection-diffusion analysis initially yields convergence order k + 1/2, but an upwind numerical flux can improve it to optimal order.
- Mixed boundary conditions: For mixed boundary conditions, the alternative scheme imposes the condition naturally, but its theoretical analysis is more complicated and is not developed.Computational results indicate excellent behavior and optimal convergence.
5. Numerical examples.
The numerical examples validate the spatial discretization, explicit time integration, convergence analysis, and dissipative behavior of the fractional models.
- Numerical setup: The semi-discrete formulation treats the spatial dimension, while a fourth-order low-storage explicit Runge–Kutta method advances the solution in time.The time discretization uses LSERK coefficients, with unknown vector u_h.
- Numerical setup: Stability of the explicit method requires a timestep satisfying Δt ≤ CΔx^α_min, with 0 < C < 1, in the examples.
- Example 1: The fractional diffusion example uses u_0(x) = x^6(1−x)^6 and an exact solution u(x,t) = e^−tx^6(1−x)^6.
- Example 1: O(h^k+1) convergence is observed across 1 < α < 2 for the fractional Laplacian problem.
- Example 2: O(h^k+1) convergence is likewise observed across 1 < α < 2 for the fractional Burgers’ equation.
- Example 3: Increasing α produces a stronger dissipative effect in the discontinuous-initial-condition Burgers’ example, approaching the classical case at α = 2.
6. Concluding remarks.
The paper proposes a local discontinuous Galerkin method for fractional convection-diffusion problems and derives stability and error estimates. Numerical examples confirm the analysis for different values of α.
- A local discontinuous Galerkin method is proposed for convection-diffusion problems with fractional diffusion defined through a fractional Laplacian.The fractional Laplacian is rewritten using first-order derivatives and integrals, yielding a low-order system.
- Stability and error estimates are derived, establishing optimal convergence for the proposed method.
- Numerical examples confirm the analysis for different values of α, including α = 1.5 and α = 2.0.