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A priori error estimates of Runge-Kutta discontinuous Galerkin schemes to smooth solutions of fractional conservation laws

Fabio Leotta, Jan Giesselmann

TL;DR

This work analyzes a second-order in time, fully explicit Runge-Kutta discontinuous Galerkin (RKDG) scheme for scalar fractional conservation laws in one space dimension, incorporating a nonlocal diffusion operator $g_\lambda$ with $\lambda\in(0,1)$. The authors extend the hyperbolic RKDG error theory by introducing a novel upwind projection that adapts in time to the evolving solution and flux, enabling an energy-method-based a priori error estimate in $L^2$ and $H^{\lambda/2}$-type norms. Under suitable regularity and CFL-type time-step restrictions, they prove convergence with rates $h^{k+1-1/\alpha}$, $h^{k+1-\lambda/2}$, and $\tau^2$, with optimality for $\alpha \ge 2/\lambda$, and show the proof hinges on a careful balance between consistency, projection differences, and Gronwall-type growth. The results extend high-order RKDG convergence theory from purely hyperbolic problems to fractional conservation laws, providing rigorous guidance for numerical simulations in smooth regions where shocks are absent or weak.

Abstract

We give a priori error estimates of second order in time fully explicit Runge-Kutta discontinuous Galerkin schemes using upwind fluxes to smooth solutions of scalar fractional conservation laws in one space dimension. Under the time step restrictions $τ\leq c h$ for piecewise linear and $τ\lesssim h^{4/3}$ for higher order finite elements, we prove a convergence rate for the energy norm $\|\cdot\|_{L^\infty_tL^2_x}+|\cdot|_{L^2_tH^{λ/2}_x}$ that is optimal for solutions and flux functions that are smooth enough. Our proof relies on a novel upwind projection of the exact solution.

A priori error estimates of Runge-Kutta discontinuous Galerkin schemes to smooth solutions of fractional conservation laws

TL;DR

This work analyzes a second-order in time, fully explicit Runge-Kutta discontinuous Galerkin (RKDG) scheme for scalar fractional conservation laws in one space dimension, incorporating a nonlocal diffusion operator with . The authors extend the hyperbolic RKDG error theory by introducing a novel upwind projection that adapts in time to the evolving solution and flux, enabling an energy-method-based a priori error estimate in and -type norms. Under suitable regularity and CFL-type time-step restrictions, they prove convergence with rates , , and , with optimality for , and show the proof hinges on a careful balance between consistency, projection differences, and Gronwall-type growth. The results extend high-order RKDG convergence theory from purely hyperbolic problems to fractional conservation laws, providing rigorous guidance for numerical simulations in smooth regions where shocks are absent or weak.

Abstract

We give a priori error estimates of second order in time fully explicit Runge-Kutta discontinuous Galerkin schemes using upwind fluxes to smooth solutions of scalar fractional conservation laws in one space dimension. Under the time step restrictions for piecewise linear and for higher order finite elements, we prove a convergence rate for the energy norm that is optimal for solutions and flux functions that are smooth enough. Our proof relies on a novel upwind projection of the exact solution.
Paper Structure (10 sections, 13 theorems, 75 equations)

This paper contains 10 sections, 13 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. The RKDG method
  4. The difference between numerical and physical flux
  5. Analysis of the scheme
  6. Consistency, error equations and energy identity
  7. The upwind projection operator
  8. Time step differences of projection errors
  9. Bounding the RHS of the energy identity
  10. The Gronwall argument

Key Result

Theorem 2.1

Let $\alpha\in\mathbb{N}$ with $\alpha\geq 3$, $f\in C^{\alpha+1}(\mathbb{R})$ and $u\in C^\alpha([0,T];H^{k+1}(\mathbb{R}))$ be the exact solution to the fractional conservation law (FCL). Let $u_h^n$, a piecewise discontinuous polynomial of degree $k\geq 1$, be the numerical solution of the RKDG s for all $n\in\{1,\ldots,N\}$, where $u^n:=u(t^n,\cdot)$.

Theorems & Definitions (31)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Lemma 4.1: Consistency
  • proof
  • Remark 4.2
  • Lemma 4.3: Error equations
  • Lemma 4.4: Energy identity
  • proof
  • ...and 21 more