Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analysis of a local discontinuous Galerkin scheme for fractional Korteweg-de Vries equation

Mukul Dwivedi, Tanmay Sarkar

Abstract

We propose a local discontinuous Galerkin (LDG) method for the fractional Korteweg-de Vries (KdV) equation, involving the fractional Laplacian with exponent $α\in (1,2)$ in one and multiple space dimensions. By decomposing the fractional Laplacian into first-order derivatives and a fractional integral, we prove the $L^2$-stability of the semi-discrete LDG scheme incorporating suitable interface and boundary fluxes. We derive the optimal error estimate for linear flux and demonstrate an error estimate with an order of convergence $\mathcal{O}(h^{k+\frac{1}{2}})$ for general nonlinear flux utilizing the Gauss-Radau projections. Moreover, we extend the stability and error analysis to the multiple space dimensional case. Additionally, we discretize time using the Crank-Nicolson method to devise a fully discrete stable LDG scheme, and obtain a similar order error estimate as in the semi-discrete scheme. Numerical illustrations are provided to demonstrate the efficiency of the scheme, confirming an optimal order of convergence.

Analysis of a local discontinuous Galerkin scheme for fractional Korteweg-de Vries equation

Abstract

We propose a local discontinuous Galerkin (LDG) method for the fractional Korteweg-de Vries (KdV) equation, involving the fractional Laplacian with exponent in one and multiple space dimensions. By decomposing the fractional Laplacian into first-order derivatives and a fractional integral, we prove the -stability of the semi-discrete LDG scheme incorporating suitable interface and boundary fluxes. We derive the optimal error estimate for linear flux and demonstrate an error estimate with an order of convergence for general nonlinear flux utilizing the Gauss-Radau projections. Moreover, we extend the stability and error analysis to the multiple space dimensional case. Additionally, we discretize time using the Crank-Nicolson method to devise a fully discrete stable LDG scheme, and obtain a similar order error estimate as in the semi-discrete scheme. Numerical illustrations are provided to demonstrate the efficiency of the scheme, confirming an optimal order of convergence.
Paper Structure (15 sections, 18 theorems, 173 equations, 4 figures, 5 tables)

This paper contains 15 sections, 18 theorems, 173 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminary results and fractional calculus
  3. LDG scheme for the one-dimensional fractional KdV equation
  4. Stability estimates
  5. Error analysis
  6. LDG for multi-dimensional fractional KdV equation
  7. Error estimate
  8. Stability and Convergence analysis of fully-discrete LDG scheme
  9. Numerical Examples
  10. Crank-Nicolson LDG Scheme
  11. Higher Order LDG scheme
  12. Compare with classical KdV
  13. Fractional Case
  14. Examples of multiple dimensional equation
  15. Concluding remarks

Key Result

Lemma 2.1

Let $u,$$v \in C^n(\mathbb R)$, $n\in \mathbb N$ be such that $\frac{d^j}{dx^j}v(x)= 0$ and $\frac{d^j}{dx^j}u(x)= 0$ as $x \to \pm\infty,$$\forall ~0\leq j\leq n$. Then the fractional integrals and derivatives defined in leftfrac-rightfracD satisfy the following properties:

Figures (4)

  • Figure 6.1: The exact solutions and approximate solution of \ref{['fkdv']} at $T=1$ with $N=80$, $k=3$ and fractional exponent $\alpha=1.950$, 1.970 and 1.999.
  • Figure 6.2: Approximate solution of fractional KdV equation at $T=1$ with $N=320$, $k=3$ for the fractional values $\alpha= 1.2, 1.4, 1.6$ and 1.8, by choosing smooth initial condition $V_0$ in \ref{['smoothfrac']}.
  • Figure 6.3: Approximate solution at $T=0.001$ with $N=320$, $k=3$ and initial condition $W_0$ of \ref{['fkdv']}.
  • Figure 6.4: Approximate solution at $T=0.005$ with $N=320$, $k=3$ and initial condition $W_0$ of \ref{['fkdv']}.

Theorems & Definitions (34)

  • Lemma 2.1: See podlubny1998fractional
  • Definition 2.2: Fractional space ervin2006variational
  • Remark 2.3
  • Lemma 2.4: See ervin2006variational
  • Lemma 2.5: See xu2014discontinuous
  • Definition 2.6
  • Lemma 2.7: See kilbas2006theory
  • Lemma 2.8: Fractional Poincaré-Friedrichs ervin2006variational
  • Lemma 2.9: See Theorem 2.8 in deng2013local
  • Remark 3.1
  • ...and 24 more