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Fully discrete finite difference schemes for the Fractional Korteweg-de Vries equation

Mukul Dwivedi, Tanmay Sarkar

Abstract

In this paper, we present and analyze fully discrete finite difference schemes designed for solving the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation involving the fractional Laplacian. We design the scheme by introducing the discrete fractional Laplacian operator which is consistent with the continuous operator, and posses certain properties which are instrumental for the convergence analysis. Assuming the initial data (u_0 \in H^{1+α}(\mathbb{R})), where (α\in [1,2)), our study establishes the convergence of the approximate solutions obtained by the fully discrete finite difference schemes to a classical solution of the fractional KdV equation. Theoretical results are validated through several numerical illustrations for various values of fractional exponent $α$. Furthermore, we demonstrate that the Crank-Nicolson finite difference scheme preserves the inherent conserved quantities along with the improved convergence rates.

Fully discrete finite difference schemes for the Fractional Korteweg-de Vries equation

Abstract

In this paper, we present and analyze fully discrete finite difference schemes designed for solving the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation involving the fractional Laplacian. We design the scheme by introducing the discrete fractional Laplacian operator which is consistent with the continuous operator, and posses certain properties which are instrumental for the convergence analysis. Assuming the initial data (u_0 \in H^{1+α}(\mathbb{R})), where (α\in [1,2)), our study establishes the convergence of the approximate solutions obtained by the fully discrete finite difference schemes to a classical solution of the fractional KdV equation. Theoretical results are validated through several numerical illustrations for various values of fractional exponent . Furthermore, we demonstrate that the Crank-Nicolson finite difference scheme preserves the inherent conserved quantities along with the improved convergence rates.
Paper Structure (13 sections, 10 theorems, 123 equations, 4 figures, 4 tables)

This paper contains 13 sections, 10 theorems, 123 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Numerical discretization and discrete operators
  3. Discretization of fractional Laplacian
  4. Fully discrete semi-implicit scheme
  5. Convergence
  6. Crank-Nicolson finite difference scheme
  7. Stability Lemma
  8. Convergence of approximate solution
  9. Numerical experiment
  10. Benjamin-Ono equation
  11. Fractional KdV equation
  12. KdV equation
  13. Concluding remarks

Key Result

Lemma 2.1

Let $\alpha\in[1,2)$ and $\mathbb{D}^{\alpha}$ be the discrete fractional Laplacian defined by DisLap. Then, for any $u\in\mathcal{S}(\mathbb R)$,

Figures (4)

  • Figure 5.1: The exact solution $u_1$ and approximate solutions $u_{\Delta x}^{EI}$ and $u_{\Delta x}^{CN}$ at $t=20$, $t=100$ and $t=120$ with $N=512$ and $\alpha=1$.
  • Figure 5.2: The reference and approximate solutions at $t=5$ with the initial data $u_0(x) = 0.5\sin(x)$, $N=1000$ grids and $\alpha=1.5$.
  • Figure 5.3: The exact(KdV) and numerical solution at $t=20$ with the initial data $w_2(x,-10)$ with $N=32000$ for Euler implicit.
  • Figure 5.4: The exact and numerical solution obtained by Crank nicolson at $t=40$ with the initial data $w_2(x,-20)$ with $N=1000$.

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 3.1
  • ...and 16 more