Exponential approximation space reconstruction WENO scheme for dispersive PDEs

TL;DR

This paper introduces a fifth-order WENO scheme using exponential approximation space for dispersive PDEs, improving accuracy and reducing oscillations by optimizing a tension parameter.

Contribution

It develops a novel exponential approximation space within a WENO framework, enhancing solution accuracy for dispersive equations compared to traditional polynomial methods.

Findings

Achieves fifth-order convergence in dispersive PDEs

Reduces spurious oscillations with optimized tension parameter

Validated through 1D and 2D numerical examples

Abstract

In this work, we construct a fifth-order weighted essentially non-oscillatory (WENO) scheme with exponential approximation space for solving dispersive equations. A conservative third-order derivative formulation is developed directly using WENO spatial reconstruction procedure and third-order TVD Runge- Kutta scheme is used for the evaluation of time derivative. This exponential approximation space consists a tension parameter that may be optimized to fit the specific feature of the charecteristic data, yielding better results without spurious oscillations compared to the polynomial approximation space. A detailed formulation is presented for the the construction of conservative flux approximation, smoothness indicators, nonlinear weights and verified that the proposed scheme provides the required fifth convergence order. One and two-dimensional numerical examples are presented to support the theoretical claims.