Spectral Analysis of Node- and Cell-Centered Higher-Order Compact Schemes for Fully Discrete One and Two-Dimensional Convection-Dispersion Equation

TL;DR

This paper conducts a detailed spectral analysis of high-order compact schemes for the convection-dispersion equation, revealing stability limits and dispersion errors to guide reliable numerical simulations.

Contribution

It provides a comprehensive spectral analysis of sixth and eighth order compact schemes, including node and cell-centered formulations, for convection-dispersion equations, with practical stability and accuracy insights.

Findings

Identifies critical dispersion thresholds and Courant numbers for stability.

Characterizes dispersion errors and energy transport inaccuracies.

Validates theoretical predictions with numerical experiments.

Abstract

In this study, we present a comprehensive global spectral analysis of the convection dispersion equation, which is also referred to in specific contexts as the Korteweg de Vries (KdV) equation, to investigate the behaviour of high order numerical schemes across a wide range of nondimensional parameters. The motivation for this analysis stems from the equation's importance in modeling wave propagation and transport phenomena, where accurate resolution of dispersive effects is critical, and traditional numerical schemes often suffer from spurious artifacts. We analyze one sixth order and two eighth order compact spatial discretization schemes, encompassing both node centered and cell centered formulations, combined with a third order strong stability preserving Runge Kutta (SSPRK3) time integrator. The analysis is performed in terms of key nondimensional parameters such as the wavenumber, Courant Friedrichs Lewy number $N_c$, and dispersion number $D_{\alpha}$ over the full spectral plane for both one and two dimensional cases. Key numerical indicators, including the amplification factor, normalized phase speed, and normalized group velocity, are evaluated to characterize stability, dispersion error, errors in energy transport, and directional anisotropy. Critical dispersion thresholds and Courant numbers are identified, beyond which numerical instability and nonphysical phenomena such as spurious q waves and reversed phase or energy transport arise. Theoretical predictions are validated through numerical experiments involving linear and nonlinear one and two dimensional test problems, including cases with exact solutions and established benchmark results. This comprehensive analysis uncovers subtle numerical errors and offers practical guidance for selecting reliable discretization parameters, ensuring accurate and stable simulations of convection dispersion systems.