Improved approximate dispersion relation analysis using deep neural network

Abstract

In this work, we revisit and propose some improvements in the combined spatial-temporal approximate dispersion analysis (ADR). This analysis serves the purpose of quantifying spectral errors in the numerical solutions. The improvements we propose significantly enhance the predictive capabilities of ADR analysis, particularly in the context of multi-step and multi-level methods such as the Runge–Kutta (RK) and the Adam-Bashforth techniques. We deduce the equations governing numerical dissipation and group velocity, and we present corresponding plots for various multi-step methods based on our proposed approach. Additionally, we introduce a novel strategy for minimizing dissipation, optimizing it through the utilization of an evolutionary algorithm. While the linear analysis exhibits certain limitations when extended to nonlinear scenarios, we employ a deep neural network to determine stability equations for nonlinear problems. We conduct a comprehensive analysis to demonstrate that our proposed method exhibits minimal errors when compared to alternative advection equation schemes, both in one and two spatial dimensions.