A stability preserved time-integration method for nonlinear advection-diffusion models

TL;DR

This paper introduces a new implicit-explicit local differential transform method (IELDTM) for stable, high-order time integration of nonlinear advection-diffusion models, specifically applied to the Burgers equation, improving accuracy and stability.

Contribution

The paper develops a stability-preserving, high-order time integrator using IELDTM combined with Chebyshev spectral collocation for nonlinear advection-diffusion equations, addressing limitations of classical methods.

Findings

The proposed method enhances accuracy over MATLAB solvers ode45 and ode15s.

It effectively eliminates accuracy and stability issues of classical thods.

Numerical examples demonstrate superior performance in 1D and 2D Burgers equations.

Abstract

A new implicit-explicit local differential transform method (IELDTM) is derived here for time integration of the nonlinear advection-diffusion processes represented by (2+1)-dimensional Burgers equation. The IELDTM is adaptively constructed as stability preserved and high order time integrator for spatially discretized Burgers equation. For spatial discretization of the model equation, the Chebyshev spectral collocation method (ChCM) is utilized. A robust stability analysis and global error analysis of the IELDTM are presented with respect to the direction parameter \theta. With the help of the global error analysis, adaptivity equations are derived to minimize the computational costs of the algorithms. The produced method is shown to eliminate the accuracy disadvantage of the classical \theta-method and the stability disadvantages of the existing DTM-based methods. Two examples of the Burgers equation in one and two dimensions have been solved via the ChCM-IELDTM hybridization, and the produced results are compared with the literature. The present time integrator has been proven to produce more accurate numerical results than the MATLAB solvers, ode45 and ode15s.