A Directional-ODE Framework for Discretization of Advection-Diffusion Equations

TL;DR

This paper introduces a directional-ODE framework that reformulates advection-diffusion equations as directional ODEs, leading to more stable, efficient, and high-resolution numerical solutions, especially for complex systems with uncertainties.

Contribution

It presents a novel ODE-based approach to discretize ADEs, improving stability and efficiency, and extends to uncertain parameter systems, advancing numerical methods in scientific computing.

Findings

Enhanced stability of numerical solutions

Improved computational efficiency

High spatiotemporal resolution achieved

Abstract

We present a novel approach that redefines the traditional interpretation of explicit and implicit discretization methods for solving a general class of advection-diffusion equations (ADEs) featuring nonlinear advection, diffusion operators, and potential source terms. By reformulating the discrete ADEs as directional ordinary differential equations (ODEs) along temporal or spatial dimensions, we derive analytical solutions that lead to novel update formulas. In essence, the information of discrete ADEs is compressed into these directional ODEs, which we refer to as representative ODEs. The analytical update formulas derived from the representative ODEs significantly enhance stability, computational efficiency, and spatiotemporal resolution. Furthermore, we extend the framework to systems with uncertain parameters and coefficients, showcasing its versatility in addressing complex ADEs encountered in modeling and simulation across diverse scientific and engineering disciplines.