Space-time-unified GIRM (Generalized Integral Representation Method) for unsteady advective diffusion

TL;DR

This paper introduces a space-time-unified GIRM approach for unsteady advective diffusion, comparing explicit and implicit methods, with numerical validation showing the implicit method's efficiency and stability considerations.

Contribution

It develops a space-time-unified GIRM framework for unsteady advective diffusion and analyzes the efficiency of implicit versus explicit time evolution methods.

Findings

Implicit GIRM offers more efficient computation.

Explicit GIRM is simpler to implement.

Neumann problems require smaller time steps.

Abstract

The Generalized Integral Representation Method (GIRM) for Space-Time-Separated Method (STSM) and Space-Time-Unified Method (STUM) are discussed. STSM and STUM give explicit and implicit time evolutions, respectively. The algorithm of STSM is much simpler than STUM. However, the implicit time evolution of STUM could give us much more efficient computation. Numerical calculations using STUM for Dirichlet and Neumann problems in 2D space-time are conducted using a Traditional Fundamental Solution (TFS). The results seem very satisfactory. However, in case of Neumann problem, the time increment must be smaller than in case of Dirichlet problem.