The finite precision computation and the nonconvergence of difference scheme

TL;DR

This paper demonstrates that finite precision arithmetic can cause difference schemes to fail in converging to the true solution, challenging the assumption that stable schemes are always reliable in numerical computations.

Contribution

The authors introduce a theoretical framework to analyze the impact of round-off errors on the convergence of difference schemes, showing that convergence cannot be guaranteed in finite precision environments.

Findings

Round-off errors can break the consistency of difference schemes.

Stable schemes may not guarantee convergence in finite precision computations.

Experimental results show non-convergence as time step approaches zero.

Abstract

The authors show that the round-off error can break the consistency which is the premise of using the difference equation to replace the original differential equations. We therefore proposed a theoretical approach to investigate this effect, and found that the difference scheme can not guarantee the convergence of the actual compute result to the analytical one. A conservation scheme experiment is applied to solve a simple linear differential equation satisfing the LAX equivalence theorem in a finite precision computer. The result of this experiment is not convergent when time step-size decreases trend to zero, which proves that even the stable scheme can't guarantee the numerical convergence in finite precision computer. Further the relative convergence concept is introduced.