Anomaly in Numerical Integrations of the KPZ Equation and Improved Discretization

TL;DR

This paper identifies issues with traditional numerical schemes for the KPZ equation caused by microscopic roughness and introduces a new discretization method that improves the accuracy and reliability of direct numerical integration.

Contribution

The paper reveals limitations of conventional finite difference schemes for the KPZ equation and proposes a novel discretization that overcomes these issues in 1+1 dimensions.

Findings

Conventional schemes do not accurately approximate the KPZ continuum due to microscopic roughness.

The effective diffusion coefficient in traditional methods is inconsistent with the expected value.

The proposed discretization enhances the reliability of numerical integration for the KPZ equation.

Abstract

We demonstrate and explain that conventional finite difference schemes for direct numerical integration do not approximate the continuum Kardar-Parisi-Zhang (KPZ) equation due to microscopic roughness. The effective diffusion coefficient is found to be inconsistent with the nominal one. We propose a novel discretization in 1+1 dimensions which does not suffer from this deficiency and elucidates the reliability and limitations of direct integration approaches.