On Spatial Discretization of Evolutionary Differential Equations on the Periodic Domain with a Mixed Derivative

TL;DR

This paper introduces a reformulation technique for evolutionary PDEs with mixed derivatives, enabling a unified numerical approach and demonstrating the suitability of the average-difference method for spatial discretization.

Contribution

It proposes a novel reformulation method for PDEs with mixed derivatives, facilitating analysis and numerical discretization, and establishes the well-posedness of the sine-Gordon equation.

Findings

The reformulation enables a unified framework for such PDEs.

The average-difference method is effective for discretizing mixed derivatives.

The sine-Gordon equation is proven to be globally well-posed.

Abstract

Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper, we aim at the unified framework of numerical methods for such PDEs. However, due to the presence of the mixed derivative, we cannot discuss numerical methods without some appropriate reformulation, which is mathematically challenging itself. Therefore, we first propose a novel procedure for the reformulation of target PDEs into a standard form of evolutionary equations. This contribution may become an important basis not only of numerical analysis, but also of PDE-theory. In order to illustrate this point, we establish the global well-posedness of the sine-Gordon equation. After that, we classify and discuss the spatial discretizations based on the proposed reformulation technique. As a result, we show the average-difference method is suitable for the discretization of the mixed derivative.