On Spatial Discretization of Evolutionary Differential Equations on the Periodic Domain with a Mixed Derivative
For researchers studying PDEs with mixed derivatives, this provides a foundational reformulation technique that enables numerical methods and theoretical analysis.
The paper proposes a novel reformulation procedure for evolutionary PDEs with a mixed derivative into a standard form, enabling numerical analysis. It establishes global well-posedness of the sine-Gordon equation and shows the average-difference method is suitable for discretization of the mixed derivative.
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.