A finite difference method with symmetry properties for the high-dimensional Bratu equation

TL;DR

This paper introduces a symmetric finite difference method that leverages symmetry properties to accurately solve the high-dimensional 3D Bratu equation, providing detailed bifurcation diagrams and stability analysis.

Contribution

The paper presents a novel symmetric finite difference method that improves accuracy and efficiency in solving high-dimensional Bratu equations, including bifurcation analysis and stability assessment.

Findings

SFDM outperforms previous methods for 3D Bratu equation

Successfully solves on grids up to 301^3 points

Accurately identifies bifurcation points across dimensions

Abstract

Solving the three-dimensional (3D) Bratu equation is highly challenging due to the presence of multiple and sharp solutions. Research on this equation began in the late 1990s, but there are no satisfactory results to date. To address this issue, we introduce a symmetric finite difference method (SFDM) which embeds the symmetry properties of the solutions into a finite difference method (FDM). This SFDM is primarily used to obtain more accurate solutions and bifurcation diagrams for the 3D Bratu equation. Additionally, we propose modifying the Bratu equation by incorporating a new constraint that facilitates the construction of bifurcation diagrams and simplifies handling the turning points. The proposed method, combined with the use of sparse matrix representation, successfully solves the 3D Bratu equation on grids of up to $301^3$ points. The results demonstrate that SFDM outperforms all previously employed methods for the 3D Bratu equation. Furthermore, we provide bifurcation diagrams for the 1D, 2D, 4D, and 5D cases, and accurately identify the first turning points in all dimensions. All simulations indicate that the bifurcation diagrams of the Bratu equation on the cube domains closely resemble the well-established behavior on the ball domains described by Joseph and Lundgren [1]. Furthermore, when SFDM is applied to linear stability analysis, it yields the same largest real eigenvalue as the standard FDM despite having fewer equations and variables in the nonlinear system.