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.

翻译：求解三维布拉图方程极具挑战性，因其存在多个尖锐解。关于该方程的研究始于20世纪90年代末，但至今尚无令人满意的结果。为解决此问题，我们引入了一种对称有限差分方法，该方法将解的对称性嵌入到有限差分格式中。此SFDM主要用于获取三维布拉图方程更精确的解及分岔图。此外，我们提出通过引入新约束来修正布拉图方程，这有助于构建分岔图并简化对转折点的处理。所提出的方法结合稀疏矩阵表示，成功在网格规模高达$301^3$个点的网格上求解了三维布拉图方程。结果表明，SFDM在三维布拉图方程的求解上优于所有先前使用的方法。此外，我们提供了一维、二维、四维和五维情况下的分岔图，并准确识别了所有维度上的首个转折点。所有模拟均表明，立方体域上布拉图方程的分岔图与Joseph和Lundgren [1] 描述的球域上已确立的经典行为高度相似。进一步地，当SFDM应用于线性稳定性分析时，尽管非线性系统的方程和变量数量更少，其仍能产生与标准FDM相同的最大实特征值。