In this paper, in order to improve the spatial accuracy, the exponential integrator Fourier Galerkin method (EIFG) is proposed for solving semilinear parabolic equations in rectangular domains. In this proposed method, the spatial discretization is first carried out by the Fourier-based Galerkin approximation, and then the time integration of the resulting semi-discrete system is approximated by the explicit exponential Runge-Kutta approach, which leads to the fully-discrete numerical solution. With certain regularity assumptions on the model problem, error estimate measured in $H^2$-norm is explicitly derived for EIFG method with two RK stages. Several two and three dimensional examples are shown to demonstrate the excellent performance of EIFG method, which are coincident to the theoretical results.

翻译：本文为提高空间精度，提出了指数积分器傅里叶伽辽金方法（EIFG），用于求解矩形区域上的半线性抛物型方程。在该方法中，首先通过基于傅里叶的伽辽金近似进行空间离散，随后采用显式指数Runge-Kutta方法对所得半离散系统进行时间积分，从而得到全离散数值解。在模型问题满足特定正则性假设的条件下，本文为具有两个RK阶段的EIFG方法显式推导了以$H^2$范数度量的误差估计。通过多个二维与三维算例验证了EIFG方法的优异性能，其结果与理论分析一致。