In this paper, we investigate the stability and time-step constraints for solving advection-diffusion equations using exponential time differencing (ETD) Runge-Kutta (RK) methods in time and discontinuous Galerkin (DG) methods in space. We demonstrate that the resulting fully discrete scheme is stable when the time-step size is upper bounded by a constant. More specifically, when central fluxes are used for the advection term, the schemes are stable under the time-step constraint tau <= tau_0 * d / a^2, while when upwind fluxes are used, the schemes are stable if tau <= max{tau_0 * d / a^2, c_0 * h / a}. Here, tau is the time-step size, h is the spatial mesh size, and a and d are constants for the advection and diffusion coefficients, respectively. The constant c_0 is the CFL constant for the explicit RK method for the purely advection equation, and tau_0 is a constant that depends on the order of the ETD-RK method. These stability conditions are consistent with those of the implicit-explicit RKDG method. The time-step constraints are rigorously proved for the lowest-order case and are validated through Fourier analysis for higher-order cases. Notably, the constant tau_0 in the fully discrete ETD-RKDG schemes appears to be determined by the stability condition of their semi-discrete (continuous in space, discrete in time) ETD-RK counterparts and is insensitive to the polynomial degree and the specific choice of the DG method. Numerical examples, including problems with nonlinear convection in one and two dimensions, are provided to validate our findings.

翻译：本文研究了在时间上采用指数时间差分（ETD）龙格-库塔（RK）方法、在空间上采用间断伽辽金（DG）方法求解对流-扩散方程时的稳定性与时间步长约束。我们证明，当时间步长上界为一个常数时，所得的全离散格式是稳定的。具体而言，当对流项采用中心通量时，格式在时间步长约束条件 τ ≤ τ₀ * d / a² 下稳定；而当采用迎风通量时，格式在 τ ≤ max{τ₀ * d / a², c₀ * h / a} 条件下稳定。其中，τ 为时间步长，h 为空间网格尺寸，a 和 d 分别为对流系数和扩散系数常数。常数 c₀ 为纯对流方程显式 RK 方法的 CFL 常数，τ₀ 为依赖于 ETD-RK 方法阶数的常数。这些稳定性条件与隐式-显式 RKDG 方法的一致。时间步长约束在最低阶情形下得到了严格证明，并通过傅里叶分析对高阶情形进行了验证。值得注意的是，全离散 ETD-RKDG 格式中的常数 τ₀ 似乎由其半离散（空间连续、时间离散）ETD-RK 对应格式的稳定性条件决定，并且对多项式阶数及 DG 方法的具体选择不敏感。文中提供了一维和二维非线性对流问题的数值算例以验证我们的结论。