Stability and Time-Step Constraints of Exponential Time Differencing Runge--Kutta Discontinuous Galerkin Methods for Advection-Diffusion Equations

TL;DR

The paper develops and analyzes fully discrete ETD-RKDG schemes for advection-diffusion equations, establishing sharp time-step constraints that ensure stability. Through Fourier (von Neumann) analysis and dimensionless reformulations, it shows that central fluxes yield a stable bound $\tau \leq \tau_0 \frac{d}{a^2}$ while upwind fluxes require $\tau \leq \max\{\tau_0 \frac{d}{a^2}, c_0 \frac{h}{a}\}$, with $\tau_0$ depending on the ETD-RK order and independent of the DG polynomial degree. The key insight is that $\tau_0$ is dictated by the semidiscrete ETD-RK and remains robust across spatial discretizations; higher-order ETD-RKDG schemes achieve stability with the same $\tau_0$ values as their semi-discrete counterparts. Numerical experiments in 1D and 2D, including nonlinear convection, corroborate the theory, demonstrating accurate convergence and large allowable time steps. Overall, the work connects ETD time integration with DG spatial discretization to deliver stable, efficient schemes for advection-diffusion problems, comparable to IMEX approaches while enabling larger time steps.

Abstract

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.

Figures (9)

• Figure 4.1: The square of growth factor $|\widehat{G}(\tau_0,\xi)|^2$ versus $\xi$. Since $|\widehat{G}(\tau_0,\xi)|^2$ is an even function with respect to $\xi$, we present only $\xi>0$. The value of $\tau_0=2$ for ETD-RK1 is sharp. The searched values of $\tau_0$ for ETD-RK2, ETD-RK3, and ETD-RK4 are valid up to the last digit shown. For example, if $\tau_0=3.93$, then $\tau=3.93$ satisfies the stability condition, but $\tau=3.94$ does not.
• Figure 4.2: The square of growth factor $\rho(\widehat{G}(\tau_0,h,\xi))^2$ versus $\xi=\omega h$ for a specific spatial discretization setting: central DG flux for the advection combined with LDG discretization for diffusion, using a $\mathbb{P}^4$ polynomial space and $h = \frac{\pi}{10^{6}}$. The results for other spatial discretization choices are close.
• Figure 5.1: Example \ref{['ex:stability']}. Stability test. The growth of the maximum norm of the solutions for the advection-dominated problem over time using different ETD-RKDG methods. A uniform mesh with $h=\frac{\pi}{1000}$ is employed. The time-step sizes are set to the stable values $\tau=\tau_0\frac{d}{a^2}$ in the left column and the unstable values $\tau=1.1\times\tau_0 \frac{d}{a^2}$ in the right column, where $\tau_0 = 2, 3.93, 4.55$, and $4.81$ for ETD-RK1, ETD-RK2, ETD-RK3, and ETD-RK4, respectively.
• Figure 5.2: Example \ref{['ex:adaptive']}. $h$ variation test. The growth of the maximum norm of the solutions for the advection-dominated problem over time using different ETD-RKDG methods. A nonuniform mesh with $\Delta x_{2m-1}:\Delta x_{2m}=1:9$ for $m=1,2,\ldots,1000$ is employed. The time-step sizes are set to the stable values $\tau=\tau_0\frac{d}{a^2}$ in the left column and the unstable values $\tau=1.1\times\tau_0 \frac{d}{a^2}$ in the right column, where $\tau_0 = 2, 3.93, 4.55$, and $4.81$ for ETD-RK1, ETD-RK2, ETD-RK3, and ETD-RK4, respectively.
• Figure 5.3: Example \ref{['ex:adaptive']}. $p$ variation test. The growth of the maximum norm of the solutions for the advection-dominated problem over time using different ETD-RKDG methods. A uniform mesh with $h=\frac{\pi}{1000}$ is employed. Nonuniform polynomials of degrees $r$ and $k$ are used on the cells $I_{2m-1}$ and $I_{2m}$, respectively, for $m=1,2,\ldots,1000$ in the ETD-RK$r$ methods, where $r=1,2,3,4$ and $k=0,1,\ldots,5$. The time-step sizes are set to the stable values $\tau=\tau_0\frac{d}{a^2}$ in the left column and the unstable values $\tau=1.1\times\tau_0\frac{d}{a^2}$ in the right column, where $\tau_0 = 2, 3.93, 4.55$, and $4.81$ for ETD-RK1, ETD-RK2, ETD-RK3, and ETD-RK4, respectively.

Theorems & Definitions (22)

• Theorem 3.1
• proof
• Corollary 3.1
• Theorem 3.2
• proof
• Corollary 3.2
• Theorem 4.1
• proof
• Remark 4.1
• Example 1