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

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 $$\tau \le \tau _0 {d}/{a^2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>τ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , while when upwind fluxes are used, the schemes are stable if $$\tau \le \max \left\{ \tau _0 {d}/{a^2}, c_0 h/a\right\} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>≤</mml:mo> <mml:mo>max</mml:mo> <mml:mfenced> <mml:msub> <mml:mi>τ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>d</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>h</mml:mi> <mml:mo>/</mml:mo> <mml:mi>a</mml:mi> </mml:mfenced> </mml:mrow> </mml:math> . Here, $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> 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 $$c_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is the CFL constant for the explicit RK method for the purely advection equation, and $$\tau _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>τ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> 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 $$\tau _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>τ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> in the fully discrete ETD-RK-DG schemes appears to be determined by the stability condition of their semidiscrete (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.