The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Diffusion Problems

Abstract

In this paper, a fully discrete stability analysis is carried out for the direct discontinuous Galerkin (DDG) methods coupled with Runge-Kutta-type implicit-explicit time marching, for solving one-dimensional linear convection-diffusion problems. In the spatial discretization, both the original DDG methods and the refined DDG methods with interface corrections are considered. In the time discretization, the convection term is treated explicitly and the diffusion term implicitly. By the energy method, we show that the corresponding fully discrete schemes are unconditionally stable, in the sense that the time-step $\tau$ is only required to be upper bounded by a constant which is independent of the mesh size h. Optimal error estimate is also obtained by the aid of a special global projection. Numerical experiments are given to verify the stability and accuracy of the proposed schemes.

Appendix A

Proof of Lemma 5

We only show the proof for the original DDG operator. The proof for the refined DDG operator is the similar. By the definition of $\mathcal {D}(\cdot ,\cdot )$ and $\widehat{u_x}$ (5), we get

$$\begin{aligned} \mathcal {D}(u,v)-\mathcal {D}(v,u)=&\,d\sum _{j=1}^N\left[ \{u_x\}_{j+\frac{1}{2}} + \sum _{m=1}^{\lfloor k/2\rfloor } \beta _m {\tilde{h}_j}^{2m-1} [\partial ^{2m}_x u]_{j+\frac{1}{2}} \right] [v]_{j+\frac{1}{2}} \\&\, -d\sum _{j=1}^N\left[ \{v_x\}_{j+\frac{1}{2}} + \sum _{m=1}^{\lfloor k/2\rfloor } \beta _m {\tilde{h}_j}^{2m-1} [\partial ^{2m}_x v]_{j+\frac{1}{2}} \right] [u]_{j+\frac{1}{2}}. \end{aligned}$$

Using the Cauchy-Schwarz inequality and the Young’s inequality, we have

$$\begin{aligned} |\mathcal {D}(u,v)-\mathcal {D}(v,u)|\leqslant&\, d\varepsilon (h^{-1}|\![u]\!|^2+h^{-1}|\![v]\!|^2)\\&\,+d\frac{h}{4\varepsilon } \sum _{j=1}^N\left[ \{u_x\}_{j+\frac{1}{2}} + \sum _{m=1}^{\lfloor k/2\rfloor } \beta _m {\tilde{h}_j}^{2m-1} [\partial ^{2m}_x u]_{j+\frac{1}{2}} \right] ^2\\&\, + d\frac{h}{4\varepsilon } \sum _{j=1}^N\left[ \{v_x\}_{j+\frac{1}{2}} + \sum _{m=1}^{\lfloor k/2\rfloor } \beta _m {\tilde{h}_j}^{2m-1} [\partial ^{2m}_x v]_{j+\frac{1}{2}} \right] ^2 \end{aligned}$$

for arbitrary positive $\varepsilon$. By the aid of the inverse inequalities (11) and (12), we get

$$\begin{aligned} |\mathcal {D}(u,v)-\mathcal {D}(v,u)|\leqslant&\, d\varepsilon (h^{-1}|\![u]\!|^2+h^{-1}|\![v]\!|^2) +d \mu \frac{\lfloor k/2\rfloor +1}{8\varepsilon }(\Vert u_x\Vert ^2+\Vert v_x\Vert ^2)\\&\,+d\mu \frac{\lfloor k/2\rfloor +1}{2\varepsilon } \sum _{m=1}^{\lfloor k/2\rfloor } \beta _m^2 h^{4m-2} (\Vert \partial ^{2m}_x u\Vert ^2+\Vert \partial ^{2m}_x v\Vert ^2)\\ \leqslant&\,d\varepsilon (h^{-1}|\![u]\!|^2+h^{-1}|\![v]\!|^2)\\&\,+ \frac{\lfloor k/2\rfloor +1}{8\varepsilon }d\mu \left[ 1+ 4\mu ^2\sum _{m=1}^{\lfloor k/2\rfloor } \beta _m^2 \right] (\Vert u_x\Vert ^2+\Vert v_x\Vert ^2). \end{aligned}$$

Thus, by taking $\varepsilon =\sqrt{\frac{\lfloor k/2\rfloor +1}{8}\left[ 1+ 4\mu ^2\sum\limits_{m=1}^{\lfloor k/2\rfloor } \beta _m^2 \right] }=C_{\mu }$, we get the desired result.