Implicit–Explicit Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Convection–Diffusion Problems

Abstract

Local discontinuous Galerkin methods with generalized alternating numerical fluxes coupled with implicit–explicit time marching for solving convection–diffusion problems is analyzed in this paper, where the explicit part is treated by a strong-stability-preserving Runge–Kutta scheme, and the implicit part is treated by an L-stable diagonally implicit Runge–Kutta method. Based on the generalized alternating numerical flux, we establish the important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient, which plays a key role in obtaining the unconditional stability of the proposed schemes. Also by the aid of the generalized Gauss–Radau projection, optimal error estimates can be shown. Numerical experiments are given to verify the stability and accuracy of the proposed schemes with different numerical fluxes.

7 Appendix

Proof of (4.7)

Firstly, by Taylor’s expansion

$$\begin{aligned} U^{n+1}=U^n + \tau U_t^n +\frac{\tau ^2}{2}U_{tt}^n + \frac{\tau ^3}{6}U_{ttt}(t_{\xi }), \end{aligned}$$

where $t_{\xi }\in (t^n,t^{n+1})$ and we omit the argument x for simplicity. Secondly, notice that

$$\begin{aligned} U^{n,1}&=\,U^n +\gamma \tau \sqrt{d}Q_x^{n,1},\\ U^{n,2}&=\,U^n + \tau U_t^{n,1} -2\gamma \tau \sqrt{d}Q_{x}^{n,1} + \gamma \tau \sqrt{d}Q_{x}^{n,2}, \end{aligned}$$

and then

$$\begin{aligned}&\,U^{n}-\frac{\tau }{2}c(U^{n,1}_x+U^{n,2}_x)+\frac{\tau }{2}\sqrt{d}(Q^{n,1}_x+Q^{n,2}_x) \\&\quad =\,U^n + \frac{\tau }{2}U_t^{n,1}+\frac{\tau }{2}U_t^{n,2} \\&\quad =\,U^n + \tau U_t^n +\frac{\tau ^2}{2}U_{tt}^{n,1}+\frac{\gamma \tau ^2}{2}(Q_{xt}^{n,2}-Q_{xt}^{n,1}) \\&\quad =\,U^n + \tau U_t^n +\frac{\tau ^2}{2}U_{tt}^{n}+\frac{\gamma \tau ^3}{2}\sqrt{d}Q_{xtt}^{n,1} +\frac{\gamma \tau ^2}{2} \sqrt{d}(Q_{xt}^{n,2}-Q_{xt}^{n,1}). \end{aligned}$$

As a result

$$\begin{aligned} \zeta ^n&=\,\frac{\tau ^3}{6}U_{ttt}(x,t_{\xi })-\frac{\gamma \tau ^3}{2}\sqrt{d}Q_{xtt}^{n,1} -\frac{\gamma \tau ^2}{2}\sqrt{d}(Q_{xt}^{n,2}-Q_{xt}^{n,1}) \\&=\, \frac{\tau ^3}{6}U_{ttt}(x,t_{\xi })-\frac{\gamma \tau ^3}{2}( U_{ttt}^{n,1}+cU_{xtt}^{n,1}) -\frac{\gamma \tau ^2}{2} [(U^{n,2}-U^{n,1})_{tt}+c(U^{n,2}-U^{n,1})_{xt}). \end{aligned}$$

Since

$$\begin{aligned} U^{n,2}-U^{n,1}&=\, \tau U_t^{n,1} -3\gamma \tau \sqrt{d}Q_x^{n,1} +\gamma \tau \sqrt{d}Q_x^{n,2} \\&=\, \tau U_t^{n,1} -3\gamma \tau (U_t^{n,1}+cU_x^{n,1})+\gamma \tau (U_t^{n,2}+cU_x^{n,2}). \end{aligned}$$

We get $\Vert \zeta ^n\Vert = {\mathcal {O}}(\tau ^3)$ if $U_{ttt},U_{xtt},U_{xxt}\in L^{\infty }(0,T;L^2)$.

Proof of Lemma 4.2

By taking $v=2\xi _u^{n,1}$ in (4.12a) and $v=2\xi _u^{n,2}$ in (4.12b), we get from (2.10) and (4.13b) that

$$\begin{aligned} \Vert \xi _u^{n,1}\Vert ^2+\Vert \xi _u^{n,1}-\xi _u^n\Vert ^2 - \Vert \xi _u^n\Vert ^2&=\,-2\gamma \tau \Vert \xi _q^{n,1}\Vert ^2 +2(\eta _u^{n,1}-\eta _u^n,\xi _u^{n,1}), \end{aligned}$$

(7.1)

$$\begin{aligned} \Vert \xi _u^{n,2}\Vert ^2+\Vert \xi _u^{n,2}-\xi _u^n\Vert ^2 - \Vert \xi _u^n\Vert ^2&=\,2\tau {\mathcal {H}}(\xi _u^{n,1},\xi _u^{n,2}) +2(\eta _u^{n,2}-\eta _u^n,\xi _u^{n,2}) \nonumber \\&\quad -2(1-2\gamma )\tau (\xi _q^{n,1},\xi _q^{n,2}) -2\gamma \tau \Vert \xi _q^{n,2}\Vert ^2 \nonumber \\&\quad +2(1-2\gamma )\tau (\eta _q^{n,1},\xi _q^{n,2}) +2\gamma \tau (\eta _q^{n,2},\xi _q^{n,2}). \end{aligned}$$

(7.2)

Hence applying Young’s inequality yields

$$\begin{aligned} \Vert \xi _u^{n,1}\Vert ^2 \le 2( \Vert \xi _u^n\Vert ^2 - 2\gamma \tau \Vert \xi _q^{n,1}\Vert ^2) + C h^{2k+2}\tau ^2. \end{aligned}$$

(7.3)

Using (4.16) for the term $2\tau {\mathcal {H}}(\xi _u^{n,1},\xi _u^{n,2})$, applying Cauchy–Schwarz inequality and Young’s inequality for the remaining terms, we get

$$\begin{aligned} \Vert \xi _u^{n,2}\Vert ^2&\le \,\Vert \xi _u^n\Vert ^2 + C \frac{c}{\sqrt{d}} \tau (\Vert \xi _q^{n,2}\Vert +h^{k+1})\Vert \xi _u^{n,1}\Vert +C h^{k+1}\tau \Vert \xi _u^{n,2}\Vert \nonumber \\&\quad -2(1-2\gamma )\tau (\xi _q^{n,1},\xi _q^{n,2})-2\gamma \tau \Vert \xi _q^{n,2}\Vert ^2 + C h^{k+1}\tau \Vert \xi _q^{n,2}\Vert \nonumber \\&\le \, \Vert \xi _u^n\Vert ^2 +\Vert \xi _u^{n,1}\Vert ^2 + \frac{C c^2\tau }{2d}\tau \Vert \xi _q^{n,2}\Vert ^2 + \frac{1}{2}\Vert \xi _u^{n,2}\Vert ^2+Ch^{2k+2}\tau \nonumber \\&\quad -2(1-2\gamma )\tau (\xi _q^{n,1},\xi _q^{n,2})-(2\gamma -\varepsilon ) \tau \Vert \xi _q^{n,2}\Vert ^2. \end{aligned}$$

(7.4)

So taking $\varepsilon =\frac{\gamma }{2}$ and letting $\frac{C c^2\tau }{2d} \le \frac{\gamma }{2}$, we get

$$\begin{aligned} \Vert \xi _u^{n,2}\Vert ^2&\le \, 6\Vert \xi _u^n\Vert ^2 -2\tau [4\gamma \Vert \xi _q^{n,1}\Vert ^2+2(1-2\gamma )(\xi _q^{n,1},\xi _q^{n,2}) +\gamma \Vert \xi _q^{n,2}\Vert ^2]+ Ch^{2k+2}\tau \nonumber \\&\le \, 6\Vert \xi _u^{n}\Vert ^2+Ch^{2k+2}\tau , \end{aligned}$$

(7.5)

if $\gamma \in [\gamma _1,\gamma _2]$. Thus the lemma is proved. $\square$