Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems

Abstract

We present an efficient and easy to implement approach to solving the semidiscrete equation systems resulting from time discretization of nonlinear parabolic problems with discontinuous Galerkin methods of order $r$. It is based on applying Newton’s method and decoupling the Newton update equation, which consists of a coupled system of $r+1$ elliptic problems. In order to avoid complex coefficients which arise inevitably in the equations obtained by a direct decoupling, we decouple not the exact Newton update equation but a suitable approximation. The resulting solution scheme is shown to possess fast linear convergence and consists of several steps with same structure as implicit Euler steps. We construct concrete realizations for order one to three and give numerical evidence that the required computing time is reduced significantly compared to assembling and solving the complete coupled system by Newton’s method.

Appendix A: Complete time stepping schemes for dG(2) and dG(3)

1.1 A.1 Scheme for dG(2)

First, the solution component $\widetilde{W}_2^l$ is computed by the three linear solution steps

$$\begin{aligned} \left( M + \root 3 \of {\frac{1}{60}} \, k_m\bar{A} \right) \widetilde{V}_1^l&= R_0^l+R_1^l+R_2^l + \frac{1}{10} k_m\bar{A} M^{-1} \left(-4R_0^l+R_1^l +6R_2^l \right) \nonumber \\&\qquad + \frac{1}{40} k_m^2 \bar{A} M^{-1} \bar{A} M^{-1} \left(2R_0^l-R_1^l+6R_2^l \right), \nonumber \\ \left( M + \root 3 \of {\frac{1}{60}} \, k_m\bar{A} \right) \widetilde{V}_2^l&= M \widetilde{V}_1^l, \\ \left( M + \root 3 \of {\frac{1}{60}} \, k_m\bar{A} \right) \widetilde{W}_2^l&= M \widetilde{V}_2^l,\nonumber \end{aligned}$$

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where $\widetilde{V}_1^l$ and $\widetilde{V}_2^l$ are temporary values. For the other two components, backward substitution gives the linear equations

$$\begin{aligned} \begin{aligned} \left( M +\frac{1}{10}k_m\bar{A} \right) \widetilde{W}_0^l&= \frac{6}{5} R_0^l - \frac{3}{10} R_1^l + \frac{6}{5} R_2^l -\frac{1}{5} \left(M + \frac{1}{2} k_m\bar{A} \right) \widetilde{W}_2^l,\\ \left( M +\frac{1}{10}k_m\bar{A} \right) \widetilde{W}_1^l&= \frac{3}{8} R_1^l-\frac{3}{2} R_2^l +5 M \left(\widetilde{W}_0^l + \widetilde{W}_2^l \right)+ k_m\bar{A} \left(- \frac{3}{4} \widetilde{W}_0^l +\frac{7}{4} \widetilde{W}_2^l \right). \end{aligned} \end{aligned}$$

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1.2 A.2 Scheme for dG(3)

To obtain the solution component $\widetilde{W}_3^l$, we solve the four linear equations

$$\begin{aligned} \left( M + \root 4 \of {\frac{1}{840}} \, k_m\bar{A} \right) \widetilde{V}_1^l&= R_0^l+ R_1^l+ R_2^l + R_3^l \nonumber \\&\quad + \frac{1}{21} k_m\bar{A} M^{-1} \left(-9 R_0^l-2 R_1^l + 5 R_2^l + 12 R_3^l \right) \nonumber \\&\quad + \frac{1}{126} \left(k_m\bar{A} M^{-1} \right)^2 \left(9 R_0^l- 2 R_1^l+ R_2^l + 18 R_3^l \right)\nonumber \\&\quad {+} \frac{1}{5670} \left(k_m\bar{A} M^{-1}\! \right)^3 \left(\! -27 R_0^l +8 R_1^l-17 R_2^l + 108 R_3^l \!\right), \nonumber \\ \left( M + \root 4 \of {\frac{1}{840}} \, k_m\bar{A} \right) \widetilde{V}_2^l&= M \widetilde{V}_1^l, \\ \left( M + \root 4 \of {\frac{1}{840}} \, k_m\bar{A} \right) \widetilde{V}_3^l&= M \widetilde{V}_2^l, \nonumber \\ \left( M + \root 4 \of {\frac{1}{840}} \, k_m\bar{A} \right) \widetilde{W}_3^l&= M \widetilde{V}_3^l, \nonumber \end{aligned}$$

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with temporary variables $\widetilde{V}_1^l$, $\widetilde{V}_2^l$ and $\widetilde{V}_3^l$. The remaining components are given by

$$\begin{aligned} \begin{aligned} \left(M + \frac{2}{13+\sqrt{29}} \, k_m\bar{A} \right) \widetilde{V}^{l}_{4}&= G_0^l,\quad \left(M + \frac{2}{13-\sqrt{29}} \, k_m\bar{A}\right) \widetilde{W}_0^l = M\widetilde{V}_4^l, \\ \left(M + \frac{2}{13-\sqrt{29}} \, k_m\bar{A} \right)\widetilde{W}_1^l&= G_1^l,\quad \left(M + \frac{2}{13-\sqrt{29}} \, k_m\bar{A} \right)\widetilde{W}_2^l = G_2^l \end{aligned} \end{aligned}$$

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with another temporary value $\widetilde{V}_4^l$ and the right hand sides

$$\begin{aligned} G_0^l&:= \frac{26}{35}R_0^l -\frac{628}{945} R_1^l + \frac{142}{945} R_2^l - \frac{44}{35} R_3^l \\&+k_m\bar{A} M^{-1} \left( \frac{18}{35} R_0^l - \frac{4}{45} R_1^l+\frac{22}{315} R_2^l - \frac{12}{35} R_3^l \right) \\&+\frac{9}{35} M \widetilde{W}_3^l + k_m\bar{A} \left( \frac{4}{35} \widetilde{W}_3^l + \frac{1}{70} k_mM^{-1} \bar{A} \widetilde{W}_3^l \right), \\ G_1^l&:= \frac{1844-52 \sqrt{29}}{945} R_0^l + \frac{4568+1256 \sqrt{29}}{25515} R_1^l \\&+ \frac{5548-284 \sqrt{29}}{25515} R_2^l + \frac{-536+88\sqrt{29}}{945} R_3^l \\&+ M \left(\frac{-26+2 \sqrt{29}}{27} \widetilde{W}_0^l + \frac{11-18 \sqrt{29}}{945} \widetilde{W}_3^l \right)\\&+ k_m\bar{A} \left(- \frac{4}{27} \widetilde{W}_0^l + \frac{23-9 \sqrt{29}}{1890} \widetilde{W}_3^l \right),\\ G_2^l&:= \left(\frac{116}{11025} -\frac{2\sqrt{29}}{1225} \right)R_0^l + \left(\frac{116252}{297675}+\frac{1168\sqrt{29}}{99225} \right)R_1^l \\&+ \left(\frac{196822}{297675}+\frac{3548 \sqrt{29}}{99225} \right)R_2^l + \left(-\frac{9404}{11025} + \frac{64 \sqrt{29}}{3675} \right)R_2^l\\&+ M \left(\frac{-257-9\sqrt{29}}{280} \widetilde{W}_1^l + \frac{-6317+2691 \sqrt{29}}{88200} \widetilde{W}_3^l \right)\\&+ k_m\bar{A} \left(\frac{1349+23 \sqrt{29}}{9800} \widetilde{W}_1^l + \frac{-3023+279 \sqrt{29}}{88200} \widetilde{W}_3^l \right). \end{aligned}$$