Error Boundedness of Discontinuous Galerkin Methods with Variable Coefficients

Abstract

For practical applications, the long time behaviour of the error of numerical solutions to time-dependent partial differential equations is very important. Here, we investigate this topic in the context of hyperbolic conservation laws and flux reconstruction schemes, focusing on the schemes in the discontinuous Galerkin spectral element framework. For linear problems with constant coefficients, it is well-known in the literature that the choice of the numerical flux (e.g. central or upwind) and the selection of the polynomial basis (e.g. Gauß–Legendre or Gauß–Lobatto–Legendre) affects both the growth rate and the asymptotic value of the error. Here, we extend these investigations of the long time error to variable coefficients using both Gauß–Lobatto–Legendre and Gauß–Legendre nodes as well as several numerical fluxes. We derive conditions guaranteeing that the errors are still bounded in time. Furthermore, we analyse the error behaviour under these conditions and demonstrate in several numerical tests similarities to the case of constant coefficients. However, if these conditions are violated, the error shows a completely different behaviour. Indeed, by applying central numerical fluxes, the error increases without upper bound while upwind numerical fluxes can still result in uniformly bounded numerical errors. An explanation for this phenomenon is given, confirming our analytical investigations.

Appendix

1.1 Technical Explanation of the Investiagtion in Sect. 4

We presented the ideas how to reach (35) from (31). Applying the interpolation operator together with discrete norms results in¹²

$$\begin{aligned} \left\langle \partial _t \mathbb {I}^N(u^k),\varphi ^k \right\rangle&= \left( \partial _t \underline{\mathbb {I}^N(u^k)}, \underline{\varphi }^k \right) _N \nonumber \\&\quad +\,\left\{ \left\langle \partial _t \mathbb {I}^N(u^k), \varphi ^k \right\rangle - \left( \partial _t \underline{\mathbb {I}^N(u^k)}, \underline{\varphi }^k \right) _N \right\} , \end{aligned}$$

(32)

$$\begin{aligned} \frac{1}{2}\left\langle a^k \mathbb {I}^N(u^k), \partial _\xi \varphi ^k\right\rangle&=\frac{1}{2} \left( \underline{\underline{a^k}} \underline{\mathbb {I}^N(u^k)} , \partial _\xi \underline{\varphi }^k\right) _N \nonumber \\&\quad +\frac{1}{2} \left\{ \left\langle a^k \mathbb {I}^N(u^k), \partial _\xi \varphi ^k\right\rangle - \left( \underline{\underline{a^k}} \underline{\mathbb {I}^N(u^k)} , \partial _\xi \underline{\varphi }^k\right) _N \right\} , \end{aligned}$$

(63)

$$\begin{aligned} \frac{1}{2}\left\langle a^k \partial _\xi \mathbb {I}^N(u^k),\varphi ^k\right\rangle&= \frac{1}{2} \left( \underline{\underline{a^k}} \partial _\xi \underline{\mathbb {I}^N(u^k) } , \underline{\varphi }^k \right) _N\nonumber \\&\quad +\frac{1}{2} \left\{ \left\langle a^k \partial _\xi \mathbb {I}^N(u^k) , \varphi ^k \right\rangle - \left( \underline{\underline{a^k}} \partial _\xi \underline{\mathbb {I}^N(u^k)} , \underline{\varphi }^k \right) _N \right\} , \end{aligned}$$

(64)

$$\begin{aligned} \frac{1}{2}\left\langle \mathbb {I}^N(u^k) \partial _\xi a^k, \varphi ^k \right\rangle&= \frac{1}{2} \left( \underline{\underline{\mathbb {I}^N(u^k)}} \partial _\xi \underline{a}^k , \underline{\varphi }^k \right) _N\nonumber \\&\quad +\frac{1}{2} \left\{ \left\langle \mathbb {I}^N(u^k) \partial _\xi a^k,\varphi ^k \right\rangle - \left( \underline{\underline{\mathbb {I}^N(u^k)}} \partial _\xi \underline{a}^k ,\underline{\varphi }^k \right) _N \right\} . \end{aligned}$$

(65)

It is well known [5, Section 5.4.3] that the integration error arising from the use of Gauß quadrature (Gauß–Legendre and Gauß–Lobatto–Legendre) decays spectrally fast. Indeed, for all $\varphi \in \mathbb {P}^N$ and $m\ge 1$,

$$\begin{aligned} \left| \left\langle u,\varphi \right\rangle -(\underline{u},\underline{\varphi })_N \right| \le C N^{-m}|u|_{H^{m,N-1}(-1,1)}||\varphi ||_{\mathbf{L}^2(-1,1)}, \end{aligned}$$

where C is a constant independent of m and u. The curly brackets of (32), (63)–(65) have to be reformulated. Using

$$\begin{aligned} \left\langle \partial _t \mathbb {I}^N(u^k), \varphi ^k \right\rangle - \left( \partial _t \underline{\mathbb {I}^N(u^k)}, \underline{\varphi }^k \right) _N= & {} \left\langle \underbrace{ \partial _t \left( \mathbb {I}^N(u^k)-P^m_{N-1} \left( \mathbb {I}^N(u^k)\right) \right) }_{=:Q(u^k)}, \varphi ^k \right\rangle \nonumber \\&- \left( \partial _t \left( \underline{\mathbb {I}^N(u^k)}-\underline{P^m_{N-1} \left( \mathbb {I}^N(u^k)\right) } \right) , \underline{\varphi }^k \right) _N , \end{aligned}$$

(66)

where $P^m_{N-1}$ is the orthogonal projection of u onto $\mathbb {P}^{N-1}$ using the inner product of $H^m(e^k)$, gives a new formulation for (32). The projection operator is defined by the classical truncated Fourier series $P^{N-1}u=\sum _{k=0}^{N-1} \hat{u}_k \varPhi _k$ up to order $N-1$ where Sobolev type orthogonal polynomials $\{\varPhi _k \}$ are used as basis functions in the Hilbert space $H^m(e^k)$. The coefficients are calculated using the inner product of $H^m(e^k)$ given by

$$\begin{aligned} \left\langle u,v\right\rangle _m=\sum _{k=0}^m \int _{e_k} \frac{{\text {d}}^k u}{{\text {d}}x^k}(x)\frac{{\text {d}}^k v}{{\text {d}}x^k}(x) {\text {d}}x. \end{aligned}$$

For more details about the projection operator and about approximation results, we strongly recommend [5, Section 5] and also [2, 3]. An analogous approach as (66) leads to terms with $Q_1$ for (63), $Q_2$ for (64) and $Q_3$ for (65). The $Q_j$ measure the projection error of a polynomial of degree N to a polynomial of degree $N-1$. Since u and a are bounded, also these values have to be bounded. This values can be introduced and finally one obtains (35).

Later, in this section the error of the fluxes hase to be calulated. We obtain for the left and right boundary:

$$\begin{aligned} \text {left:} \quad -\mathbf{E}_L^1 \left( f^{\mathrm {num},1}_L - \frac{1}{2} a_L^1\mathbf{E}_L^1 \right)= & {} -\mathbf{E}_L^1 \left( \left( a_L^1\frac{0+\mathbf{E}_L^1 }{2} - \sigma a_L^1 \frac{\mathbf{E}_L^1 }{2} \right) -\frac{a_L^1 \mathbf{E}_L^1 }{2} \right) \\= & {} \frac{\sigma a_L^1}{2}\left( \mathbf{E}_L^1\right) ^2, \\ \text {right:}\quad \mathbf{E}_R^K \left( f^{\mathrm {num},K}_R - \frac{1}{2} a_R^K\mathbf{E}^K_R \right)= & {} \mathbf{E}^K_R \left( \left( a_R^K\frac{0+\mathbf{E}^K_R }{2} +\frac{1}{2} \sigma a_R^K \mathbf{E}^K_R \right) -\frac{\mathbf{E}^K_R a_R^K}{2}\right) \\= & {} \frac{\sigma a_R^K}{2} \left( \mathbf{E}^K_R \right) ^2. \end{aligned}$$

1.2 Technical Steps of the Development in Sect. 5

Here, we are presenting the main steps to reach (48).

$$\begin{aligned}&\frac{\varDelta x_k}{2} \left\langle \partial _t \mathbb {I}^N(u^k), \varphi ^k \right\rangle +\frac{1}{2} \Bigg ( a^k \mathbb {I}^N(u^k) \varphi ^k \Bigg |_{-1}^1 - \left\langle a^k \mathbb {I}^N(u^k),\partial _\xi \varphi ^k \right\rangle + \left\langle \partial _\xi \mathbb {I}^N(u^k), a^k \varphi ^k \right\rangle \\&\quad + {\left\langle \partial _\xi a^k , \varphi ^k \mathbb {I}^N(u^k)\right\rangle } \Bigg ) = -\frac{\varDelta x_k}{2} \left\langle \partial _t \varepsilon _p^k, \varphi ^k \right\rangle -\frac{1}{2} \left( a^k \varepsilon _p^k \varphi ^k \Bigg |_{-1}^1 \right) +\frac{1}{2} \left\langle a^k \varepsilon _p^k, \partial _\xi \varphi ^k \right\rangle \\&\quad -\frac{1}{2} \left\langle \partial _\xi \varepsilon _p^k, a^k \varphi ^k \right\rangle -\frac{1}{2} \left\langle \varphi ^k \varepsilon _p^k, \partial _\xi a^k \right\rangle . \end{aligned}$$

Integration-by-parts yields

$$\begin{aligned} -\frac{1}{2} \left( a^k \varepsilon _p^k \varphi ^k \Bigg |_{-1}^1 -\left\langle a^k\varepsilon _p^k, \partial _\xi \varphi ^k \right\rangle \right) = -\frac{1}{2}\left\langle \partial _\xi (a^k \varepsilon _p^k), \varphi ^k \right\rangle . \end{aligned}$$

With (32),(63)– (65), one obtains

$$\begin{aligned}&\frac{\varDelta x_k}{2} \left( \partial _t \underline{\mathbb {I}^N (u^k)}, \underline{\varphi }^k \right) _N +\underline{\varphi }^{k,T} \underline{\underline{R}}^{T} \underline{\underline{B}} \left( \underline{f}^{\mathrm {num},k}\left( \mathbb {I}^N(u^k)^-, \mathbb {I}^N(u^k)^+\right) - \frac{1}{2} \left( \underline{\underline{R}}\underline{a^k} \right) \cdot \left( \underline{\underline{R}} \underline{u} \right) \right) \nonumber \\&\qquad +\,\underbrace{\left( \frac{1}{2} a^k \mathbb {I}^N(u^k)\varphi ^k\Bigg |_{-1}^1 - \underline{\varphi }^{k,T} \underline{\underline{R}}^{T} \underline{\underline{B}} \left( \underline{f}^{\mathrm {num},k}\left( \mathbb {I}^N(u^k)^-, \mathbb {I}^N(u^k)^+\right) - \frac{1}{2} \left( \underline{\underline{R}}\underline{a^k} \right) \cdot \left( \underline{\underline{R}} \underline{u} \right) \right) \right) }_{=:\varepsilon _2^k(a^k)}\nonumber \\&\qquad - \frac{1}{2} \left( \underline{\underline{a^k}} \underline{\mathbb {I}^N(u^k)} , \partial _\xi \underline{\varphi }^k \right) _N +\frac{1}{2} \left( \partial _\xi \underline{\mathbb {I}^N(u^k)}, \underline{\underline{a^k}} \underline{\varphi }^k \right) _N +\frac{1}{2} \left( \partial _\xi \underline{a}^k , \underline{\underline{\mathbb {I}^N(u^k)}} \underline{\varphi }^k \right) _N \nonumber \\&\quad = \frac{\varDelta x_k}{2} \left\langle \hat{T}^k(u^k) , \varphi ^k\right\rangle +\frac{\varDelta x_k}{4} \left\langle Q_1(u^k), \partial _x \varphi ^k \right\rangle \nonumber \\&\qquad +\frac{\varDelta x_k}{4} \Big \{\left( \underline{Q(u^k)}, \underline{\varphi }^k \right) _N - \left( \underline{Q_1(u^k)}, \partial _x \underline{\varphi }^k \right) _N + { \left( \underline{Q_2(u^k)}, \underline{\underline{a}}^k \underline{\varphi }^k \right) _N } \nonumber \\&\qquad + \left( \partial _x \underline{a}^k, \underline{\underline{Q_3(u^k)}}\underline{\varphi }^k \right) _N \Big \} \end{aligned}$$

(47)

with

$$\begin{aligned} \hat{T}^(u^k)&:= -\Bigg \{\partial _t \varepsilon _p^k+\frac{1}{2} \left( \partial _x \left( a^k \varepsilon _p^k\right) +\varepsilon _p^k \partial _x a^k \partial _x \varepsilon _p^k \right) \\&\qquad +\frac{1}{2} \left( Q(u^k) +a^kQ_2(u^k) +Q_3(u^k) \partial _x a^k\right) \Bigg \}. \end{aligned}$$

We transposed every term in (29) and subtracted it from equation (47). Using $\varepsilon _1^k=\mathbb {I}^N(u^k)-U^k$ yields

$$\begin{aligned}&\frac{\varDelta x_k}{2} \left( \partial _t \underline{\varepsilon }_1^k, \underline{\varphi }^k \right) +\underline{\varphi }^{k,T} \underline{\underline{R}}^{T} \underline{\underline{B}} \left( \underline{f}^{\mathrm {num},k}\left( (\varepsilon _1^k)^-, (\varepsilon _1^k)^+\right) - \frac{1}{2} \left( \underline{\underline{R}}\underline{a^k} \right) \cdot \left( \underline{\underline{R}} \varepsilon _1^k \right) \right) \\&\qquad +\,\varepsilon _2^k(a^k)- \frac{1}{2} \left( \underline{\underline{a^k}} \underline{\varepsilon }_1^k , \partial _\xi \underline{\varphi }^k \right) _N +\frac{1}{2} \left( \partial _\xi \underline{\varepsilon }_1^k , \underline{\underline{a^k}} \underline{\varphi }^k \right) _N +\frac{1}{2} \left( \partial _\xi \underline{a}^k , \underline{\underline{\varepsilon }}_1^k \underline{\varphi }^k \right) _N\\&\quad = \frac{\varDelta x_k}{2} \left\langle \hat{T}^k(u^k) , \varphi ^k\right\rangle +\frac{\varDelta x_k}{4} \left\langle Q_1(u^k), \partial _x \varphi ^k \right\rangle +\frac{\varDelta x_k}{4} \Big \{\left( \underline{Q(u^k)}, \underline{\varphi }^k \right) _N \\&\qquad -\, \left( \underline{Q_1(u^k)}, \partial _x \underline{\varphi }^k \right) _N + {\left( \underline{Q_2(u^k)}, \underline{\underline{a}}^k \underline{\varphi }^k \right) _N } + \left( \partial _x \underline{a}^k, \underline{\underline{Q_3(u^k)}}\underline{\varphi }^k \right) _N \Big \} . \end{aligned}$$

Putting $\varphi ^k=\varepsilon _1^k$ results in the energy equation similar to (37):

$$\begin{aligned}&\frac{\varDelta x_k}{4} \frac{{\text {d}}}{{\text {d}}t}||\varepsilon _1^k||_N^2 + \underline{\varepsilon }_1^{k,T} \underline{\underline{R}}^{T} \underline{\underline{B}} \left( \underline{f}^{\mathrm {num},k}\left( (\varepsilon _1^k)^-, (\varepsilon _1^k)^+\right) - \frac{1}{2} \left( \underline{\underline{R}}\underline{a^k} \right) \cdot \left( \underline{\underline{R}} \underline{\varepsilon }_1^k \right) \right) \\&\qquad +\,\varepsilon _2^k(a^k)- \frac{1}{2} \left( \underline{\underline{a^k}} \underline{\varepsilon }_1^k , \partial _\xi \underline{\varepsilon }_1^k \right) _N +\frac{1}{2} \left( \partial _\xi \underline{\varepsilon }_1^k , \underline{\underline{a^k}} \underline{\varepsilon }_1^k \right) _N +\frac{1}{2} \left( \partial _\xi \underline{a}^k , \underline{\underline{\varepsilon }}_1^k \underline{\varepsilon }_1^k \right) _N\\&\quad = \frac{\varDelta x_k}{2} \left\langle \hat{T}^k(u^k) , \varepsilon _1^k\right\rangle +\frac{\varDelta x_k}{4} \left\langle Q_1(u^k), \partial _x \varepsilon _1^k \right\rangle \\&\qquad +\,\underbrace{\frac{\varDelta x_k}{4} \Big \{ \left( \underline{Q(u^k)}, \underline{\varepsilon }_1^k \right) _N -\left( \underline{Q_1(u^k)}, \partial _x \underline{\varepsilon }_1^k \right) _N +{ \left( \underline{Q_2(u^k)}, \underline{\underline{a}}^k\underline{\varepsilon }_1^k \right) _N } +\left( \partial _x \underline{a}^k, \underline{\underline{Q_3(u^k)}} \underline{\varepsilon }_1^k \right) _N\Big \}}_{\hat{Q}^k}. \end{aligned}$$

Together with (38), one obtains

$$\begin{aligned}&\frac{\varDelta x_k}{4} \frac{{\text {d}}}{{\text {d}}t}||\varepsilon _1^k||_N^2 +\underline{\varepsilon }_1^{k,T} \underline{\underline{R}}^{T} \underline{\underline{B}} \left( \underline{f}^{\mathrm {num},k}\left( (\varepsilon _1^k)^-, (\varepsilon _1^k)^+\right) -\frac{1}{2} \left( \underline{\underline{R}}\underline{a^k} \right) \cdot \left( \underline{\underline{R}} \underline{\varepsilon }_1^k \right) \right) \nonumber \\&\quad +\,\varepsilon _2^k(a^k) +\frac{1}{2} \left( \partial _\xi \underline{a}^k , \underline{\underline{\varepsilon }}_1^k \underline{\varepsilon }_1^k \right) _N = \frac{\varDelta x_k}{2} \left\langle \hat{T}^k(u^k) , \varepsilon _1^k\right\rangle +\frac{\varDelta x_k}{4} \left\langle Q_1(u^k), \partial _x \varepsilon _1^k \right\rangle +\hat{Q}^k. \end{aligned}$$

(67)

Summing this up over all elements results in

$$\begin{aligned}&\frac{1}{2} \frac{{\text {d}}}{{\text {d}}t} \sum _{k=1}^K \frac{\varDelta x_k}{2}||\varepsilon _1^k||_N^2+\sum _{k=1}^K \underline{\varepsilon }_1^{k,T} \underline{\underline{R}}^{T} \underline{\underline{B}} \Bigg (\underline{f}^{\mathrm {num},k}\left( (\varepsilon _1^k )^{-},(\varepsilon _1^k )^{+}\right) \\&\qquad - \frac{1}{2 } \left( \underline{\underline{R}}\underline{a^k} \right) \left( \underline{\underline{R}} \underline{\varepsilon }_1^k \right) \Bigg ) +\frac{1}{2} \sum _{k=1}^K \frac{\varDelta x_k}{2} {\left( \partial _x \underline{a}^k , \underline{\underline{\varepsilon }}_1^k \underline{\varepsilon }_1^k \right) _N } +\sum _{k=1}^K \frac{\varDelta x_k}{4} \varepsilon _2^k(a^k) \\&\quad = \sum _{k=1}^K\frac{\varDelta x_k}{2} \left\langle \hat{T}^k(u^k), \varepsilon _1^k\right\rangle + \sum _{k=1}^K\frac{\varDelta x_k}{4} \left\langle Q_1(u^k), \partial _x \varepsilon _1^k\right\rangle + \sum _{k=1}^K \hat{Q}^k . \end{aligned}$$

Applying the same approach like in Eqs. (41)–(42) and the fact that $\varepsilon _1 \in \mathbb {P}^N$, it is$||\partial _x \underline{\varepsilon }_1^k||_N^2 \le c_1 N^2 ||\underline{\varepsilon }_1^k||_N^2$ and we get finally (48).

1.2.1 Calculating the Fluxes from Table 2

• Split central flux $f^{\mathrm {num}}(u_-, u_+)= \frac{a_-u_-+a_+u_+}{2}$: One obtains

  $$\begin{aligned}&\frac{1}{2}\left( a_R^{k-1}\mathbf{E}^{k-1}_R+a_L^{k}\mathbf{E}^k_L \right) \left( \mathbf{E}^{k-1}_R-\mathbf{E}_L^k \right) -\frac{1}{2} \left( a_R^{k-1}\left( \mathbf{E}_R^{k-1 } \right) ^2 -a_L^k \left( \mathbf{E}^k_L\right) ^2 \right) \\&\quad = \frac{1}{2} \left( a_R^{k-1} \left( \mathbf{E}_R^k\right) ^2-a_R^{k-1}\mathbf{E}_L^k \mathbf{E}^{k-1}_R + a_L^k \mathbf{E}^k_L \mathbf{E}^{k-1}_R-a_L^k \left( \mathbf{E}_L^k \right) ^2 \right) \\&\qquad -\frac{1}{2} \left( a_R^{k-1}\left( \mathbf{E}_R^{k-1 } \right) ^2 -a_L^k \left( \mathbf{E}^k_L \right) ^2 \right) = \frac{1}{2} \mathbf{E}_L^k \mathbf{E}_R^{k-1} \left( a_L^k-a_R^{k-1} \right) = 0 \end{aligned}$$

  and

  $$\begin{aligned} \begin{array}{ll} \text { left:} &{} -\mathbf{E}_L^1 \left( f^{\mathrm {num},1}_L -\frac{1}{2}a_L^1\mathbf{E}_L^1 \right) = -\mathbf{E}_L^1 \left( \frac{a^1_L}{2}\mathbf{E}_L^1-\frac{a_L^1}{2} \mathbf{E}_L^1 \right) =0, \\ \text { right:} &{} \mathbf{E}^K_R \left( f^{\mathrm {num},K}_R -\frac{1}{2}a_R^K \mathbf{E}^K_R \right) = \frac{1}{2} \left( \mathbf{E}_R^K \right) ^2 \left( a^K_R-a_R^K \right) =0 . \end{array} \end{aligned}$$

• Edge based upwind flux $f^{\mathrm {num}}(u_-,u_-)=a(x)u_-$: It is

  

  and

  $$\begin{aligned} \begin{array}{ll} \text { left:}&{} \ -\mathbf{E}_L^1 \left( f^{\mathrm {num},1}_L -\frac{1}{2}a_L^1\mathbf{E}_L^1 \right) =\frac{1}{2} \left( \mathbf{E}^1_L \right) a_L^1, \\ \text { right:}&{} \ \mathbf{E}^K_R \left( f^{\mathrm {num},K}_R -\frac{1}{2}a_R^K \mathbf{E}^K_R \right) =\left( \mathbf{E}^k_R \right) ^2 \left( a^K(x_R)-\frac{a_R^{K}}{2} \right) = \left( \mathbf{E}^k_R \right) ^2 \left( \frac{a_R^{K}}{2} \right) . \end{array} \end{aligned}$$

• Split upwind flux $f^{\mathrm {num}}(u_-,u_-)=a_-u_-$: It is

  

  where we used in the last step the assumption about the exactness of the interpolation and the continuity of a. At the boundaries we get

  $$\begin{aligned}&\text { left:}\qquad \frac{a_L^1 }{2} \left( \mathbf{E}^1_L\right) ^2,\\&\text { right:}\qquad \frac{a_R^K }{2} \left( \mathbf{E}^K_R\right) ^2. \end{aligned}$$