Discontinuous Galerkin method for hyperbolic equations involving $\delta$-singularities: negative-order norm error estimates and applications

Abstract

In this paper, we develop and analyze discontinuous Galerkin (DG) methods to solve hyperbolic equations involving $\delta$-singularities. Negative-order norm error estimates for the accuracy of DG approximations to $\delta$-singularities are investigated. We first consider linear hyperbolic conservation laws in one space dimension with singular initial data. We prove that, by using piecewise $k$th degree polynomials, at time $t$, the error in the $H^{-(k+2)}$ norm over the whole domain is $(k+1/2)$th order, and the error in the $H^{-(k+1)}(\mathbb R \backslash \mathcal R _t)$ norm is $(2k+1)$th order, where $\mathcal R _t$ is the pollution region due to the initial singularity with the width of order $\mathcal O (h^{1/2} \log (1/h))$ and $h$ is the maximum cell length. As an application of the negative-order norm error estimates, we convolve the numerical solution with a suitable kernel which is a linear combination of B-splines, to obtain $L^2$ error estimate of $(2k+1)$th order for the post-processed solution. Moreover, we also obtain high order superconvergence error estimates for linear hyperbolic conservation laws with singular source terms by applying Duhamel’s principle. Numerical examples including an acoustic equation and the nonlinear rendez-vous algorithms are given to demonstrate the good performance of DG methods for solving hyperbolic equations involving $\delta$-singularities.

Appendix A: Proof of Lemma 3.1

In this appendix we prove Lemma 3.1. The main line of proof is based on the idea in [6, 31]. For simplicity, we only consider $\delta$-singularity in equation (1.2), hence $u_0(x)=\delta (x)+f(x)$, where $f(x)$ is sufficiently smooth and has a compact support on the computational domain $\Omega$.

1.1 A.1 The weight function

Let $\varphi (x)$ be a positive bounded function, which can be taken as a weight function. For any function $q\in H^1_h,$ we define the weighted $L^2$-norm as

$$\begin{aligned} \Vert q\Vert _{\varphi ,D}=\left(\,\int _Dq^2\varphi dx\right)^{\frac{1}{2}} \end{aligned}$$

in the domain $D.$ If $\varphi =1$ or $D=\Omega ,$ the corresponding subscript will be omitted.

In this paper, we will consider two weight functions $\varphi ^1(x,t)$ and $\varphi ^{-1}(x,t)$, respectively, in order to determine the left-hand and right-hand boundary of the region $\mathcal R _T$ such that, outside this region, we can resume the $(k+1)$th order accuracy in the $L^2$-norm. Both weight functions are related to the cut-off of the exponent function $\phi (r)\in C^1:\Omega \rightarrow \mathbb R$,

$$\begin{aligned} \phi (r)=\left\{ \begin{array}{l@{\quad }l}2-e^r,&r<0,\\ e^{-r},&r>0, \end{array}\right. \end{aligned}$$

and they are defined as the solutions of the linear hyperbolic problem,

$$\begin{aligned}&\varphi _t^a+\varphi _x^a=0,\end{aligned}$$

(7.1)

$$\begin{aligned}&\varphi ^a(x,0)=\phi \left(\frac{a(x-x_c)}{\gamma h^\sigma }\right), \end{aligned}$$

(7.2)

where $\gamma >0,\,0<\sigma <1$ and $x_c$ are three parameters which will be chosen later. We always assume $\gamma h^{\sigma -1}\ge 1$ in this section.

In [31], the authors have listed several properties about the two weight functions. Here, we state some of them that will be used.

Proposition 7.1

For each of the weight function $\varphi ^a(x,t)$, the following properties hold

$$\begin{aligned}&1\le \varphi ^a(x,t)\le 2,&\quad a(x-x_c-t)\le 0,\end{aligned}$$

(7.3)

$$\begin{aligned}&0<\varphi ^a(x,t)<h^s,&\quad a(x-x_c-t)>s \, \log (1/h)\gamma h^\sigma . \end{aligned}$$

(7.4)

Lemma 7.1

Let $\mathbb V$ be a Gauss–Radau projection, either $\mathbb P _-$ or $\mathbb P _+$. For any sufficiently smooth function $p(x)$, there exists a positive constant $C$ independent of $h$ and $p$, such that

$$\begin{aligned} \Vert \mathbb V ^\bot p\Vert _{\varphi ,D}&\le Ch^{k+1}\Vert \partial _x^{k+1}p\Vert _{\varphi ,D},\end{aligned}$$

(7.5)

$$\begin{aligned} \Vert \mathbb V ^\bot (\varphi v_h)\Vert _{\varphi ^{-1},D}&\le C\gamma ^{-1}h^{1-\sigma }\Vert v_h\Vert _{\varphi ,D},\end{aligned}$$

(7.6)

$$\begin{aligned} \Vert \mathbb V (\varphi v_h)\Vert _{\varphi ^{-1},D}&\le C\Vert v_h\Vert _{\varphi ,D}. \end{aligned}$$

(7.7)

where $D$ is either the single cell $I_j$ or the whole computational domain $\Omega$.

Lemma 7.2

For any function $v\in V_h$ there holds the following identity

$$\begin{aligned} \mathcal H (v,\varphi v)=-\frac{1}{2}\sum _j\varphi _{j+\frac{1}{2}}[v]^2_{j+\frac{1}{2}}+\frac{1}{2}(v,\varphi _x v). \end{aligned}$$

(7.8)

1.2 A.2 The smooth solution

We consider the following problem

$$\begin{aligned} v_t+v_x&= 0,\end{aligned}$$

(7.9)

$$\begin{aligned} v(x,0)&= v_0(x), \end{aligned}$$

(7.10)

where the initial condition $v_0(x)$, is a sufficiently smooth function modified from the original initial condition $u_0(x)=\delta (x)+f(x)$ such that it agrees with $u_0(x)$ for all $x\in \Omega \backslash I_i$, and satisfies

$$\begin{aligned} |\partial _x^\alpha v_0(x)|\le Ch^{-\alpha -1},\quad x\in I_i, \end{aligned}$$

where $I_i$ is the cell containing $x=0$.

1.3 A.3 Error representation and error equations

Denote the error by $e=v-u_h$, where $u_h$ approximates to equation (1.2) or equation(7.9). Clearly, $e$ also satisfies the scheme (2.2) with $g(x,t)=0$. We divide the error into the form $e=\eta -\xi$, where

$$\begin{aligned} \eta =v-\mathbb P _-v=\mathbb P _-^\bot v,\quad \mathrm{and}\quad \xi =u_h-\mathbb P _-v. \end{aligned}$$

Then following [31], we obtain

$$\begin{aligned} \frac{d\Vert \xi \Vert _\varphi ^2}{dt}&= 2\left(\xi _t,\mathbb P _+^\bot (\varphi \xi )\right)+2\left(\eta _t,\mathbb P _+(\varphi \xi )\right)+2\mathcal H (\xi ,\varphi \xi )-(\xi ,\varphi _x\xi )\\&= 2\Pi _1+2\Pi _2-\Pi _3, \end{aligned}$$

where

$$\begin{aligned} \Pi _1=\left(\xi _t,\mathbb P _+^\bot (\varphi \xi )\right),\quad \Pi _2=\left(\eta _t,\mathbb P _+(\varphi \xi )\right),\quad \Pi _3=\sum _j\varphi _{j+\frac{1}{2}}[\xi ]^2_{j+\frac{1}{2}}. \end{aligned}$$

First we estimate $\Pi _1$. Denote $w=\xi _t-\mathbb P _{k-1}\xi _t.$ From the scheme (2.3), we have

$$\begin{aligned} (\xi _t,w)_j=(\eta _t,w)_j-(e_t,w)_j=(\eta _t,w)_j-[\xi ]_{j-\frac{1}{2}}w^+_{j-\frac{1}{2}}. \end{aligned}$$

Plugging the above into $\Pi _1$ and defining $\psi =\sqrt{\varphi },$ we obtain

$$\begin{aligned} (\xi _t,\mathbb P _+^\bot (\varphi \xi ))_j&= \left(\frac{(\xi _t,w)_j}{\Vert w\Vert _{I_j}^2}w,\mathbb P _+^\bot (\varphi \xi )\right)_j\\&= \left(\left((\eta _t,w)_j-[\xi ]_{j-1/2}w^+_{j-\frac{1}{2}}\right)\frac{w}{\Vert w\Vert _{I_j}^2},\mathbb P _+^\bot (\varphi \xi )\right)_j\\&\le \frac{C}{\Vert w\Vert _{I_j}}\left(\left|(\psi \eta _t,w)_j\right|+\left|[\psi \xi ]_{j-1/2}w^+_{j-\frac{1}{2}}\right|\right)\ \left\Vert\psi ^{-1}\mathbb P _+^\bot (\varphi \xi )\right\Vert_{I_j}\\&\le \frac{Ch^{1-\sigma }}{\gamma }\left(\Vert \eta _t\Vert _{\varphi ,I_j}^2+\Vert \xi \Vert _{\varphi ,I_j}^2\right) +\frac{Ch^{1/2-\sigma }}{\gamma }\left(\varphi _{j-1/2}[\xi ]_{j-1/2}^2+\Vert \xi \Vert _{\varphi ,I_j}^2\right). \end{aligned}$$

Summing up with respect to $j$, we obtain

$$\begin{aligned} (\xi _t,\mathbb P _+^\bot (\varphi \xi ))\!\le \!\frac{Ch^{1-\sigma }}{\gamma }\left(\Vert \eta _t\Vert _\varphi ^2\!+\!\Vert \xi \Vert _\varphi ^2\right) \!+\!\frac{Ch^{1/2-\sigma }}{\gamma }\left(\sum _j\varphi _{j-1/2}[\xi ]_{j-1/2}^2\!+\!\Vert \xi \Vert _\varphi ^2\right). \end{aligned}$$

For $\Pi _2$, it is not difficult to find out that

$$\begin{aligned} \Pi _2\le C\Vert \eta _t\Vert _\varphi \ \Vert \xi \Vert _\varphi \le C(\Vert \eta _t\Vert _\varphi ^2+\Vert \xi \Vert _\varphi ^2). \end{aligned}$$

Then if $\gamma$ is large enough and $\sigma =\frac{1}{2}$, we have

$$\begin{aligned} 2\Pi _1+2\Pi _2-\Pi _3\le C\left(\Vert \eta _t\Vert _\varphi ^2+\Vert \xi \Vert _\varphi ^2\right). \end{aligned}$$

By Gronwall’s inequality,

$$\begin{aligned} \Vert \xi (T)\Vert _\varphi ^2\le C\int _0^T\Vert \eta _t\Vert _\varphi ^2dt+C\Vert \xi (0)\Vert _\varphi ^2. \end{aligned}$$

(7.11)

1.4 A.4 The final estimate

This part is almost the same as in [31]. We will only discuss the left-hand boundary of $\mathcal R _T$ since the discussion for the right one is similar. Denote $x_L(t)=t+x_c$ with

$$\begin{aligned} x_c=-2s\log (1/h)\gamma h^\sigma , \end{aligned}$$

where $s$ and $\gamma$ are sufficiently large and $\sigma =1/2$. As we have mentioned before, the $\delta$-singularity in the initial datum is contained in the cell $I_i$. Then by proposition A.1, we obtain $0<\phi (x)<h^s$ for any $x\in I_i$. We choose $v_0$ to satisfy $\mathbb P _kv_0=\mathbb P _ku_0=u_h(0)$, then

$$\begin{aligned} \Vert \xi (0)\Vert _\varphi \le \Vert \xi (0)\Vert _{\varphi ,L^2(\mathbb R \backslash I_i)}+\Vert \xi (0)\Vert _{\varphi ,L^2(I_i)}\le Ch^{k+1}\Vert f\Vert _{k+2}+Ch^{s-1/2}. \end{aligned}$$

If $s$ is large enough, then $\Vert \xi (0)\Vert _\varphi \le Ch^{k+1}.$

Define the domain $\mathcal R _T^+=(x_L(T),\infty )$, then

$$\begin{aligned} \Vert u_h-v\Vert _\mathbb{R \backslash \mathcal R _T^+}\le \Vert u_h-v\Vert _{\varphi ,\mathbb R \backslash \mathcal R _T^+}\le \Vert \eta \Vert _{\varphi ,\mathbb R \backslash \mathcal R _T^+}+\Vert \xi \Vert _\varphi \le Ch^{k+1}\Vert f\Vert _{k+1}+\Vert \xi \Vert _\varphi . \end{aligned}$$

To estimate the second term on the right hand side, we need to use (7.11). Denote

$$\begin{aligned} w(t)=\max \left\{ x_{j+\frac{1}{2}}:x_{j-\frac{1}{2}}<t+\frac{1}{2}x_c,\forall j\right\} , \end{aligned}$$

and $\mathcal R _1(t)=(-\infty ,w(t)),\,\mathcal R _2(t)=\mathbb R \backslash \mathcal R _1(t)=(w(t),\infty )$. If $\gamma h^{\sigma -1}$ is large enough, $\mathcal R _1(t)$ stays away from the bad interval $[t-h,t+h]$ where $v(x,t)\ne u(x,t),$ then we have

$$\begin{aligned} \Vert \eta _t\Vert _{\varphi ,\mathcal R _1(t)}\le Ch^{k+1}\Vert f\Vert _{k+2}. \end{aligned}$$

Now we proceed to estimate $\Vert \eta _t\Vert _{\varphi ,\mathcal R _2(t)}$. Since $\mathcal R _2$ contains the whole bad region, we will use the property of the weight function. By (7.4) we have $\varphi \le h^s$ in this zone. Then we obtain

$$\begin{aligned} \Vert \eta _t\Vert _{\varphi ,{\mathcal{R }_{2}(t)}}&\le Ch^{s/2}\Vert \eta _t\Vert _{\mathcal{R }_{2}(t)}\le Ch^{s/2+k+1}\Vert \partial _x^{k+2}v\Vert _{\mathcal{R }_{2}(t)} \\&\le Ch^{(s-3)/2}+Ch^{s/2+k+1}\Vert f\Vert _{k+2,{\mathcal{R }_{2}(t)}}. \end{aligned}$$

Similarly, we can estimate the right-hand side of the non-smooth region. If we take s large enough, we have

$$\begin{aligned} \Vert u_h\!-\!u(x,T)\Vert _\mathbb{R \backslash \mathcal R _T^+}\!=\!\Vert u_h-v(x,T)\Vert _\mathbb{R \backslash \mathcal R _T^+}\!\le \! Ch^{k+1}\Vert f\Vert _{k+2}\!+\!Ch^{(s-3)/2}\!\le \! Ch^{k+1}. \end{aligned}$$