Skip to main content

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


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. Bramble, J.H., Schatz, A.H.: High order local accuracy by averaging in the finite element method. Math. Comput. 31, 94–111 (1977)

    MathSciNet  MATH  Article  Google Scholar 

  2. Canuto, C., Fagnani, F., Tilli, P.: An Eulerian approach to the analysis of Rendez-vous algorithms. In: Chung, M.J., Misra, P. (eds.) 17th IFAC World Congress 2008, vol. 17, Part 1. Curran, Red Hook, New York (2009)

  3. Chalaby, A.: On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms. Math. Comput. 66, 527–545 (1997)

    Article  Google Scholar 

  4. Ciarlet, P.G.: Finite Element Method For Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  5. Cockburn, B.: A Introduction to the discontinuous Galerkin methods for convection dominated problems. In: Barth, T., Deconink, H. (eds.) High-Order Method for Computational Physics. Lecture Notes in Computational Science and Engineering, vol. 9, pp. 69–224. Springer, Berlin (1999)

  6. Cockburn, B., Guzmán, J.: Error estimate for the Runge-Kutta discontinuous Galerkin method for the transport equation with discontinuous initial data. SIAM J. Numer. Anal. 46, 1364–1398 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  7. Cockburn, B., Hou, S., Shu, C.W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)

    MathSciNet  MATH  Google Scholar 

  8. Cockburn, B., Lin, S.Y., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84, 90–113 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  9. Cockburn, B., Luskin, M., Shu, C.W., Süli, E.: Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Math. Comput. 72, 577–606 (2003)

    MATH  Google Scholar 

  10. Cockburn, B., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)

    MathSciNet  MATH  Google Scholar 

  11. Cockburn, B., Shu, C.W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  12. Greenberg, J.M., Leroux, A.Y., Baraille, R., Noussair, A.: Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34, 1980–2007 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  13. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  14. Hou, S., Liu, X.D.: Solutions of multi-dimensional hyperbolic systems of conservation laws by square entropy condition satisfying discontinuous Galerkin method. J. Sci. Comput. 31, 127–151 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  15. Jiang, G.S., Shu, C.W.: On a cell entropy inequality for discontinuous Galerkin methods. Math. Comput. 62, 531–538 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  16. John, F.: Partial different equations. Springer, New York (1971)

    Book  Google Scholar 

  17. Johnson, C., Nävert, U., Pitkäranta, J.: Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 45, 285–312 (1984)

    MATH  Article  Google Scholar 

  18. Johnson, C., Pitkäranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46, 1–26 (1986)

    MATH  Article  Google Scholar 

  19. Johnson, C., Schatz, A., Walhbin, L.: Crosswind smear and pointwise errors in streamline diffusion finite element methods. Math. Comput. 49, 25–38 (1987)

    MATH  Article  Google Scholar 

  20. Koren, B.: A robust upwind discretization method for advection, diffusion and source terms. Notes on Numerical Fluid Mechanics, vol. 45, pp. 117–138. Vieweg, Braunschweig (1993)

  21. LeVeque, R.J., Yee, H.C.: A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys. 86, 187–210 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  22. Noussair, A.: Analysis of nonlinear resonance in conservation laws with point sources and well-balanced scheme. Stud. Appl. Math. 104, 313–352 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  23. de Oliveira, P., Santos, J.: On a class of high resolution methods for solving hyperbolic conservation laws with source terms. In: Sequeira, A., da Veiga, H.B., Videman, J.H. (eds.) Applied Nonlinear Analysis, pp. 432–445. Kluwer academic publishers, New York (1999)

    Google Scholar 

  24. Reed, W.H., Hill, T.R.: Triangular mesh methods for the Neutron transport equation. Los Alamos Scientific Laboratory Report LA-UR-73-479, Los Alamos, NM (1973)

  25. Ryan, J.K., Cockburn, B.: Local derivative post-processing for the discontinuous Galerkin method. J. Comput. Phys. 228, 8642–8664 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  26. Ryan, J.K., Shu, C.W.: On a one-sided post-processing technique for the discontinuous Galerkin methods. Methods Appl. Anal. 10, 295–308 (2003)

    MathSciNet  MATH  Google Scholar 

  27. Santos, J., de Oliveira, P.: A converging finite volume scheme for hyperbolic conservation laws with source terms. J. Comput. Appl. Math. 111, 239–251 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  28. Schroll, H.J., Winther, R.: Finite-difference Schemes for scalar conservation laws with source terms. IMA J. Numer. Anal. 16, 201–215 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  29. Slingerland, P.V., Ryan, J.K., Vuik, C.: Position-dependent smooth-increasing accuracy-conserving (SIAC) filtering for improving discontinuous Galerkin solutions. SIAM J. Sci. Comput. 33, 802–825 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  30. Walfisch, D., Ryan, J.K., Kirby, R.M., Haimes, R.: One-sided smoothness-increasing accuracy-conserving filtering for enhanced streamline integration through discontinuous fields. J. Sci. Comput. 38, 164–184 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  31. Zhang, Q., Shu, C.W.: Error estimates for the third order explicit Runge-Kutta discontinuous Galerkin method for linear hyperbolic equation in one-dimension with discontinuous initial data. Numerische Mathematik. (2013, submitted)

  32. Zhang, X., Shu, C.W.: On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Chi-Wang Shu.

Additional information

This research was supported by DOE grant DE-FG02-08ER25863 and NSF grant DMS-1112700.

Appendix A: Proof of Lemma 3.1

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 \(\Omega \).

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,

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

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

$$\begin{aligned}&1\le \varphi ^a(x,t)\le 2,&\quad a(x-x_c-t)\le 0,\end{aligned}$$
$$\begin{aligned}&0<\varphi ^a(x,t)<h^s,&\quad a(x-x_c-t)>s \, \log (1/h)\gamma h^\sigma . \end{aligned}$$

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

$$\begin{aligned} \Vert \mathbb V ^\bot p\Vert _{\varphi ,D}&\le Ch^{k+1}\Vert \partial _x^{k+1}p\Vert _{\varphi ,D},\end{aligned}$$
$$\begin{aligned} \Vert \mathbb V ^\bot (\varphi v_h)\Vert _{\varphi ^{-1},D}&\le C\gamma ^{-1}h^{1-\sigma }\Vert v_h\Vert _{\varphi ,D},\end{aligned}$$
$$\begin{aligned} \Vert \mathbb V (\varphi v_h)\Vert _{\varphi ^{-1},D}&\le C\Vert v_h\Vert _{\varphi ,D}. \end{aligned}$$

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

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

A.2 The smooth solution

We consider the following problem

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

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 \(x=0\).

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

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


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

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,

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

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

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

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Yang, Y., Shu, CW. Discontinuous Galerkin method for hyperbolic equations involving \(\delta \)-singularities: negative-order norm error estimates and applications. Numer. Math. 124, 753–781 (2013).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification

  • 65M60
  • 65M15