Abstract
Linear wave equations sourced by a dirac delta distribution δ(x) and its derivative(s) can serve as a model for many different phenomena. We describe a discontinuous Galerkin (DG) method to numerically solve such equations with source terms proportional to ∂nδ∕∂xn. Despite the presence of singular source terms, which imply discontinuous or potentially singular solutions, our DG method achieves global spectral accuracy even at the source’s location. Our DG method is developed for the wave equation written in fully first-order form. The first-order reduction is carried out using a distributional auxiliary variable that removes some of the source term’s singular behavior. While this is helpful numerically, it gives rise to a distributional constraint. We show that a time-independent spurious solution can develop if the initial constraint violation is proportional to δ(x). Numerical experiments verify this behavior and our scheme’s convergence properties by comparing against exact solutions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The term \(\dot {\phi }(0,x)\) is found from evaluation of the evolution equation (15), \(\dot {\phi } = -\pi {}'- \dot {F}(t) \delta {}(x)\), at t = 0.
References
Cáceres, M.J., Carrillo, J.A., Perthame, B.: Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states. J. Math. Neurosci. 1(1), 1–33 (2011)
Canizares, P., Sopuerta, C.F.: Simulations of extreme-mass-ratio inspirals using pseudospectral methods. In: Journal of Physics: Conference Series, vol. 154, p. 012053. IOP Publishing (2009)
Canizares, P., Sopuerta, C.F., Jaramillo, J.L.: Pseudospectral collocation methods for the computation of the self-force on a charged particle: Generic orbits around a schwarzschild black hole. Phys. Rev. D 82(4), 044023 (2010)
Casti, A.R.R., Omurtag, A., Sornborger, A., Kaplan, E., Knight, B., Victor, J., Sirovich, L.: A population study of integrate-and-fire-or-burst neurons. Neural Comput. 14(5), 957–986 (2002)
Cockburn, B.: An introduction to the discontinuous Galerkin method for convection-dominated problems. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations: Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, June 23–28, 1997. Lecture Notes in Physics, pp. 150–268. Springer, Berlin (1998). https://doi.org/10.1007/BFb0096353
Cockburn, B.: Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws. J. Comput. Appl. Math. 128, 187 (2001). https://doi.org/10.1016/S0377-0427(00)00512-4
Cockburn, B., Shu, C.W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V. Multidimensional systems. J. Comput. Phys. 141, 199 (1998). https://doi.org/10.1006/jcph.1998.5892
Cockburn, B., Karniadakis, G.E., Shu, C.W.: The development of discontinuous Galerkin methods. In: Cockburn, B., Karniadakis, G.E., Shu, C.W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, pp. 3–50. Springer, Berlin (2000). https://doi.org/10.1007/978-3-642-59721-3_1
Engquist, B., Tornberg, A.K., Tsai, R.: Discretization of dirac delta functions in level set methods. J. Comput. Phys. 207(1), 28–51 (2005)
Fan, K., Cai, W., Ji, X.: A generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions. J. Comput. Phys. 227(4), 2387–2410 (2008)
Field, S.E., Hesthaven, J.S., Lau, S.R.: Discontinuous Galerkin method for computing gravitational waveforms from extreme mass ratio binaries. Class. Quantum Grav. 26, 165010 (2009). https://doi.org/10.1088/0264-9381/26/16/165010
Field, S.E., Hesthaven, J.S., Lau, S.R.: Persistent junk solutions in time-domain modeling of extreme mass ratio binaries. Phys. Rev. D 81(12), 124030 (2010)
Field, S.E., Gottlieb, S., Grant, Z.J., Isherwood, L.F., Khanna, G.: A GPU-accelerated mixed-precision WENO method for extremal black hole and gravitational wave physics computations. Commun. Appl. Math. Comput., 1–19 (2021). https://doi.org/10.1007/s42967-021-00129-2
Friedlander, F.G., Joshi, M.S., Joshi, M., Joshi, M.C.: Introduction to the Theory of Distributions. Cambridge University Press (1998)
Hesthaven, J., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2008). https://doi.org/10.1007/978-0-387-72067-8
Jung, J.H., Khanna, G., Nagle, I.: A spectral collocation approximation for the radial-infall of a compact object into a Schwarzschild black hole. Int. J. Mod. Phys. C 20(11), 1827–1848 (2009)
Lousto, C.O.: A time-domain fourth-order-convergent numerical algorithm to integrate black hole perturbations in the extreme-mass-ratio limit. Classical Quantum Gravity 22(15), S543 (2005)
Petersson, N.A., Sjogreen, B.: Stable grid refinement and singular source discretization for seismic wave simulations. Commun. Comput. Phys. 8(5), 1074–1110 (2010), 8(LLNL-JRNL-419382) (2009)
Poisson, E., Pound, A., Vega, I.: The motion of point particles in curved spacetime. Living Rev. Relativity 14(1), 1–190 (2011)
Reed, W., Hill, T.: Triangular mesh methods for the neutron transport equation. In: Conference: National Topical Meeting on Mathematical Models and Computational Techniques for Analysis of Nuclear Systems, Ann Arbor, Michigan, USA, 8 Apr 1973 LA-UR–73-479, CONF-730414–2 (1973). http://www.osti.gov/scitech/servlets/purl/4491151
Shearer, P.M.: Introduction to Seismology. Cambridge University Press (2019)
Sopuerta, C.F., Laguna, P.: Finite element computation of the gravitational radiation emitted by a pointlike object orbiting a nonrotating black hole. Phys. Rev. D 73(4), 044028 (2006)
Stakgold, I., Holst, M.J.: Green’s Functions and Boundary Value Problems, vol. 99. Wiley (2011)
Sundararajan, P.A., Khanna, G., Hughes, S.A.: Towards adiabatic waveforms for inspiral into kerr black holes: A new model of the source for the time domain perturbation equation. Phys. Rev. D 76(10), 104005 (2007)
Sundararajan, P.A., Khanna, G., Hughes, S.A., Drasco, S.: Towards adiabatic waveforms for inspiral into kerr black holes. II. Dynamical sources and generic orbits. Phys. Rev. D 78(2), 024022 (2008)
Tornberg, A.K., Engquist, B.: Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200(2), 462–488 (2004)
Yang, Y., Shu, C.W.: Discontinuous Galerkin method for hyperbolic equations involving δ-singularities: negative-order norm error estimates and applications. Numer. Math. 124(4), 753–781 (2013)
Acknowledgements
We thank Manas Vishal for providing an independent check of Theorem 1 using Mathematica. The authors acknowledge support of NSF Grants No. PHY-2010685 (G.K) and No. DMS-1912716 (S.F, S.G, and G.K), AFOSR Grant No. FA9550-18-1-0383 (S.G) and Office of Naval Research/Defense University Research Instrumentation Program (ONR/DURIP) Grant No. N00014181255. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while a subset of the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Advances in Computational Relativity program.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Field, S.E., Gottlieb, S., Khanna, G., McClain, E. (2023). Discontinuous Galerkin Method for Linear Wave Equations Involving Derivatives of the Dirac Delta Distribution. In: Melenk, J.M., Perugia, I., Schöberl, J., Schwab, C. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1. Lecture Notes in Computational Science and Engineering, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-031-20432-6_19
Download citation
DOI: https://doi.org/10.1007/978-3-031-20432-6_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-20431-9
Online ISBN: 978-3-031-20432-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)