A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form

Abstract

We present a stable discontinuous Galerkin (DG) method with a perfectly matched layer (PML) for three and two space dimensional linear elastodynamics, in velocity-stress formulation, subject to well-posed linear boundary conditions. First, we consider the elastodynamics equation, in a cuboidal domain, and derive an unsplit PML truncating the domain using complex coordinate stretching. The hyperbolic structure of the underlying system enables the construction of continuous energy estimates, in the time domain for the elastic wave equation, and in the Laplace space for a sequence of PML model problems, with variations in one, two and three space dimensions, respectively. They correspond to PMLs normal to boundary faces, along edges and in corners. Second, we develop a DG numerical method for the linear elastodynamics equation using physically motivated numerical flux and penalty parameters, which are compatible with all well-posed, internal and external, boundary conditions. When the PML damping vanishes in all directions, by construction, our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. Third, to ensure numerical stability of the discretization when PML damping is present, it is necessary to extend the numerical DG fluxes, and the numerical inter-element and boundary procedures, to the PML auxiliary differential equations. This is crucial for deriving discrete energy estimates analogous to the continuous energy estimates. Numerical solutions are evolved in time using the high order arbitrary derivative (ADER) time stepping scheme of the same order of accuracy with the spatial discretization. By combining the DG spatial approximation with the high order ADER time stepping scheme and the accuracy of the PML we obtain an arbitrarily high-order accurate wave propagation solver in the time domain. Numerical experiments are presented in two and three space dimensions corroborating the theoretical results.

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References

  1. 1.

    Achenbach, V.V.: Wave Propagation in Elastic Solids. Applied Mathematics and Mechanics, vol. 6. North-Holland, Amsterdam (1973)

    Google Scholar 

  2. 2.

    Appelö, D., Kreiss, G.: A new absorbing layer for elastic waves. J. Comput. Phys. 215, 642–660 (2006)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Petersson, N.A., Sjögreen, B.: Perfectly matched layer for Maxwell’s equation in second order formulation. J. Comput. Phys. 209, 19–46 (2005)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Baffet, D., Bielak, J., Givoli, D., Hagstrom, T., Rabinovich, D.: Long-time stable high-order absorbing boundary conditions for elastodynamics. Comput. Methods Appl. Mech. Engrg. 241–244, 20–37 (2012)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bécache, E., Fauqueux, S., Joly, P.: Stability of perfectly matched layers, group velocities and anisotropic waves. J. Comput. Phys. 188, 399–433 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bérenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Chew, W., Weedon, W.: A 3-d perfectly matched medium from modified Maxwell’s equations with stretched coordinates. IEEE Microw. Opt. Technol. Lett. 7, 599–604 (1994)

    Article  Google Scholar 

  8. 8.

    de la Puente, J., Ampuero, J.-P., Käser, M.: Dynamic rupture modeling on unstructured meshes using a discontinuous Galerkin method. J. Geophys. Res. 114, B10302 (2009)

    Article  Google Scholar 

  9. 9.

    Diaz, J., Joly, P.: A time domain analysis of pml models in acoustics. Comput. Methods Appl. Mech. Eng. 195, 3820–3853 (2006)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Dumbser, M., Käser, M.: An arbitrary high-order discontinuous galerkin method for elastic waves on unstructured meshes— I. The two-dimensional isotropic case with external source terms. Geophys. J. Int. 166, 855–877 (2006)

    Article  Google Scholar 

  11. 11.

    Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: High order ader schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. J. Comput. Phys 5, 824–862 (2016)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Duru, K.: Perfectly matched layers and high order difference methods for wave equations. PhD Thesis, Uppsala University Sweden (2012)

  13. 13.

    Duru, K.: The role of numerical boundary procedures in the stability of perfectly matched layers. SIAM J. Sci. Comput. 38, A1171–A1194 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Duru, K., Gabriel, A.-A., Kreiss, G.: On energy stable discontinuous galerkin spectral element approximations of the perfectly matched layer for the wave equation. Comput. Methods Appl. Mech. Eng. 350, 898–937 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Duru, K., Kozdon, J.E., Kreiss, G.: Boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form. J. Comput. Phys. 303, 372–395 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Duru, K., Kreiss, G.: Boundary waves and stability of the perfectly matched layer for the two space dimensional elastic wave equation in second order form. SIAM Numer. Anal. 52, 2883–2904 (2014)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Duru, K., Rannabauer, L., Gabriel, A.-A., Igel, H.: A new discontinuous Galerkin spectral element method for elastic waves with physically motivated numerical fluxes (2017). arXiv:1802.06380

  18. 18.

    Duru, K., Rannabauer, L., Gabriel, A.-A., Ling, O.K.A., Igel, H., Bader, M.: A stable discontinuous galerkin method for linear elastodynamics in geometrically complex media using physics based numerical fluxes (2019). arXiv:1907.02658

  19. 19.

    Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Proc. Natl. Acad. Sci. USA 74(5), 1765–1766 (1977)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Givoli, D.: High-order local non-reflecting boundary conditions: a review. Wave Motion 39, 319–326 (2004)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley, New York (1995)

    Google Scholar 

  22. 22.

    Hagstrom, T.: New results on absorbing layers and radiation boundary conditions. In: Ainsworth, M., et al. (eds.) Topics in Computational Wave Propagation. Springer, Berlin (2003)

    Google Scholar 

  23. 23.

    Halpern, L., Petit-Bergez, S., Rauch, J.: The analysis of matched layers. Conflu. Math. 3, 159–236 (2011)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181, 186–221 (2002)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Métivier, L., Tago, J., Virieux, J.: Smart layers: a simple and robust alternative to PML approaches for elastodynamics. Geophys. J. Int. 199, 700–706 (2014)

    Article  Google Scholar 

  26. 26.

    Kristeková, M., Kristek, J., Moczo, P.: Time-frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals. Geophys. J. Int. 178, 813–825 (2009)

    Article  Google Scholar 

  27. 27.

    Kristeková, M., Kristek, J., Moczo, P., Day, S.M.: Misfit criteria for quantitative comparison of seismograms. Bull. Seism. Soc. Am. 96, 1836–1850 (2006)

    Article  Google Scholar 

  28. 28.

    Kuzuoglu, M., Mittra, R.: Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic. IEEE Microw. Guided Wave Lett. 6, 447–449 (1996)

    Article  Google Scholar 

  29. 29.

    Lysmer, J., Kuhlemeyer, R.L.: Finite dynamic model for infinite media. J. Eng. Mech. Div. ASCE 95, 859–877 (1969)

    Google Scholar 

  30. 30.

    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover Publications Inc., New York (1994)

    Google Scholar 

  31. 31.

    Modave, A., Lambrechts, J., Geuzaine, C.: Perfectly matched layers for convex truncated domains with discontinuous Galerkin time domain simulations. Comput. Math Appl. 73, 684–700 (2017)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Pelties, C., de la Puente, J., Ampuero, J.-P., Brietzke, G.B., Käser, M.: Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes. J. Geophys. Res. 117, B02309 (2012)

    Google Scholar 

  33. 33.

    Petersson, N.A., O’Reilly, O., Sjögreen, B., Bydlon, S.: Discretizing singular point sources in hyperbolic wave propagation problems. J. Comput. Phys. 321, 532–555 (2016)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Reinarz, A., Charrier, D.E., Bader, M., Bovard, L., Dumbser, M., Fambri, F., Duru, K., Gabriel, A.-A., Gallard, J.-M., Köppel, S., Krenz, L., Rannabauer, L., Rezzolla, L., Samfass, P., Tavelli, M., Weinzierl, T.: Exahype: an engine for parallel dynamically adaptive simulations of wave problems. Comput. Phys. Comm. 254, 107251 (2020)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Roden, J.A., Gedney, S.D.: Convolution PML (CPML): an efficient fdtd implementation of the CFS-PML for arbitrary media. IEEE Microw. Opt. Technol. Lett. 27, 334–339 (2000)

    Article  Google Scholar 

  36. 36.

    Skelton, E.A., Adams, S.D.M., Craster, R.V.: Guided elastic waves and perfectly matched layers. Wave Motion 44, 573–592 (2007)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Sun, Q., Zhang, R., Zhan, Q., Liu, Q.H.: A novel coupling algorithm for perfectly matched layer with wave equation-based discontinuous galerkin time-domain method. IEEE Trans. Antennas Propag. 66, 255–261 (2018)

    Article  Google Scholar 

  38. 38.

    Toro, E.F.: The equations of fluid dynamics. In: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin, Heidelberg (2009). https://doi.org/10.1007/b79761_1

  39. 39.

    Uphoff, C., Rettenberger, S., Bader, M., Madden, E.H., Ulrich, T., Wollherr, S., Gabriel, A.-A.: Extreme scale multi-physics simulations of the tsunamigenic 2004 sumatra megathrust earthquake. In: Proceeding SC ’17 Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (2018)

  40. 40.

    Xie, Z., Martin, R., Komatitsch, D., Matzen, R.: Improved forward wave propagation and adjoint-based sensitivity kernel calculations using a numerically stable finite-element pml. Geophys. J. Int. 198, 1714–1747 (2014)

    Article  Google Scholar 

  41. 41.

    Zeng, C., Xia, J., Miller, R., Tsoflias, G.: Application of the multi-axial perfectly matched layer (M-PML) to near-surface seismic modeling with Rayleigh waves. Geophysics 76, 43–52 (2011)

    Article  Google Scholar 

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Acknowledgements

The work presented in this paper was enabled by funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No 671698 (ExaHyPE). A.-A.G. acknowledges additional support by the German Research Foundation (DFG) (Projects No. KA 2281/4-1, GA 2465/2-1, GA 2465/3-1), by BaCaTec (Project No. A4) and BayLat, by KONWIHR—the Bavarian Competence Network for Technical and Scientific High Performance Computing (Project NewWave), by KAUST-CRG (GAST, Grant No. ORS-2016-CRG5-3027 and FRAGEN, Grant No. ORS-2017-CRG6 3389.02), by the European Union’s Horizon 2020 research and innovation program (ChEESE, Grant No. 823844 and TEAR, Grant No. 852992).

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Appendices

Other receivers

In this section we present the seismograms for the remaining receivers, namely: Receiver 1, 2, 3, 7 and 8. In Fig. 12 we display the results for the HHS problem. Figure 13 shows the result for LOH1 problem with \(N = 3\) degree polynomial approximation, and Fig. 14 shows the result for LOH1 problem with \(N = 5\) degree polynomial approximation. Note the effective absorption properties of the PML, and the spectral accuracy of the DG method.

Fig. 12
figure12

The homogeneous half space problem

Fig. 13
figure13

The LOH1 benchmark problem with degree \(N = 3\) polynomial approximation

Fig. 14
figure14

The LOH1 benchmark problem with degree \(N = 5\) polynomial approximation

Source terms

We will derive a representation of the modified source terms in the physical space. Let \(t\ge 0\) denote time, and f(t) and g(t) are real functions of exponential order. We denote their corresponding Laplace transforms \({\widetilde{f}}(s) = {\mathscr {L}}\left( f(t)\right) \), \({\widetilde{g}}(s) = {\mathscr {L}}\left( g(t)\right) \). We also define the convolution operator \(*\),

$$\begin{aligned} g(t)*f(t) = \int _{0}^{t} g(t-\tau )f(\tau ) d\tau . \end{aligned}$$

Note that \({\mathscr {L}}\left( g(t)*f(t)\right) = {\widetilde{f}}(s){\widetilde{g}}(s)\). To succeed we will use the Dirac delta distribution \(\delta (t)\)

$$\begin{aligned} \int _{0}^{+\infty } \delta (t) dt = 1, \end{aligned}$$

with

$$\begin{aligned} \delta (t)*f(t) = f(t), \quad \delta ^{\prime }(t)*f(t) = f^{\prime }(t) + \delta (t)f(0). \end{aligned}$$

We also recall

$$\begin{aligned} {\mathscr {L}}\left( \delta ^{n}(t)\right) = s^n, \quad n = 0, 1, 2, \cdots , \end{aligned}$$

where \(\delta ^{n}(t)\) is the n-th derivative of \(\delta (t)\).

We have

$$\begin{aligned} {\mathscr {L}}\left( \delta ^{\prime }(t)*f(t) \right) = s{\widetilde{f}}(s). \end{aligned}$$

Consider the modified source terms in the Laplace domain

$$\begin{aligned} \widetilde{{\mathbf {F}}} = {\mathbf {P}}\left( S_x \widetilde{{\mathbf {F}}}_{Q} - \sum _{\xi = x,y, z} \frac{d_{\xi }S_x }{s + \alpha _{\xi } + d_{\xi }} \widetilde{{\mathbf {F}}}_{w_\xi }\right) . \end{aligned}$$
(117)

If \(d_{\xi } = d_x = d \ge 0\) and \(\alpha _{\xi } = \alpha _{x} = \alpha \ge 0\), then

$$\begin{aligned} S_x \widetilde{{\mathbf {F}}}_{Q}= \left( 1 + \frac{d}{s + \alpha } \right) \widetilde{{\mathbf {F}}}_{Q}, \quad \frac{S_x }{s + \alpha _{\xi } + d_{\xi }} \widetilde{{\mathbf {F}}}_{w_\xi } = \frac{1 }{s + \alpha } \widetilde{{\mathbf {F}}}_{w_\xi }, \end{aligned}$$
(118)

and

$$\begin{aligned}&{\mathscr {L}}^{-1}\left( S_x \widetilde{{\mathbf {F}}}_{Q}\right) = \left( \delta (t) + {d} e^{-\alpha t} \right) * {{\mathbf {F}}}_{Q}\left( x,y,z, t\right) , \nonumber \\&{\mathscr {L}}^{-1}\left( \frac{1 }{s + \alpha } \widetilde{{\mathbf {F}}}_{w_\xi }\right) = e^{-\alpha t} * {{\mathbf {F}}}_{w_\xi } \left( x,y,z, t\right) , \quad \text {Re}\{s\} > - \alpha . \end{aligned}$$
(119)

Proof of Theorem 2

Proof

As before, multiply (53) with \(\widetilde{{\mathbf {Q}}}^{\dagger }\) from the left and integrate over the whole spatial domain. Integration-by-parts gives

$$\begin{aligned} \begin{aligned}&\int _{\varOmega }\left( \left( sS_x\right) \widetilde{{\mathbf {Q}}}^{\dagger } {\mathbf {P}}^{-1} \widetilde{{\mathbf {Q}}} \right) dxdy dz \\&\quad = \sum _{\xi = x, y}\frac{1}{2}\int _{\varOmega }\left( \left[ \widetilde{{\mathbf {Q}}}^\dagger \left( {\mathbf {A}}_{\xi }\frac{\partial {\widetilde{{\mathbf {Q}}}}}{\partial \xi }\right) -\left( {\mathbf {A}}_{\xi }\frac{\partial {\widetilde{{\mathbf {Q}}}}}{\partial \xi }\right) ^\dagger {\mathbf {Q}}\right] \right) dxdydz \\&\qquad + \sum _{\xi = x, y}\int _{{\widetilde{\varGamma }}}\left( \left[ \widetilde{{\mathbf {Q}}}^{\dagger }{A}_\xi \widetilde{{\mathbf {Q}}}\right] \Big |_{-1}^{1}\right) \frac{dxdydz}{d\xi } + \int _{\varOmega }\left( \widetilde{{\mathbf {Q}}}^{\dagger } {\mathbf {P}}^{-1}\widetilde{{\mathbf {F}}} \right) dxdydz. \end{aligned} \end{aligned}$$
(120)

Adding (120) to its complex conjugate, the spatial derivative terms vanish, yielding

$$\begin{aligned} \begin{aligned} \Vert \sqrt{{\mathrm{Re}}(s S_x)}\widetilde{{\mathbf {Q}}} (\cdot ,\cdot ,\cdot , s)\Vert _{P}^2&= \frac{1}{2}\left[ \left( \widetilde{{\mathbf {Q}}},\widetilde{{\mathbf {F}}} \right) _{{P}} + \left( \widetilde{{\mathbf {F}}},\widetilde{{\mathbf {Q}}} \right) _{{P}}\right] \\&\quad + \sum _{\xi = x, y}\int _{{\widetilde{\varGamma }}}\left( \left[ \widetilde{{\mathbf {Q}}}^{\dagger }{A}_x \widetilde{{\mathbf {Q}}}\right] \Big |_{-1}^{1}\right) \frac{dxdydz}{d\xi } . \end{aligned} \end{aligned}$$
(121)

Using Cauchy-Schwarz inequality and the fact (41) in the right hand side of (121) completes the proof. \(\square \)

Proof of Theorem 3

Proof

As above, multiply (55) with \(\widetilde{{\mathbf {Q}}}^{\dagger }\) from the left and integrate over the whole spatial domain. Integration-by-parts gives

$$\begin{aligned}&\int _{\varOmega }\left( \left( sS_x\right) \widetilde{{\mathbf {Q}}}^{\dagger } {\mathbf {P}}^{-1} \widetilde{{\mathbf {Q}}} \right) dxdy dz \nonumber \\&\quad = \sum _{\xi = x, y,z}\frac{1}{2}\int _{\varOmega }\left( \left[ \widetilde{{\mathbf {Q}}}^\dagger \left( {\mathbf {A}}_{\xi } \frac{\partial {\widetilde{{\mathbf {Q}}}}}{\partial \xi }\right) -\left( {\mathbf {A}}_{\xi }\frac{\partial {\widetilde{{\mathbf {Q}}}}}{\partial \xi }\right) ^\dagger {\mathbf {Q}}\right] \right) dxdydz \nonumber \\&\qquad + \sum _{\xi = x, y,z}\int _{{\widetilde{\varGamma }}} \left( \left[ \widetilde{{\mathbf {Q}}}^{\dagger }{A}_\xi \widetilde{{\mathbf {Q}}}\right] \Big |_{-1}^{1}\right) \frac{dxdydz}{d\xi } + \int _{\varOmega }\left( \widetilde{{\mathbf {Q}}}^{\dagger } {\mathbf {P}}^{-1} \widetilde{{\mathbf {F}}} \right) dxdydz. \end{aligned}$$
(122)

Adding the complex conjugate of the product, the spatial derivative terms vanish, yielding

$$\begin{aligned} \begin{aligned} \Vert \sqrt{{\mathrm{Re}}(s S_x)}\widetilde{{\mathbf {Q}}} (\cdot ,\cdot ,\cdot , s)\Vert _{P}^2&= \frac{1}{2}\left[ \left( \widetilde{{\mathbf {Q}}},\widetilde{{\mathbf {F}}} \right) _{{P}} + \left( \widetilde{{\mathbf {F}}},\widetilde{{\mathbf {Q}}} \right) _{{P}}\right] \\&\quad + \sum _{\xi = x, y, z}\int _{{\widetilde{\varGamma }}}\left( \left[ \widetilde{{\mathbf {Q}}}^{\dagger }{A}_\xi \widetilde{{\mathbf {Q}}}\right] \Big |_{-1}^{1}\right) \frac{dxdydz}{d\xi } , \quad \text {Re}\{s\} > 0. \end{aligned}\nonumber \\ \end{aligned}$$
(123)

Using Cauchy-Schwarz inequality and the fact (41) in the right hand side of (123) completes the proof. \(\square \)

Proof of Theorem 8

We introduce the boundary terms

$$\begin{aligned} \mathscr {BT}_s\left( \widehat{{\widetilde{v}}}_{\eta }^{-}, \widehat{{\widetilde{T}}}_{\eta }^{-}\right)&= \frac{\varDelta {x}}{2} \frac{\varDelta {z}}{2} \sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( \left( \widehat{{\widetilde{T}}}_\eta ^{-*} \widehat{{\widetilde{v}}}_\eta ^{-} \right) \Big |_{r = 1} - \left( \widehat{{\widetilde{T}}}_\eta ^{-*} \widehat{{\widetilde{v}}}_\eta ^{-} \right) \Big |_{r = -1}\right) _{i k} h_{i} h_{k} \\&\quad - \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2}\sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( \left( \widehat{{\widetilde{T}}}_\eta ^{-*} \widehat{{\widetilde{v}}}_\eta ^{-} \right) \Big |_{q = -1}\right) _{i k} h_{i} h_{k} \le 0,\\ \mathscr {BT}_s\left( \widehat{{\widetilde{v}}}_{\eta }^{+}, \widehat{{\widetilde{T}}}_{\eta }^{+}\right)&= \frac{\varDelta {x}}{2} \frac{\varDelta {z}}{2} \sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( \left( \widehat{{\widetilde{T}}}_\eta ^{+*} \widehat{{\widetilde{v}}}_\eta ^{+} \right) \Big |_{r = 1} - \left( \widehat{{\widetilde{T}}}_\eta ^{+*} \widehat{{\widetilde{v}}}_\eta ^{+} \right) \Big |_{r = -1}\right) _{i k} h_{i} h_{k} \\&\quad + \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2}\sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( \left( \widehat{{\widetilde{T}}}_\eta ^{+*} \widehat{{\widetilde{v}}}_\eta ^{+} \right) \Big |_{q = 1}\right) _{i k} h_{i} h_{k} \le 0, \end{aligned}$$

the interface term

$$\begin{aligned} \mathscr {IT}_s\left( \widehat{{\widetilde{v}}}^{\pm }, \widehat{{\widetilde{T}}}^{\pm } \right) = - \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2}\sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( {\widehat{{\widetilde{T}}}}_\eta \llbracket {{\widehat{{\widetilde{v}}}}_\eta \rrbracket }\right) _{i k} h_{i} h_{k} \equiv 0, \end{aligned}$$

and the fluctuation term

$$\begin{aligned} {\mathscr {F}}_{luc}\left( {\widetilde{G}}, Z\right)&= - \sum _{\xi = x, y}\frac{\varDelta {x}}{2} \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2} \frac{2}{\varDelta {\xi }}\\&\quad \sum _{\eta = x, y, z}\sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\left( \left( \frac{1}{Z_{\eta }}|{\widetilde{G}}_\eta |^2\right) \Big |_{ -1} + \left( \frac{1}{Z_{\eta }}|{\widetilde{G}}_\eta |^2\right) \Big |_{ 1} \right) _{i, k} \le 0. \end{aligned}$$

Proof

From the left, multiply (100) with \(\widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}\), we have

$$\begin{aligned} \begin{aligned}&\left( sS_x\right) \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {Q}}}\\&\quad = \sum _{\xi = x, y}\frac{1}{2}\widetilde{{\mathbf {Q}}}^\dagger \left( {\mathbf {A}}_{\xi }\left( {\mathbf {H}}{\mathbf {D}}_{\xi }\right) - \left( {\mathbf {H}}{\mathbf {D}}_{\xi }\right) ^T{\mathbf {A}}_{\xi } \right) \widetilde{{\mathbf {Q}}} + \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {F}}} \\&\qquad +\widetilde{{\mathbf {Q}}}^\dagger {\mathbf {H}}{\mathbf {H}}_{\xi }^{-1} \left( \frac{1}{2}{\mathbf {A}}_{x}\left( {\mathbf {B}}_{\xi }\left( 1,1\right) - {\mathbf {B}}_{\xi }\left( -1,-1\right) \right) \widetilde{{\mathbf {Q}}} -\left( {{\mathbf {e}}_{\xi }(-1)} \widetilde{\mathbf {FL}}_{\xi } + {{\mathbf {e}}_{\xi }(1)} \widetilde{\mathbf {FR}}_{\xi } \right) \right) \end{aligned}\nonumber \\ \end{aligned}$$
(124)

Using the identity (95) gives

$$\begin{aligned} \left( sS_x\right) \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {Q}}}&= \sum _{\xi = x, y}\frac{1}{2}\widetilde{{\mathbf {Q}}}^\dagger \left( {\mathbf {A}}_{\xi }\left( {\mathbf {H}}{\mathbf {D}}_{\xi }\right) - \left( {\mathbf {H}}{\mathbf {D}}_{\xi }\right) ^T{\mathbf {A}}_{\xi } \right) \widetilde{{\mathbf {Q}}} + \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {F}}} \nonumber \\&\quad - \sum _{\xi = x, y}\frac{\varDelta {x}}{2} \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2} \frac{2}{\varDelta {\xi }} \sum _{i=1}^{P+1}\sum _{k= 1}^{P+1}\left( \sum _{\eta = x,y,z} \left( \frac{1}{Z_\eta }|{\widetilde{G}}_\eta |^2 - \widehat{{\widetilde{T}}}_\eta ^* \widehat{{\widetilde{v}}}_\eta \right) \Big |_{ 1} \right. \nonumber \\&\quad \left. + \sum _{\eta = x,y,z} \left( \frac{1}{Z_\eta }|{\widetilde{G}}_\eta |^2 + \widehat{{\widetilde{T}}}_\eta ^* \widehat{{\widetilde{v}}}_\eta \right) \Big |_{-1}\right) _{ik} h_ih_k \end{aligned}$$
(125)

Next we add (125) to its complex conjugate, and the spatial derivative terms vanish, we have

$$\begin{aligned} {\mathrm{Re}}\left( sS_x\right) \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {Q}}}= & {} \frac{1}{2}\left( \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {F}}} + \widetilde{{\mathbf {F}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {Q}}} \right) \nonumber \\&- \sum _{\xi = x, y}\frac{\varDelta {x}}{2} \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2} \frac{2}{\varDelta {\xi }} \sum _{i=1}^{P+1}\sum _{k= 1}^{P+1}\left( \sum _{\eta = x,y,z} \left( \frac{1}{Z_\eta }|{\widetilde{G}}_\eta |^2 - \widehat{{\widetilde{T}}}_\eta ^* \widehat{{\widetilde{v}}}_\eta \right) \Big |_{ 1} \right. \nonumber \\&\left. + \sum _{\eta = x,y,z} \left( \frac{1}{Z_\eta }|{\widetilde{G}}_\eta |^2 + \widehat{{\widetilde{T}}}_\eta ^* \widehat{{\widetilde{v}}}_\eta \right) \Big |_{-1}\right) _{ik} h_ih_k \end{aligned}$$
(126)

Using Cauchy-Schwarz inequality for the source term in (126) and collecting contributions from both sides of the elements completes the proof. \(\square \)

Proof of Theorem 9

We introduce the boundary terms

$$\begin{aligned} \mathscr {BT}_s\left( \widehat{{\widetilde{v}}}_{\eta }^{-}, \widehat{{\widetilde{T}}}_{\eta }^{-}\right)&= \frac{\varDelta {x}}{2} \frac{\varDelta {y}}{2} \sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( \left( \widehat{{\widetilde{T}}}_\eta ^{-*} \widehat{{\widetilde{v}}}_\eta ^{-} \right) \Big |_{s = 1} - \left( \widehat{{\widetilde{T}}}_\eta ^{-*} \widehat{{\widetilde{v}}}_\eta ^{-} \right) \Big |_{s = -1}\right) _{i k} h_{i} h_{k} \\&\quad + \frac{\varDelta {x}}{2} \frac{\varDelta {z}}{2} \sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( \left( \widehat{{\widetilde{T}}}_\eta ^{-*} \widehat{{\widetilde{v}}}_\eta ^{-} \right) \Big |_{r = 1} - \left( \widehat{{\widetilde{T}}}_\eta ^{-*} \widehat{{\widetilde{v}}}_\eta ^{-} \right) \Big |_{r = -1}\right) _{i k} h_{i} h_{k}\\&\quad - \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2}\sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( \left( \widehat{{\widetilde{T}}}_\eta ^{-*} \widehat{{\widetilde{v}}}_\eta ^{-} \right) \Big |_{q = -1}\right) _{i k} h_{i} h_{k} \le 0,\\ \mathscr {BT}_s\left( \widehat{{\widetilde{v}}}_{\eta }^{+}, \widehat{{\widetilde{T}}}_{\eta }^{+}\right)&= \frac{\varDelta {x}}{2} \frac{\varDelta {y}}{2} \sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( \left( \widehat{{\widetilde{T}}}_\eta ^{+*} \widehat{{\widetilde{v}}}_\eta ^{+} \right) \Big |_{s = 1} - \left( \widehat{{\widetilde{T}}}_\eta ^{+*} \widehat{{\widetilde{v}}}_\eta ^{+} \right) \Big |_{s = -1}\right) _{i k} h_{i} h_{k} \\&\quad + \frac{\varDelta {x}}{2} \frac{\varDelta {z}}{2} \sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( \left( \widehat{{\widetilde{T}}}_\eta ^{+*} \widehat{{\widetilde{v}}}_\eta ^{+} \right) \Big |_{r = 1} - \left( \widehat{{\widetilde{T}}}_\eta ^{+*} \widehat{{\widetilde{v}}}_\eta ^{+} \right) \Big |_{r = -1}\right) _{i k} h_{i} h_{k} \\&\quad + \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2}\sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( \left( \widehat{{\widetilde{T}}}_\eta ^{+*} \widehat{{\widetilde{v}}}_\eta ^{+} \right) \Big |_{q = 1}\right) _{i k} h_{i} h_{k} \le 0, \end{aligned}$$

the interface term

$$\begin{aligned} \mathscr {IT}_s\left( \widehat{{\widetilde{v}}}^{\pm }, \widehat{{\widetilde{T}}}^{\pm } \right) = - \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2}\sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\sum _{\eta = x,y,z}\left( {\widehat{{\widetilde{T}}}}_\eta \llbracket {{\widehat{{\widetilde{v}}}}_\eta \rrbracket }\right) _{i k} h_{i} h_{k} \equiv 0, \end{aligned}$$

and the fluctuation term

$$\begin{aligned} {\mathscr {F}}_{luc}\left( {\widetilde{G}}, Z\right)&= - \sum _{\xi = x, y, z}\frac{\varDelta {x}}{2} \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2} \frac{2}{\varDelta {\xi }}\\&\quad \sum _{\eta = x, y, z}\sum _{i = 1}^{P+1}\sum _{k = 1}^{P+1}\left( \left( \frac{1}{Z_{\eta }}|{\widetilde{G}}_\eta |^2\right) \Big |_{ -1} + \left( \frac{1}{Z_{\eta }}|{\widetilde{G}}_\eta |^2\right) \Big |_{ 1} \right) _{i, k} \le 0. \end{aligned}$$

Proof

From the left, multiply (102) with \(\widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}\), we have

$$\begin{aligned} \begin{aligned}&\left( sS_x\right) \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {Q}}} \\&\quad = \sum _{\xi = x, y, z}\frac{1}{2}\widetilde{{\mathbf {Q}}}^\dagger \left( {\mathbf {A}}_{\xi }\left( {\mathbf {H}}{\mathbf {D}}_{\xi }\right) - \left( {\mathbf {H}}{\mathbf {D}}_{\xi }\right) ^T{\mathbf {A}}_{\xi } \right) \widetilde{{\mathbf {Q}}} + \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {F}}} \\&\qquad + \widetilde{{\mathbf {Q}}}^\dagger {\mathbf {H}}{\mathbf {H}}_{\xi }^{-1} \left( \frac{1}{2}{\mathbf {A}}_{x}\left( {\mathbf {B}}_{\xi }\left( 1,1\right) - {\mathbf {B}}_{\xi }\left( -1,-1\right) \right) \widetilde{{\mathbf {Q}}} -\left( {{\mathbf {e}}_{\xi }(-1)} \widetilde{\mathbf {FL}}_{\xi } + {{\mathbf {e}}_{\xi }(1)} \widetilde{\mathbf {FR}}_{\xi } \right) \right) \end{aligned}\nonumber \\ \end{aligned}$$
(127)

Using the identity (95) gives

$$\begin{aligned} \left( sS_x\right) \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {Q}}}= & {} \sum _{\xi = x, y, z}\frac{1}{2}\widetilde{{\mathbf {Q}}}^\dagger \left( {\mathbf {A}}_{\xi }\left( {\mathbf {H}}{\mathbf {D}}_{\xi }\right) - \left( {\mathbf {H}}{\mathbf {D}}_{\xi }\right) ^T{\mathbf {A}}_{\xi } \right) \widetilde{{\mathbf {Q}}} + \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {F}}} \nonumber \\&- \sum _{\xi = x, y}\frac{\varDelta {x}}{2} \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2} \frac{2}{\varDelta {\xi }} \sum _{i=1}^{P+1}\sum _{k= 1}^{P+1}\left( \sum _{\eta = x,y,z} \left( \frac{1}{Z_\eta }|{\widetilde{G}}_\eta |^2 - \widehat{{\widetilde{T}}}_\eta ^* \widehat{{\widetilde{v}}}_\eta \right) \Big |_{ 1} \right. \nonumber \\&\quad \left. + \sum _{\eta = x,y,z} \left( \frac{1}{Z_\eta }|{\widetilde{G}}_\eta |^2 + \widehat{{\widetilde{T}}}_\eta ^* \widehat{{\widetilde{v}}}_\eta \right) \Big |_{-1}\right) _{ik} h_ih_k \end{aligned}$$
(128)

Next we add (128) to its complex conjugate, and the spatial derivative terms vanish, we have

$$\begin{aligned} {\mathrm{Re}}\left( sS_x\right) \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {Q}}}= & {} \frac{1}{2}\left( \widetilde{{\mathbf {Q}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {F}}} + \widetilde{{\mathbf {F}}}^\dagger {{\mathbf {H}}}{\mathbf {P}}^{-1} \widetilde{{\mathbf {Q}}} \right) \nonumber \\&- \sum _{\xi = x, y}\frac{\varDelta {x}}{2} \frac{\varDelta {y}}{2} \frac{\varDelta {z}}{2} \frac{2}{\varDelta {\xi }} \sum _{i=1}^{P+1}\sum _{k= 1}^{P+1}\left( \sum _{\eta = x,y,z} \left( \frac{1}{Z_\eta }|{\widetilde{G}}_\eta |^2 - \widehat{{\widetilde{T}}}_\eta ^* \widehat{{\widetilde{v}}}_\eta \right) \Big |_{ 1} \right. \nonumber \\&\left. + \sum _{\eta = x,y,z} \left( \frac{1}{Z_\eta }|{\widetilde{G}}_\eta |^2 + \widehat{{\widetilde{T}}}_\eta ^* \widehat{{\widetilde{v}}}_\eta \right) \Big |_{-1}\right) _{ik} h_ih_k \end{aligned}$$
(129)

Using Cauchy-Schwarz inequality for the source term in (129) and collecting contributions from both sides of the elements completes the proof. \(\square \)

The Arbitrary DERivative (ADER) time integration

In this section we will summarize the ADER time-stepping scheme. For more elaborate discussions, we refer the reader to [8, 10, 38]. The DG semi-discrete approximation (84)–(85) can be written as a system first order ODEs

$$\begin{aligned} \begin{aligned} \frac{d \bar{{\mathbf {Q}}} }{dt} = \underbrace{D\bar{{\mathbf {Q}}}}_{\text {PDE}} + \underbrace{F\bar{{\mathbf {Q}}}}_{\text {Num. flux} \rightarrow 0}, \end{aligned} \end{aligned}$$
(130)

where \(D\bar{{\mathbf {Q}}}\) emanates from the PDE, is the discrete spatial operator involving derivatives and lower order PML terms, and \(F\bar{{\mathbf {Q}}}\) is the numerical flux fluctuation term, incorporating the boundary and interface conditions.

The numerical flux fluctuation is a very small term, \(F\bar{{\mathbf {Q}}} \approx 0\), and will vanish \(F\bar{{\mathbf {Q}}} \rightarrow 0\) in the limit of mesh refinement \(\varDelta {t} \rightarrow 0\).

We now introduce the discrete time variables \(t_k \le t \le t_{k+1}\), \(\varDelta {t}_k = t_{k+1}- t_{k}\), and the pseudo time variable \(\tau = t -t_k\) such that \(0 \le \tau \le \varDelta {t}_k\), and \({d}/{d\tau } = {d}/{dt} \). Going from the current time level \(\tau =0\) to the next time level \(\tau = \varDelta {t}_k\), we integrate the ODE (130), exactly having

$$\begin{aligned} \bar{{\mathbf {Q}}}(\varDelta {t})&= \bar{{\mathbf {Q}}}(0) + \int _0^{\varDelta {t}_k}{D\bar{{\mathbf {Q}}}} d\tau + \int _0^{\varDelta {t}_k}{F\bar{{\mathbf {Q}}}} d\tau , \quad \nonumber \\&= \bar{{\mathbf {Q}}}(0) + D\int _0^{\varDelta {t}_k}{\bar{{\mathbf {Q}}}} d\tau + F\int _0^{\varDelta {t}_k}{\bar{{\mathbf {Q}}}} d\tau , \end{aligned}$$
(131)

where the second equality follows from linearity. If we can evaluate the integrals \(\int _0^{\varDelta {t}_k}{\bar{{\mathbf {Q}}}} d\tau \) in (131) exactly, then the time integration in (131) is exact. However, exact time integration is possible only in the most trivial case where the right hand side of (130) vanish identically for all components. Now, we will make an important approximation. We assume that the time step \(\varDelta {t}\) is sufficiently small, such that

$$\begin{aligned} \frac{d\bar{{\mathbf {Q}}}(\tau )}{d\tau } \approx D\bar{{\mathbf {Q}}}, \quad F\bar{{\mathbf {Q}}} \approx 0, \end{aligned}$$
(132)

are reasonable approximations. Next we construct the predictor, \( \widetilde{\bar{{\mathbf {Q}}}}(\tau )\), by Taylor expansions of the solution around \(\tau = 0\) and replace the time derivatives with spatial operator in (132), we have

$$\begin{aligned} \widetilde{\bar{{\mathbf {Q}}}}(\tau ) = \bar{{\mathbf {Q}}}(0) + \tau \frac{d\bar{{\mathbf {Q}}}(0)}{d\tau } + \frac{\tau ^2}{2} \frac{d^2\bar{{\mathbf {Q}}}(0)}{d\tau ^2} + ... \approx \sum _{m = 0}^{P}\frac{\tau ^m}{m !} D^m \bar{{\mathbf {Q}}}(0), \end{aligned}$$
(133)

where P is the polynomial degree used in the spatial approximation. We can now approximate the integrals in (131) using the predictor. The result of this integration is called the time average, \( \bar{\bar{{\mathbf {Q}}}}(0) \),

$$\begin{aligned} \int _0^{\varDelta {t}_k}{\bar{{\mathbf {Q}}}} d\tau \approx { \bar{\bar{{\mathbf {Q}}}}(0) = \int _{0}^{\varDelta {t}_k} \widetilde{\bar{{\mathbf {Q}}}}(\tau ) d\tau = \sum _{m = 0}^{P}\frac{\varDelta {t}_k^{(m+1)}}{(m +1) !} D^m \bar{{\mathbf {Q}}}(0)}. \end{aligned}$$
(134)

By replacing the integrals in (131) with the time average \({\bar{\bar{{\mathbf {Q}}}}(0)} \), we derive a high order accurate, explicit, one-step, time integration scheme

$$\begin{aligned} \bar{{\mathbf {Q}}}(\varDelta {t})&= \bar{{\mathbf {Q}}}(0) + D\bar{\bar{{\mathbf {Q}}}}(0) + F\bar{\bar{{\mathbf {Q}}}}(0). \end{aligned}$$
(135)

Note that the numerical flux fluctuation, \(F\bar{\bar{{\mathbf {Q}}}}(0)\), is evaluated only once for any order of approximation. This is opposed to Runge–Kutta methods or standard Taylor series methods where the numerical flux fluctuation is included in the spatial operator, \(D^m \rightarrow (D + {F})^m\), to approximate higher time derivatives in the Taylor series terms. For the ADER scheme, the fact that the numerical flux fluctuation is evaluated only once for any order implies that most of the computations are performed within the element to compute the predictor in (133) and the time average in (134). This tremendously decreases computational cost of high performance computing applications, since we can design efficient communication avoiding parallel algorithms. Since the predictor \(\widetilde{\bar{{\mathbf {Q}}}}(\tau )\) is defined in the entire time interval \(0 \le \tau \le \varDelta {t}_k\), the ADER scheme, (131)–(135), is also easily amenable to local time-stepping methods. When the ADER time stepping scheme is combined with the DG spatial approximation the fully discrete scheme is often referred as the ADERDG method [10]. For a DG polynomial approximation of degree P, a stable ADERDG method is \((P+1)\)th order accurate in both space and time.

Hat-variables

We will now reformulate the boundary condition (21) and interface condition (24) by introducing transformed (hat-) variables so that we can simultaneously construct (numerical) boundary/interface data for particle velocities and tractions. The hat-variables encode the solution of the IBVP on the boundary/interface. The hat-variables will be constructed such that they preserve the amplitude of the outgoing characteristics and exactly satisfy the physical boundary conditions [17]. To be more specific, the hat-variables are solutions of the Riemann problem constrained against physical boundary conditions (21), and the interface condition (24). We refer the reader to [17, 18] for more detailed discussions. Once the hat-variables are available, we construct physics based numerical flux fluctuations by penalizing data against the incoming characteristics (20) at the element faces.

Boundary data

For \(Z_\eta > 0\), we define the characteristics

$$\begin{aligned} q_\eta = \frac{1}{2}\left( Z_\eta v_\eta + T_\eta \right) , \quad p_\eta = \frac{1}{2}\left( Z_\eta v_\eta - T_\eta \right) , \quad \eta = x, y, z. \end{aligned}$$
(136)

Here, \(q_\eta \) are the left going characteristics and \(p_\eta \) are the right going characteristics. We will construct boundary data which satisfy the physical boundary conditions (21) exactly and preserve the amplitude of the outgoing characteristics \(q_\eta \) at \(\xi = -1\), and \(p_\eta \) at \(\xi = 1\).

To begin, define the hat-variables preserving the amplitude of outgoing characteristics

$$\begin{aligned}&{q}_\eta \left( {\widehat{v}}_\eta , {\widehat{T}}_\eta , Z_\eta \right) = {q}_\eta \left( {v}_\eta , {T}_\eta , Z_\eta \right) , \quad \text {at} \quad \xi = -1, \nonumber \\&\quad \text {and} \quad {p}_\eta \left( {\widehat{v}}_\eta , {\widehat{T}}_\eta , Z_\eta \right) = {p}_\eta \left( {v}_\eta , {T}_\eta , Z_\eta \right) , \quad \text {at} \quad \xi = 1. \end{aligned}$$
(137)

Since hat-variables also satisfy the physical boundary condition (21), we must have

$$\begin{aligned}&\frac{Z_{\eta }}{2}\left( {1-\gamma _\eta }\right) {\widehat{v}}_\eta -\frac{1+\gamma _\eta }{2} {\widehat{T}}_\eta = 0, \quad \text {at} \quad \xi = -1, \nonumber \\&\quad \text {and} \quad \frac{Z_{\eta }}{2} \left( {1-\gamma _\eta }\right) {\widehat{v}} _\eta + \frac{1+\gamma _\eta }{2}{\widehat{T}}_\eta = 0, \quad \text {at} \quad \xi = 1. \end{aligned}$$
(138)

The algebraic problem for the hat-variables, defined by equations (137) and (138), has a unique solution, namely

$$\begin{aligned}&{\widehat{v}}_\eta = \frac{(1+\gamma _\eta )}{Z_{\eta }}q_\eta , \quad {\widehat{T}}_\eta = {(1-\gamma _\eta )}q_\eta , \quad \text {at} \quad \xi = -1, \nonumber \\&\quad \text {and} \quad {\widehat{v}}_\eta = \frac{(1+\gamma _\eta )}{Z_{\eta }}p_\eta , \quad {\widehat{T}}_\eta = {-(1-\gamma _\eta )}p_\eta , \quad \text {at} \quad \xi = 1. \end{aligned}$$
(139)

The expressions in (139) define a rule to update particle velocities and tractions on the physical boundaries \(\xi = -1, 1\). That is

$$\begin{aligned} v_\eta = {\widehat{v}}_\eta , \quad {T}_\eta = {\widehat{T}}_\eta , \quad \text {at} \quad \xi = -1, \quad \text {and} \quad v_\eta = {\widehat{v}}_\eta , \quad {T}_\eta = {\widehat{T}}_\eta , \quad \text {at} \quad \xi = 1. \end{aligned}$$
(140)

By construction, the hat-variables \({\widehat{v}}_\eta , {\widehat{T}}_\eta \), satisfy the following algebraic identities:

$$\begin{aligned}&{q}_\eta \left( {\widehat{v}}_\eta , {\widehat{T}}_\eta , Z_\eta \right) = {q}_\eta \left( {v}_\eta , {T}_\eta , Z_\eta \right) , \quad \text {at} \quad \xi = -1, \nonumber \\&\quad {p}_\eta \left( {\widehat{v}}_\eta , {\widehat{T}}_\eta , Z_\eta \right) = {p}_\eta \left( {v}_\eta , {T}_\eta , Z_\eta \right) , \quad \text {at} \quad \xi = 1,\end{aligned}$$
(141a)
$$\begin{aligned}&{q}_\eta ^2\left( {v}_\eta , {T}_\eta , Z_\eta \right) -{p}_\eta ^2\left( {\widehat{v}}_\eta , {\widehat{T}}_\eta , Z_\eta \right) = Z_\eta {\widehat{T}}_\eta {\widehat{v}}_\eta , \quad \text {at} \quad \xi = -1, \nonumber \\&\quad {p}_\eta ^2\left( {v}_\eta , {T}_\eta , Z_\eta \right) -{q}_\eta ^2\left( {\widehat{v}}_\eta , {\widehat{T}}_\eta , Z_\eta \right) = -Z_\eta {\widehat{T}}_\eta {\widehat{v}}_\eta , \quad \text {at} \quad \xi = 1,\end{aligned}$$
(141b)
$$\begin{aligned}&{\widehat{T}}_\eta {\widehat{v}}_\eta = \frac{1-\gamma _\eta ^2}{Z_{\eta }}{q}_\eta ^2\left( {v}_\eta , {T}_\eta , Z_\eta \right) \ge 0, \quad \text {at} \quad \xi = -1, \nonumber \\&\quad {\widehat{T}}_\eta {\widehat{v}}_\eta = -\frac{1-\gamma _\eta ^2}{Z_{\eta }}{p}_\eta ^2\left( {v}_\eta , {T}_\eta , Z_\eta \right) \le 0 , \quad \text {at} \quad \xi = 1. \end{aligned}$$
(141c)

The algebraic identities (141a)–(141c) will be crucial in proving numerical stability. Please see [17] for more details.

Interface data

To begin, define the outgoing characteristics at the interface

$$\begin{aligned} q_\eta ^{+} = \frac{1}{2}\left( Z_\eta ^{+} v^{+}_\eta + T_\eta ^{+}\right) , \quad p_\eta ^{-} = \frac{1}{2}\left( Z_\eta ^{-} v^{-}_\eta - T_\eta ^{-}\right) , \quad \eta = x, y, z, \end{aligned}$$
(142)

where \(Z_\eta ^{\pm } > 0\) are the impedances. We define the hat-variables preserving the amplitude of the outgoing characteristics at the interface

$$\begin{aligned}&{\widehat{q}}_\eta ^{+} \left( {\widehat{v}}_\eta ^{+}, {\widehat{T}}_\eta ^{+}, Z_\eta ^{+}\right) = {q}_\eta ^{+} \left( {v}_\eta ^{+}, {T}_\eta ^{+}, Z_\eta ^{+} \right) , \nonumber \\&{\widehat{p}}_\eta ^{-} \left( {\widehat{v}}_\eta ^{-}, {\widehat{T}}_\eta ^{-}, Z_\eta ^{-}\right) = {p}_\eta ^{-}\left( {v}_\eta ^{-}, {T}_\eta ^{-}, Z_\eta ^{-} \right) . \end{aligned}$$
(143)

The hat-variables also satisfy the interface conditions (24) exactly. For each setup, given \({q}_\eta ^{+}, {p}_\eta ^{-} \), the procedure will solve (143), and (24) for the hat-variables, \({\widehat{v}}_\eta ^{\pm }, {\widehat{T}}_\eta ^{\pm }\). We consider an interface separating two elastic solids. The hat-variables must satisfy (143) and the interface conditions, (24), that is, force balance, no opening and no slip conditions. Combining the two equations in (143) and ensuring force balance, \({\widehat{T}}_\eta ^{-} = {\widehat{T}}_\eta ^{+} = {\widehat{T}}_\eta \), we have

$$\begin{aligned}&{\widehat{T}}_\eta + \alpha _\eta \llbracket {{\widehat{v}}_\eta \rrbracket } = \varPhi _\eta , \quad \varPhi _\eta = \alpha _\eta \left( \frac{2}{Z_\eta ^{+}}q_\eta ^{+} - \frac{2}{Z_\eta ^{-}}p_\eta ^{-}\right) , \nonumber \\&\alpha _\eta = \frac{Z_\eta ^{+}Z_\eta ^{-}}{Z_\eta ^{+} + Z_\eta ^{-}} >0, \quad \eta = x, y, z. \end{aligned}$$
(144)

Furthermore, enforcing no opening and no slip conditions, \(\llbracket {{\widehat{v}}_\eta \rrbracket } = 0\), in (144) gives

$$\begin{aligned} {\widehat{T}}_\eta = \varPhi _\eta , \quad \llbracket {{\widehat{v}}_\eta \rrbracket } = 0, \quad \eta = x, y, z. \end{aligned}$$
(145)

We can now define explicitly the hat-variables corresponding to the particle velocities and tractions

$$\begin{aligned} {\widehat{T}}_\eta ^{-} = {\widehat{T}}_\eta ^{+} = {\widehat{T}}_\eta , \quad {\widehat{v}}_\eta ^{+} = \frac{2p_\eta ^{-}+{\widehat{T}}_\eta }{Z_\eta ^{-}} , \quad {\widehat{v}}_\eta ^{-} = \frac{2q_\eta ^{+}-{\widehat{T}}_\eta }{Z_\eta ^{+}} , \quad \quad \eta = x, y, z. \end{aligned}$$
(146)

Note that we have equivalently redefined the physical interface condition (24) as follows

$$\begin{aligned} v_{\eta }^{\pm } = {\widehat{v}}_{\eta }^{\pm }, \quad T_{\eta }^{\pm } = {\widehat{T}}_{\eta }, \quad \eta = x, y, z. \end{aligned}$$
(147)

By construction, the hat-variables \({\widehat{v}}_\eta ^{\pm }\), \({\widehat{T}}_\eta ^{\pm }\) satisfy the following algebraic identities:

$$\begin{aligned}&{q}_\eta \left( {\widehat{v}}_\eta ^{+}, {\widehat{T}}_\eta ^{+}, Z_\eta ^{+}\right) = {q}_\eta \left( {v}_\eta ^{+}, {T}_\eta ^{+}, Z_\eta ^{+} \right) , \quad {p}_\eta \left( {\widehat{v}}_\eta ^{-}, {\widehat{T}}_\eta ^{-}, Z_\eta ^{-}\right) = {p}_\eta \left( {v}_\eta ^{-}, {T}_\eta ^{-}, Z_\eta ^{-} \right) , \end{aligned}$$
(148a)
$$\begin{aligned}&\left( q_\eta ^2 \left( {v}_\eta ^{+}, {T}_\eta ^{+}, Z_\eta ^{+}\right) \right) -{p}_\eta ^2 \left( {\widehat{v}}_\eta ^{+}, {\widehat{T}}_\eta ^{+}, Z_\eta ^{+}\right) = Z_\eta ^{+}{\widehat{T}}_\eta {\widehat{v}}_\eta ^{+}, \nonumber \\&\quad p^2_\eta \left( {v}_\eta ^{-}, {T}_\eta ^{-}, Z_\eta ^{-} \right) -{q}^2_\eta \left( {\widehat{v}}_\eta ^{-}, {\widehat{T}}_\eta ^{-}, Z_\eta ^{-}\right) = -Z_\eta ^{-}{\widehat{T}}_\eta {\widehat{v}}_\eta ^{-}, \end{aligned}$$
(148b)
$$\begin{aligned}&\frac{1}{Z_\eta ^+}\left( q^2_\eta \left( {v}_\eta ^{+}, {T}_\eta ^{+}, Z_\eta ^{+} \right) - {p}^2_\eta \left( {\widehat{v}}_\eta ^{+}, {\widehat{T}}_\eta ^{+}, Z_\eta ^{+}\right) \right) + \frac{1}{Z_\eta ^-}\left( p^2_\eta \left( {v}_\eta ^{-}, {T}_\eta ^{-}, Z_\eta ^{-} \right) \right. \nonumber \\&\quad \left. -{q}^2_\eta \left( {\widehat{v}}_\eta ^{-}, {\widehat{T}}_\eta ^{-}, Z_\eta ^{-}\right) \right) = {\widehat{T}}_\eta \llbracket {{\widehat{v}}_\eta \rrbracket } \equiv 0, \end{aligned}$$
(148c)

The identities (148a)–(148b) can be easily verified, see [17], and will be useful in the prove of numerical stability.

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Duru, K., Rannabauer, L., Gabriel, AA. et al. A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form. Numer. Math. 146, 729–782 (2020). https://doi.org/10.1007/s00211-020-01160-w

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Keywords

  • Elastic waves
  • First order systems
  • Perfectly Matched layer
  • Laplace transforms
  • Boundary and interface conditions
  • Stability
  • High order accuracy
  • Discontinuous Galerkin method

Mathematics Subject Classification

  • 35F55
  • 35F46
  • 65M60
  • 65M70
  • 65M12
  • 65M15