High-Order Locally A-Stable Implicit Schemes for Linear ODEs

Abstract

Accurate simulations of wave propagation in complex media like Earth subsurface can be performed with a reasonable computational burden by using hybrid meshes stuffing fine and coarse cells. Locally implicit time discretizations are then of great interest. They indeed allow using unconditionally stable Schemes in the regions of computational domain covered by small cells. The receivable values of the time step are then increased which reduces the computational costs while limiting the dispersion effects. In this work we construct a method that combines optimized explicit Schemes and implicit Schemes to form locally implicit schemes for linear ODEs, including in particular semi-discretized wave problems that are considered herein for numerical experiments. Both the explicit and implicit schemes used are one-step methods constructed using their stability function. The stability function of the explicit schemes are computed by maximizing the time step that can be chosen. The implicit schemes used are unconditionally stable and do not necessary require the same number of stages as the explicit schemes. The performance assessment we provide shows a very good level of accuracy for locally implicit schemes. It also shows that a locally implicit scheme is a good compromise between purely explicit and purely implicit schemes in terms of computational time and memory usage.

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References

  1. 1.

    Cohen, G., Fauqueux, S.: Mixed finite elements with mass-lumping for the transient wave equation. J. Comput. Acoust. 8, 171–188 (2000)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Cohen, G., Pernet, S.: Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations. Springer, Berlin (2017)

    Google Scholar 

  3. 3.

    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods Algorithms, Analysis, and Applications. Springer, Berlin (2008)

    Google Scholar 

  4. 4.

    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford Science Publications, Oxford (2003)

    Google Scholar 

  5. 5.

    Duruflé, M.: Intégration numérique et éléments finis d’ordre élevé appliqués aux équations de Maxwell en regime harmonique. Ph.D. thesis, Université Paris Dauphine (2006)

  6. 6.

    Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I Nonstiff Problem. Springer, Berlin (2008)

    Google Scholar 

  7. 7.

    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, New York (2008)

    Google Scholar 

  8. 8.

    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II Stiff and Differential-Algebraic Problem. Springer, Berlin (2010)

    Google Scholar 

  9. 9.

    Mead, J.L., Renaut, R.A.: Optimal Runge-Kutta methods for first order pseudospectral operators. J. Comput. Phys. 56, 404–419 (1999)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gilbert, J.C., Joly, P.: Higher order time stepping for second order hyperbolic problems and optimal CFL conditions. Comput. Methods Appl. Sci. 16, 67–93 (2008)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Joly, P., Rodrìguez, J.: Optimized higher order time discretization of second order hyperbolic problems: construction and numerical study. J. Comput. Appl. Math. 234, 1953–1961 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Parsani, M., Ketcheson, D.I., Deconinck, W.: Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM J. Sci. Comput. 35, A957–A986 (2013)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Ketcheson, D.I., Ahmadia, A.J.: Optimal stability polynomials for numerical integration of initial value problems. Commun. Appl. Math. Comput. Sci. 7, 247–271 (2012)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Duruflé, M., N’diaye, M.: Optimized high-order explicit Runge-Kutta-Nyström schemes. In: Bittencourt, M., Dumont, N., Hesthaven, J.S. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM-2016, Springer (2016)

  15. 15.

    Diaz, J., Grote, M.: Energy conserving explicit local time stepping for second-order wave equations. SIAM J. Sci. Comput. 31, 1985–2014 (2009)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Grote, M., Mehlin, M., Mitkova, T.: Runge-Kutta-based explicit local time-stepping methods for wave propagation. SIAM J. Sci. Comput. 37, A747–A775 (2015)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Piperno, S.: Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems. ESAIM Math. Model. Numer. Anal. 40, 815–841 (2006)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Rodrìguez, J.: Raffinement de maillage spatio-temporel pour les équations de l’élastodynamique. Ph.D. thesis, Université Paris Dauphine (2004)

  19. 19.

    Skvortsov, L.M.: Diagonally implicit Runge-Kutta methods for stiff problems. Comput. Math. Comput. Phys. 46, 2110–2123 (2006)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Ehle, B.L.: A-stable methods and Padé approximations to the exponential. SIAM J. Numer. Anal. 4, 671–680 (1973)

    MathSciNet  Article  Google Scholar 

  21. 21.

    N’diaye, M.: On the study and development of high-order time integration schemes for ODEs applied to acoustic and electromagnetic wave propagation problems. Ph.D. thesis, Université de Pau et des Pays de l’Adour (2017)

  22. 22.

    Verwer, J.G.: Component splitting for semi-discrete Maxwell equations. BIT 51, 427–445 (2011)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Ascher, U., Ruuth, S., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time dependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Ascher, U., Ruuth, S., Wetton, R.J.: Implicit-explicit methods for time dependent PDE’s. SIAM J. Numer. Anal. 32, 797–823 (1995)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Frank, J., Hundsdorfer, W., Verwer, J.G.: On the stability of implicit-explicit linear multistep methods. Appl. Numer. Math. 25, 193–205 (1997)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Kanevsky, A., Carpenter, M.H., Gottlieb, D., Hesthaven, J.S.: Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes. J. Comput. Phys. 225, 1753–1781 (2007)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Wang, H., Zhang, Q., Shu, C.-W.: Implicitexplicit local discontinuous Galerkin methods with generalized alternating numerical fluxes for convectiondiffusion problems. J. Sci. Comput. 81, 2080–2114 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Descombes, S., Lanteri, S., Moya, L.: Locally implicit discontinuous Galerkin time domain method for electromagnetic wave propagation in dispersive media applied to numerical dosimetry in biological tissues. SIAM J. Sci. Comput. 38, A2611–A2633 (2016)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Descombes, S., Lanteri, S., Moya, L.: Locally implicit time integration strategies in a discontinuous Galerkin method for Maxwell’s equations. Tech. report, INRIA (2012)

  30. 30.

    Hochbruck, M., Sturm, A.: Error Analysis of a second-order locally implicit method for linear Maxwell’s equations. SIAM J. Numer. Anal. 54, 3167–3191 (2016)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Sturm, A.: Locally implicit time integration for linear Maxwell’s equations. Ph.D. thesis, Karlsruhe Institute of Technology (2017)

  32. 32.

    Barucq, H., Duruflé, M., N’Diaye, M.: High-order Padé and singly diagonally Runge-Kutta schemes for linear ODEs, application to wave propagation problems. Numer. Methods Partial Differ. Equ. 34, 760–798 (2017)

    Article  Google Scholar 

  33. 33.

    Kronbichler, M., Schoeder, S., Müller, C., Wall, W.A.: Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustics wave equation. Int. J. Numer. Meth. Eng. 106, 712–739 (2016)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Nguyen, N., Peraire, J., Cockburn, B.: High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics. J. Comput. Phys. 230, 3695–3718 (2011)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Kennedy, C.A., Carpenter, M.H.: Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44, 139–181 (2003)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Hernandez, V., Roman, J.E., Vidal, V.: Slepc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31, 351–362 (2005)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Amestoy, P., Duff, I., Koster, J., L’Excellent, J.Y.: A fully asynchronous multi-frontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23, 15–41 (2001)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Sanz-Serna, J.M., Verwer, J.G., Hundsdorfer, W.H.: Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations. Numer. Math. 50(4), 405–418 (1986)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

This work has been supported by the INRIA-TOTAL strategic action DIP (dip.inria.fr). Experiments presented in this work were carried out using the PlaFRIM experimental platform. Helene Barucq has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 777778 (MATHROCKS).

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Correspondence to Mamadou N’diaye.

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Remarks on the Implementation of the HDG Formulation of the Acoustic Wave Equation.

Remarks on the Implementation of the HDG Formulation of the Acoustic Wave Equation.

For the first order acoustic wave Eq. (24), the local variational formulation set on an element K in the HDG formulation is given as

$$\begin{aligned} \left\{ \begin{aligned}&\frac{d}{dt} \int _K \rho \, u \, \varphi _i \, dx + \int _K v \cdot \nabla \varphi _i \, dx - \int _{\partial K} v \cdot n \, \varphi _i \, dx + \int _{\partial K} \tau (u-\lambda ) \, \varphi _i \, dx = 0 \\&\frac{d}{dt} \int _K \mu ^{-1} v \cdot \psi _i \, dx - \int _K \nabla u \cdot \psi _i + \int _{\partial K} (u-\lambda ) \psi _i \cdot n \, dx = 0 \\&\int _{\partial K} (-v \cdot n + \tau (u- \lambda ) ) \, q \, dx = 0 \end{aligned} \right. \end{aligned}$$
(25)

where uv are volume unknowns (discontinuous) and \(\lambda \) a surface unknown. \(\varphi _i, \psi _i, q\) are the test functions associated with uv and \(\lambda \). The penalization parameter \(\tau \) is equal to \(\sqrt{\rho \, \mu }\). The last equation is modified when the boundary of K meets the boundary of the computational domain. For instance, is a first-order absorbing boundary condition is set

$$\begin{aligned} v \cdot n + \sqrt{\rho \, \mu } \, u = f_A . \end{aligned}$$

the last equation of (25) becomes:

$$\begin{aligned} \int _{\varGamma _N} (-v \cdot n + \tau (u - 2 \lambda )) \, q \, dx = -\int _{\varGamma _N} f_A \, q \, dx. \end{aligned}$$

The HDG formulation provides locally for each element K the semi-discrete system

$$\begin{aligned} \left\{ \begin{aligned}&M_u \dfrac{dU}{dt} + K_u V + C_u \varLambda = 0 \\&M_v \dfrac{dV}{dt} + K_v U + C_v \varLambda = 0 \\&C_\lambda \varLambda + C_u^T U + C_v^T V = 0 \\ \end{aligned} \right. \end{aligned}$$
(26)

\(C_\lambda \) is multiplied by two for an edge with an absorbing boundary condition. When the ODE (20) (which is set in the fine region) is solved with an implicit scheme, a linear system of the form

$$\begin{aligned} \beta Y + A P Y = F \end{aligned}$$

has to be solved for close dofs (with \(Y=(U, V)\)). For elements located in the fine region, we have

$$\begin{aligned} \left\{ \begin{aligned}&\beta M_u U + K_u V + C_u \varLambda = F_u \\&\beta M_v V + K_v U + C_v \varLambda = F_v \\&C_\lambda \varLambda + C_u^T U + C_v^T V = 0 \\ \end{aligned} \right. \end{aligned}$$
(27)

Unknowns U and V are eliminated element-wise to obtain a local system in \(\varLambda \)

$$\begin{aligned} \left[ C_\lambda - \left( C_u^T C_v^T \right) \left( \begin{array}{cc} \beta M_u &{} K_u \\ K_v &{} \beta M_v \end{array} \right) ^{-1} \left( \begin{array}{c} C_u \\ C_v \end{array} \right) \right] \varLambda = - \left( C_u^T C_v^T \right) \left( \begin{array}{cc} \beta M_u &{} K_u \\ K_v &{} \beta M_v \end{array} \right) ^{-1} \left( \begin{array}{c} F_u \\ F_v \end{array} \right) . \end{aligned}$$

This equation has to be assembled with all other elements to obtain the final system solved by \(\varLambda \). For adjacent elements (located on the coarse region), we have:

$$\begin{aligned} \left\{ \begin{aligned}&\beta M_u U + C_u \varLambda = F_u \\&\beta M_v V + C_v \varLambda = F_v \\&C_\lambda \varLambda = 0 \\ \end{aligned} \right. \end{aligned}$$
(28)

where only unknowns on edges of the fine region are concerned for \(\varLambda \). The equation to be assembled with other elements is given as

$$\begin{aligned} C_\lambda \varLambda = 0. \end{aligned}$$

As a result, when the linear system is assembled for \(\varLambda \), only unknowns located on edges (faces in 3-D) of the fine region are involved. It is actually equivalent to add the contribution \(C_\lambda \varLambda = 0\) or impose a fictitious homogeneous absorbing boundary condition on edges located at the interface between the fine and coarse region. Once \(\varLambda \) has been computed on the edges of the fine region (\(\varLambda \) is null for other edges), the unknown U and V are reconstructed element-wise, e.g. in adjacent elements:

$$\begin{aligned} U= & {} \dfrac{1}{\beta } M_u^{-1} \left( F_u - C_u \varLambda \right) \\ V= & {} \dfrac{1}{\beta } M_v^{-1} \left( F_v - C_v \varLambda \right) \end{aligned}$$

For quadrilateral/hexahedral elements, the mass matrices \(M_u\) and \(M_v\) are diagonal, and the elimination/reconstruction of U and V can be lead efficiently as detailed in [21].

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Barucq, H., Duruflé, M. & N’diaye, M. High-Order Locally A-Stable Implicit Schemes for Linear ODEs. J Sci Comput 85, 31 (2020). https://doi.org/10.1007/s10915-020-01313-x

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Keywords

  • Time integration
  • Hybrid discontinuous Galerkin method
  • Hyperbolic problems
  • Wave equations