High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation

Abstract

We present a high-order Galerkin method in both space and time for the 1D unsteady linear advection-diffusion equation. Three Interior Penalty Discontinuous Galerkin (IPDG) schemes are detailed for the space discretization, while the time integration is performed at the same order of accuracy thanks to an Arbitrary high order DERivatives (ADER) method. The orders of convergence of the three ADER-IPDG methods are carefully examined through numerical illustrations, showing that the approach is consistent, accurate, and efficient. The numerical results indicate that the symmetric version of IPDG is typically more accurate and more efficient compared to the other approaches.

Keywords

Advection-diffusion / Galerkin / Arbitrary high order DERivatives (ADER) approach / Interior Penalty Discontinuous Galerkin (IPDG) / High-order schemes / Empirical convergence rates

References

[1.] Abgrall R. Residual distribution schemes: current status and future trends. Comput. Fluids, 2006, 35(7): 641-669,

[2.] Baldassari, C.: Modélisation et simulation numérique pour la migration terrestre par équation d’ondes. Theses, Université de Pau et des Pays de l’Adour (2009). https://theses.hal.science/tel-00472810

[3.] Bergmann M, Carlino MG, Iollo A. Second order ADER scheme for unsteady advection-diffusion on moving overset grids with a compact transmission condition. SIAM J. Sci. Comput., 2022, 44(1): 524-553,

[4.] Bergmann M, Carlino MG, Iollo A, Telib H. ADER scheme for incompressible Navier-Stokes equations on overset grids with a compact transmission condition. J. Comput. Phys., 2022, 467,

[5.] Bergmann M, Fondanèche A, Iollo A. An Eulerian finite-volume approach of fluid-structure interaction problems on quadtree meshes. J. Comput. Phys., 2022, 471,

[6.] Boscheri W, Dumbser M. Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes. J. Comput. Phys., 2017, 346: 449-479,

[7.] Boscheri W, Loubère R. High order accurate direct arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for non-conservative hyperbolic systems with stiff source terms. Commun. Comput. Phys., 2017, 21(1): 271-312,

[8.] Busto S, Dumbser M, Escalante C, Favrie N, Gavrilyuk S. On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems. J. Sci. Comput., 2021, 87(2): 48,

[9.] Busto S, Toro EF, Vázquez-Cendón ME. Design and analysis of ADER-type schemes for model advection-diffusion-reaction equations. J. Comput. Phys., 2016, 327: 553-575,

[10.] Cockburn B. Discontinuous Galerkin methods. ZAMM, 2003, 83(11): 731-754,

[11.] Cockburn, B., Karniadakis, G.E., Shu, C.-W.: Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, Berlin (2000). https://doi.org/10.1007/978-3-642-59721-3

[12.] Cockburn B, Shu C-W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput., 2001, 16: 173-261,

[13.] Dumbser M, Balsara DS, Toro EF, Munz C-D. A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys., 2008, 227(18): 8209-8253,

[14.] Dumbser M, Fambri F, Tavelli M, Bader M, Weinzierl T. Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine. Axioms, 2018, 7(3): 63,

[15.] Dumbser M, Munz C-D. Building blocks for arbitrary high order discontinuous Galerkin schemes. J. Sci. Comput., 2006, 27(1–3): 215-230,

[16.] Fambri F. Discontinuous Galerkin methods for compressible and incompressible flows on space-time adaptive meshes: toward a novel family of efficient numerical methods for fluid dynamics. Arch. Comput. Methods Eng., 2020, 27(1): 199-283,

[17.] Fambri F, Dumbser M. Semi-implicit discontinuous Galerkin methods for the incompressible Navier-Stokes equations on adaptive staggered Cartesian grids. Comput. Methods Appl. Mech. Eng., 2017, 324: 170-203,

[18.] Fambri F, Dumbser M, Zanotti O. Space-time adaptive ADER-DG schemes for dissipative flows: compressible Navier-Stokes and resistive MHD equations. Comput. Phys. Commun., 2017, 220: 297-318,

[19.] Gassner G, Dumbser M, Hindenlang F, Munz C-D. Explicit one-step time discretizations for discontinuous Galerkin and finite volume schemes based on local predictors. J. Comput. Phys., 2011, 230(11): 4232-4247,

[20.] Hartmann, R.: Numerical analysis of higher order discontinuous Galerkin finite element methods. In: VKI LS 2008-08: CFD-ADIGMA Course on very High Order Discretization Methods, Oct. 13–17 (2008). https://elib.dlr.de/57074/1/Har08b.pdf

[21.] Hidalgo A, Dumbser M. ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations. J. Sci. Comput., 2011, 48(1/2/3): 173-189,

[22.] Liu H, Yan J. The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal., 2009, 47(1): 675-698,

[23.] Loubère R, Dumbser M, Diot S. A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws. Commun. Comput. Phys., 2014, 16(3): 718-763,

[24.] Ricchiuto M, Abgrall R. Explicit Runge-Kutta residual distribution schemes for time dependent problems: second order case. J. Comput. Phys., 2010, 229(16): 5653-5691,

[25.] Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. Frontiers in Applied Mathematics. SIAM, Philadelphia (2008) (OCLC: ocn226292048)

[26.] Titarev VA, Toro EF. ADER: arbitrary high order Godunov approach. J. Sci. Comput., 2002, 17(1/4): 609-618,

[27.] Toro EF, Montecinos GI. Advection-diffusion-reaction equations: hyperbolization and high-order ADER discretizations. SIAM J. Sci. Comput., 2014, 36(5): 2423-2457,

[28.] Zahran YH. Central ADER schemes for hyperbolic conservation laws. J. Math. Anal. Appl., 2008, 346(1): 120-140,