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DeC and ADER: similarities, differences and a unified framework. (English) Zbl 1459.76086
Summary: In this paper, we demonstrate that the explicit ADER approach as it is used inter alia in [O. Zanotti et al., Comput. Fluids 118, 204–224 (2015; Zbl 1390.76381)] can be seen as a special interpretation of the deferred correction (DeC) method as introduced in [A. Dutt et al., BIT 40, No. 2, 241–266 (2000; Zbl 0959.65084)]. By using this fact, we are able to embed ADER in a theoretical background of time integration schemes and prove the relation between the accuracy order and the number of iterations which are needed to reach the desired order. Next, we extend our investigation to stiff ODEs, treating these source terms implicitly. Some differences in the interpretation and implementation can be found. Using DeC yields typically a much simpler implementation, while ADER benefits from a higher accuracy, at least for our numerical simulations. Then, we also focus on the PDE case and present common space-time discretizations using DeC and ADER in closed forms. Finally, in the numerical section we investigate A-stability for the ADER approach – this is done for the first time up to our knowledge – for different order using several basis functions and compare them with the DeC ansatz. Then, we compare the performance of ADER and DeC for stiff and non-stiff ODEs and verify our analysis focusing on two basic hyperbolic problems.
76M12 Finite volume methods applied to problems in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Abgrall, R., Residual distribution schemes: current status and future trends, Comput. Fluids, 35, 7, 641-669 (2006) · Zbl 1177.76205
[2] Abgrall, R., High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices, J. Sci. Comput., 73, 2, 461-494 (2017) · Zbl 1398.65242
[3] Abgrall, R.; Bacigaluppi, P.; Tokareva, S., High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics, Comput. Math. Appl., 78, 2, 274-297 (2019) · Zbl 1442.65245
[4] Abgrall, R., Meledo, E.l., Öffner, P.: On the connection between residual distribution schemes and flux reconstruction. arXiv preprint arXiv:1807.01261 (2018)
[5] Abgrall, R., Mélédo, E.l., Öffner, P., Ranocha, H.: Error boundedness of correction procedure via reconstruction/flux reconstruction and the connection to residual distribution schemes. In: Bressan, A., Lewicka, M., Wang, D., Zheng, Y. (eds.) Hyperbolic Problems: Theory. Numerics, Applications, Volume 10 of AIMS on Applied Mathematics, pp. 215-222. American Institute of Mathematical Sciences, Springfield (2020)
[6] Abgrall, R., Nordström, J., Öffner, P., Tokareva, S.: Analysis of the SBP-SAT stabilization for finite element methods part II: entropy stability. Commun. Appl. Math. Comput. (accepted) (2020)
[7] Abgrall, R.; Torlo, D., High order asymptotic preserving deferred correction implicit-explicit schemes for kinetic models, SIAM J. Sci. Comput., 42, 3, B816-B845 (2020) · Zbl 07226088
[8] Bacaer, N., A Short History of Mathematical Population Dynamics (2011), Berlin: Springer, Berlin · Zbl 1321.92028
[9] Balsara, DS; Meyer, C.; Dumbser, M.; Du, H.; Xu, Z., Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes-speed comparisons with Runge-Kutta methods, J. Comput. Phys., 235, 934-969 (2013) · Zbl 1291.76237
[10] Busto, S.; Chiocchetti, S.; Dumbser, M.; Gaburro, E.; Peshkov, I., High order ADER schemes for continuum mechanics, Front. Phys., 8, 32 (2020)
[11] Butcher, JC, Numerical Methods for Ordinary Differential Equations (2008), Hoboken: Wiley, Hoboken · Zbl 1167.65041
[12] Christlieb, A.; Ong, B.; Qiu, J-M, Integral deferred correction methods constructed with high order Runge-Kutta integrators, Math. Comput., 79, 270, 761-783 (2010) · Zbl 1209.65073
[13] Dematté, R.; Titarev, VA; Montecinos, G.; Toro, E., ADER methods for hyperbolic equations with a time-reconstruction solver for the generalized Riemann problem: the scalar case, Commun. Appl. Math. Comput., 2, 369-402 (2020)
[14] 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., 227, 18, 8209-8253 (2008) · Zbl 1147.65075
[15] Dumbser, M.; Enaux, C.; Toro, EF, Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 227, 8, 3971-4001 (2008) · Zbl 1142.65070
[16] Dumbser, M.; Fambri, F.; Tavelli, M.; Bader, M.; Weinzierl, T., Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine, Axioms, 7, 3, 63 (2018) · Zbl 1434.65179
[17] Dutt, A.; Greengard, L.; Rokhlin, V., Spectral deferred correction methods for ordinary differential equations, BIT Numer. Math., 40, 2, 241-266 (2000) · Zbl 0959.65084
[18] Glaubitz, J.; Öffner, P., Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points, Appl. Numer. Math., 151, 98-118 (2020) · Zbl 1434.65180
[19] Glaubitz, J.; Öffner, P.; Sonar, T., Application of modal filtering to a spectral difference method, Math. Comput., 87, 309, 175-207 (2018) · Zbl 1376.65133
[20] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems (1996), Berlin: Springer, Berlin · Zbl 0859.65067
[21] Han Veiga, M., Öffner, P., Torlo, D.: ADER and DeC implementations, 02 (2020). https://git.math.uzh.ch/abgrall_group/dec-is-ader
[22] Huang, J.; Shu, C-W, Positivity-preserving time discretizations for production-destruction equations with applications to non-equilibrium flows, J. Sci. Comput., 78, 3, 1811-1839 (2019) · Zbl 1420.35190
[23] Jackson, H., On the eigenvalues of the ADER-WENO Galerkin predictor, J. Comput. Phys., 333, 409-413 (2017) · Zbl 1380.65269
[24] Ketcheson, DI, Relaxation Runge-Kutta methods: conservation and stability for inner-product norms, SIAM J. Numer. Anal., 57, 2850-2870 (2019) · Zbl 1427.65115
[25] Liu, Y.; Shu, C-W; Zhang, M., Strong stability preserving property of the deferred correction time discretization, J. Comput. Math., 26, 633-656 (2008) · Zbl 1174.65036
[26] Liu, Y.; Vinokur, M.; Wang, Z., Spectral difference method for unstructured grids i: basic formulation, J. Comput. Phys., 216, 2, 780-801 (2006) · Zbl 1097.65089
[27] Minion, ML, Semi-implicit spectral deferred correction methods for ordinary differential equations, Commun. Math. Sci., 1, 3, 471-500 (2003) · Zbl 1088.65556
[28] Nordström, J.; Lundquist, T., Summation-by-parts in time, J. Comput. Phys., 251, 487-499 (2013) · Zbl 1349.65399
[29] Öffner, P.; Ranocha, H., Error boundedness of discontinuous Galerkin methods with variable coefficients, J. Sci. Comput., 79, 3, 1572-1607 (2019) · Zbl 1418.65161
[30] Öffner, P.; Torlo, D., Arbitrary high-order, conservative and positive preserving Patankar-type deferred correction schemes, Appl. Numer. Math., 153, 15-34 (2020) · Zbl 1437.65073
[31] Rackauckas, C., Nie, Q.: DifferentialEquations.jl—a performant and feature-rich ecosystem for solving differential equations in Julia. J. Open Res. Softw. 5(1), 15 (2017)
[32] Ranocha, H.; Sayyari, M.; Dalcin, L.; Parsani, M.; Ketcheson, DI, Relaxation Runge-Kutta methods: fully-discrete explicit entropy-stable schemes for the compressible Euler and Navier-Stokes equations, SIAM J. Sci. Comput., 42, 2, A612-A638 (2020) · Zbl 1432.76207
[33] Schwartzkopff, T.; Munz, C-D; Toro, EF, ADER: a high-order approach for linear hyperbolic systems in 2d, J. Sci. Comput., 17, 1-4, 231-240 (2002) · Zbl 1022.76034
[34] Titarev, VA; Toro, EF, ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17, 1-4, 609-618 (2002) · Zbl 1024.76028
[35] Titarev, VA; Toro, EF, Analysis of ADER and ADER-WAF schemes, IMA J. Numer. Anal., 27, 3, 616-630 (2006) · Zbl 1119.65083
[36] Torlo, D.: Hyperbolic problems: high order methods and model order reduction. PhD thesis, University Zurich, (2020) · Zbl 1437.65073
[37] Toro, E., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (2009), Berlin: Springer, Berlin · Zbl 1227.76006
[38] Toro, E.F., Millington, R.C., Nejad, L.A.M.: Towards very high order Godunov schemes. In: Toro, E.F. (eds.) Godunov Methods. Springer, Boston, MA (2001). doi:10.1007/978-1-4615-0663-8_87 · Zbl 0989.65094
[39] Toro, E.; Titarev, V., Solution of the generalized Riemann problem for advection-reaction equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 458, 2018, 271-281 (2002) · Zbl 1019.35061
[40] Vanharen, J.; Puigt, G.; Vasseur, X.; Boussuge, J-F; Sagaut, P., Revisiting the spectral analysis for high-order spectral discontinuous methods, J. Comput. Phys., 337, 379-402 (2017) · Zbl 1415.76577
[41] Han Veiga, M., Velasco-Romero, D.A., Wenger, Q., Teyssier, R.: An arbitrary high-order spectral difference method for the induction equation. arXiv:2005.13563 (2020)
[42] Wanner, G.; Hairer, E., Solving Ordinary Differential Equations II (1996), Berlin: Springer, Berlin · Zbl 0859.65067
[43] Zanotti, O.; Fambri, F.; Dumbser, M.; Hidalgo, A., Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting, Comput. Fluids, 118, 204-224 (2015) · Zbl 1390.76381
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