A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity

doi:10.1007/s42967-023-00294-6

Abstract

Abstract: The deferred correction (DeC) is an iterative procedure, characterized by increasing the accuracy at each iteration, which can be used to design numerical methods for systems of ODEs. The main advantage of such framework is the automatic way of getting arbitrarily high order methods, which can be put in the Runge-Kutta (RK) form. The drawback is the larger computational cost with respect to the most used RK methods. To reduce such cost, in an explicit setting, we propose an efficient modification: we introduce interpolation processes between the DeC iterations, decreasing the computational cost associated to the low order ones. We provide the Butcher tableaux of the new modified methods and we study their stability, showing that in some cases the computational advantage does not affect the stability. The flexibility of the novel modification allows nontrivial applications to PDEs and construction of adaptive methods. The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.

Key words: Efficient deferred correction (DeC), Arbitrary high order, Stability, Adaptive methods, Hyperbolic PDEs

CLC Number:

65M12
65L20

Lorenzo Micalizzi, Davide Torlo. A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity[J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1629-1664.

TrendMD

References

1. Abgrall, R.: Residual distribution schemes: current status and future trends. Comput. Fluids 35(7), 641–669 (2006)
2. Abgrall, R.: High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. J. Sci. Comput. 73(2/3), 461–494 (2017)
3. Abgrall, R., Ivanova, K.: Staggered residual distribution scheme for compressible flow. arXiv: 2111. 10647 (2022)
4. Abgrall, R., Bacigaluppi, P., Tokareva, S.: High-order residual distribution scheme for the timedependent Euler equations of fluid dynamics. Comput. Math. Appl. 78(2), 274–297 (2019)
5. Abgrall, R., Le Mélédo, E., Öffner, P., Torlo, D.: Relaxation deferred correction methods and their applications to residual distribution schemes. SMAI J. Comput. Math. 8, 125–160 (2022)
6. Abgrall, R., Torlo, D.: High order asymptotic preserving deferred correction implicit-explicit schemes for kinetic models. SIAM J. Sci. Comput. 42(3), 816–845 (2020)
7. Bacigaluppi, P., Abgrall, R., Tokareva, S.: “A posteriori’’ limited high order and robust schemes for transient simulations of fluid flows in gas dynamics. J. Comput. Phys. 476, 111898 (2023)
8. Boscarino, S., Qiu, J.-M.: Error estimates of the integral deferred correction method for stiff problems. ESAIM Math. Model. Numer. Anal. 50(4), 1137–1166 (2016)
9. Boscarino, S., Qiu, J.-M., Russo, G.: Implicit-explicit integral deferred correction methods for stiff problems. SIAM J. Sci. Comput. 40(2), 787–816 (2018)
10. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Auckland (2016)
11. Cheli, F., Diana, G.: Advanced Dynamics of Mechanical Systems. Springer, Cham (2015)
12. Christlieb, A., Ong, B., Qiu, J.-M.: Comments on high-order integrators embedded within integral deferred correction methods. Commun. Appl. Math. Comput. Sci. 4(1), 27–56 (2009)
13. 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)
14. Ciallella, M., Micalizzi, L., Öffner, P., Torlo, D.: An arbitrary high order and positivity preserving method for the shallow water equations. Comput. Fluids 247, 105630 (2022)
15. Cohen, G., Joly, P., Roberts, J.E., Tordjman, N.: Higher order triangular finite elements with mass lumping for the wave equation. SIAM J. Numer. Anal. 38(6), 2047–2078 (2001)
16. Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT 40(2), 241–266 (2000)
17. Fox, L., Goodwin, E.: Some new methods for the numerical integration of ordinary differential equations. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 45, pp. 373–388. Cambridge University Press, Cambridge (1949)
18. Han Veiga, M., Öffner, P., Torlo, D.: DeC and ADER: similarities, differences and a unified framework. J. Sci. Comput. 87(1), 1–35 (2021)
19. Huang, J., Jia, J., Minion, M.: Accelerating the convergence of spectral deferred correction methods. J. Comput. Phys. 214(2), 633–656 (2006)
20. Jund, S., Salmon, S.: Arbitrary high-order finite element schemes and high-order mass lumping. Int. J. Appl. Math. Comput. Sci. 17(3), 375–393 (2007)
21. Ketcheson, D., Bin Waheed, U.: A comparison of high-order explicit Runge-Kutta, extrapolation, and deferred correction methods in serial and parallel. Commun. Appl. Math. Comput. Sci. 9(2), 175–200 (2014)
22. Layton, A.T., Minion, M.L.: Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics. J. Comput. Phys. 194(2), 697–715 (2004)
23. Layton, A.T., Minion, M.L.: Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations. BIT Numer. Math. 45(2), 341–373 (2005)
24. Liu, Y., Shu, C.-W., Zhang, M.: Strong stability preserving property of the deferred correction time discretization. J. Comput. Math. 26(5), 633–656 (2008)
25. Micalizzi, L., Torlo, D., Boscheri, W.: Efficient iterative arbitrary high order methods: an adaptive bridge between low and high order. arXiv: 2212. 07783 (2022)
26. Michel, S., Torlo, D., Ricchiuto, M., Abgrall, R.: Spectral analysis of continuous FEM for hyperbolic PDEs: influence of approximation, stabilization, and time-stepping. J. Sci. Comput. 89(2), 1–41 (2021)
27. Michel, S., Torlo, D., Ricchiuto, M., Abgrall, R.: Spectral analysis of high order continuous FEM for hyperbolic PDEs on triangular meshes: influence of approximation, stabilization, and time-stepping. J. Sci. Comput. 94(3), 49 (2023)
28. Minion, M.L.: Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1(3), 471–500 (2003)
29. Minion, M.L.: Semi-implicit projection methods for incompressible flow based on spectral deferred corrections. Appl. Numer. Math. 48(3/4), 369–387 (2004)
30. Minion, M.L.: A hybrid parareal spectral deferred corrections method. Commun. Appl. Math. Comput. Sci. 5(2), 265–301 (2011)
31. Öffner, P., Torlo, D.: Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes. Appl. Numer. Math. 153, 15–34 (2020)
32. Pasquetti, R., Rapetti, F.: Cubature points based triangular spectral elements: an accuracy study. J. Math. Stud. 51(1), 15–25 (2018)
33. Ricchiuto, M., Abgrall, R.: Explicit Runge-Kutta residual distribution schemes for time dependent problems: second order case. J. Comput. Phys. 229(16), 5653–5691 (2010)
34. Ricchiuto, M., Torlo, D.: Analytical travelling vortex solutions of hyperbolic equations for validating very high order schemes. arXiv: 2109. 10183 (2021)
35. Speck, R., Ruprecht, D., Emmett, M., Minion, M., Bolten, M., Krause, R.: A multi-level spectral deferred correction method. BIT Numer. Math. 55(3), 843–867 (2015)
36. Torlo, D.: Hyperbolic problems: high order methods and model order reduction. PhD thesis, University Zurich (2020)
37. Wanner, G., Hairer, E.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, vol. 375. Springer, Berlin (1996)

[1]	Mária Lukáčová-Medvid'ová, Yuhuan Yuan. Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation [J]. Communications on Applied Mathematics and Computation, 2024, 6(4): 2215-2238.
[2]	Saray Busto, Michael Dumbser. A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems [J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1742-1778.
[3]	Jingcheng Lu, Eitan Tadmor. Revisting High-Resolution Schemes with van Albada Slope Limiter [J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1924-1953.
[4]	Michel Bergmann, Afaf Bouharguane, Angelo Iollo, Alexis Tardieu. High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation [J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1954-1977.
[5]	Dinshaw S. Balsara, Deepak Bhoriya, Chi-Wang Shu, Harish Kumar. Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-conservative Products [J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 907-962.
[6]	Hailiang Liu, Jia-Hao He, Xuping Tian. Anderson Acceleration of Gradient Methods with Energy for Optimization Problems [J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 1299-1318.
[7]	Maohui Lyu, Vrushali A. Bokil, Yingda Cheng, Fengyan Li. Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media [J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 30-63.
[8]	Joseph Hunter, Zheng Sun, Yulong Xing. Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation [J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 658-687.
[9]	Changpin Li, Dongxia Li, Zhen Wang. L1/LDG Method for the Generalized Time-Fractional Burgers Equation in Two Spatial Dimensions [J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1299-1322.
[10]	Dakang Cen, Zhibo Wang, Seakweng Vong. Efficient Finite Difference/Spectral Method for the Time Fractional Ito Equation Using Fast Fourier Transform Technic [J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1591-1600.
[11]	Robin Ming Chen, Feimin Huang, Dehua Wang, Difan Yuan. On the Vortex Sheets of Compressible Flows [J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 967-986.
[12]	Xian-ting Wang, Yun-guang Lu, Naoki Tsuge. Global Existence and Stability of Solutions to River Flow System [J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1247-1255.
[13]	R. Abgrall, J. Nordström, P. Öffner, S. Tokareva. Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability [J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 573-595.
[14]	Linjin Li, Jingmei Qiu, Giovanni Russo. A High-Order Semi-Lagrangian Finite Difference Method for Nonlinear Vlasov and BGK Models [J]. Communications on Applied Mathematics and Computation, 2023, 5(1): 170-198.
[15]	F. S. Mousavinejad, M. FatehiNia, A. Ebrahimi. P-Bifurcation of Stochastic van der Pol Model as a Dynamical System in Neuroscience [J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1293-1312.