Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order

Abstract

We propose a new paradigm for designing efficient p-adaptive arbitrary high-order methods. We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match the accuracy achieved in a specific iteration with the discretization accuracy of the same iteration. Apart from the computational advantage, the newly modified methods allow to naturally perform the p-adaptivity, stopping the iterations when appropriate conditions are met. Moreover, the modification is very easy to be included in an existing implementation of an arbitrary high-order iterative scheme and it does not ruin the possibility of parallelization, if this was achievable by the original method. An application to the Arbitrary DERivative (ADER) method for hyperbolic Partial Differential Equations (PDEs) is presented here. We explain how such a framework can be interpreted as an arbitrary high-order iterative scheme, by recasting it as a Deferred Correction (DeC) method, and how to easily modify it to obtain a more efficient formulation, in which a local a posteriori limiter can be naturally integrated leading to the p-adaptivity and structure-preserving properties. Finally, the novel approach is extensively tested against classical benchmarks for compressible gas dynamics to show the robustness and the computational efficiency.

Keywords

p-Adaptivity / Arbitrary DERivative (ADER) / Arbitrary high order / Deferred Correction (DeC) / Positivity preserving / 65M60 / 65Y20 / 35L65

References

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

[2] Abgrall R. High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. J. Sci. Comput.. 2017, 73(2/3): 461-494

[3] Abgrall R, Bacigaluppi P, Tokareva S. High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics. Comput. Math. Appl.. 2019, 78(2): 274-297

[4] Abgrall R, Le Mélédo E, Öffner P, Torlo D. Relaxation deferred correction methods and their applications to residual distribution schemes. SMAI J. Comput. Math.. 2022, 8: 125-160

[5] Abgrall, R., Lukácova-Medvid’ová, M., Öffner, P.: On the convergence of residual distribution schemes for the compressible Euler equations via dissipative weak solutions. arXiv:2207.11969 (2022)

[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. 5, 573–595 (2021)

[7] Abgrall R, Öffner P, Ranocha H. Reinterpretation and extension of entropy correction terms for residual distribution and discontinuous Galerkin schemes: application to structure preserving discretization. J. Comput. Phys.. 2022, 453: 110955

[8] Abgrall R, Torlo D. High order asymptotic preserving deferred correction implicit-explicit schemes for kinetic models. SIAM J. Sci. Comput.. 2020, 42(3): 816-845

[9] Bacigaluppi, P., Abgrall, R., Tokareva, S.: “A posteriori” limited high order and robust residual distribution schemes for transient simulations of fluid flows in gas dynamics. arXiv:1902.07773 (2019)

[10] Balsara DS, Shu C-W. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys.. 2000, 160(2): 405-452

[11] Becker R. Stosswelle und detonation. Zeitschrift für Physik. 1922, 8(1): 321-362

[12] Berberich JP, Chandrashekar P, Klingenberg C. High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws. Comput. Fluids. 2021, 219: 104858

[13] Boscheri W, Balsara DS. High order direct Arbitrary-Lagrangian-Eulerian (ALE) PNPM schemes with WENO Adaptive-Order reconstruction on unstructured meshes. J. Comput. Phys.. 2019, 398: 108899

[14] Boscheri W, Dumbser M. Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes. Commun. Comput. Phys.. 2013, 14: 1174-1206

[15] 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

[16] Boscheri W, Dumbser M, Gaburro E. Continuous finite element subgrid basis functions for discontinuous Galerkin schemes on unstructured polygonal Voronoi meshes. Commun. Comput. Phys.. 2022, 32: 259-298

[17] 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: 271-312

[18] Boscheri W, Loubère R, Dumbser M. Direct arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws. J. Comput. Phys.. 2015, 292: 56-87

[19] Busto S, Chiocchetti S, Dumbser M, Gaburro E, Peshkov I. High order ADER schemes for continuum mechanics. Front. Phys.. 2020, 8: 32

[20] Busto S, Dumbser M, Río-Martín L. Staggered semi-implicit hybrid finite volume/finite element schemes for turbulent and non-Newtonian flows. Mathematics. 2021, 9(22): 2972

[21] 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

[22] Castro MJ, Parés C. Well-balanced high-order finite volume methods for systems of balance laws. J. Sci. Comput.. 2020

[23] Chen T, Shu C-W. Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys.. 2017, 345: 427-461

[24] Chen T, Shu C-W. Review of entropy stable discontinuous Galerkin methods for systems of conservation laws on unstructured simplex meshes. CSIAM Trans. Appl. Math.. 2020, 1: 1-52

[25] Cheng Y, Chertock A, Herty M, Kurganov A, Wu T. A new approach for designing moving-water equilibria preserving schemes for the shallow water equations. J. Sci. Comput.. 2019, 80(1): 538-554

[26] Chertock A, Cui S, Kurganov A, Özcan CN, Tadmor E. Well-balanced schemes for the Euler equations with gravitation: conservative formulation using global fluxes. J. Comput. Phys.. 2018, 358: 36-52

[27] Ciallella M, Micalizzi L, Öffner P, Torlo D. An arbitrary high order and positivity preserving method for the shallow water equations. Comput. Fluids. 2022, 247 105630

[28] Ciallella, M., Torlo, D., Ricchiuto, M.: Arbitrary high order WENO finite volume scheme with flux globalization for moving equilibria preservation. arXiv:2205.13315 (2022)

[29] Clain S, Diot S, Loubère R. A high-order finite volume method for systems of conservation laws-Multi-dimensional Optimal Order Detection (MOOD). J. Comput. Phys.. 2011, 230(10): 4028-4050

[30] Colonius T, Lele S, Moin P. Sound generation in a mixing layer. J. Fluid Mech.. 1997, 330: 375-409

[31] Diot S, Clain S, Loubère R. Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials. Comput. Fluids. 2012, 64: 43-63

[32] Diot S, Loubère R, Clain S. The MOOD method in the three-dimensional case: very-high-order finite volume method for hyperbolic systems. Int. J. Numer. Methods Fluids. 2013, 73: 362-392

[33] Dumbser M. Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations. Comput. Fluids. 2010, 39(1): 60-76

[34] 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

[35] Dumbser M, Munz C-D. ADER discontinuous Galerkin schemes for aeroacoustics. Comptes Rendus Mécanique. 2005, 333(9): 683-687

[36] Dumbser M, Toro EF. A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. J. Sci. Comput.. 2011, 48: 70-88

[37] Dumbser M, Zanotti O. Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations. J. Comput. Phys.. 2009, 228(18): 6991-7006

[38] Dutt A, Greengard L, Rokhlin V. Spectral deferred correction methods for ordinary differential equations. BIT. 2000, 40(2): 241-266

[39] Farhat C, Fezoui L, Lanteri S. Two-dimensional viscous flow computations on the connection machine: unstructured meshes, upwind schemes and massively parallel computations. Comput. Methods Appl. Mech. Eng.. 1993, 102(1): 61-88

[40] Friedrich L, Winters AR, Fernández DCDR, Gassner GJ, Parsani M, Carpenter MH. An entropy stable h/p non-conforming discontinuous Galerkin method with the summation-by-parts property. J. Sci. Comput.. 2018, 77(2): 689-725

[41] Gaburro, E.: A unified framework for the solution of hyperbolic PDE systems using high order direct Arbitrary-Lagrangian-Eulerian schemes on moving unstructured meshes with topology change. Arch. Comput. Methods Eng. 28(3), 1249–1321 (2021)

[42] Gaburro E, Boscheri W, Chiocchetti S, Klingenberg C, Springel V, Dumbser M. High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes. J. Comput. Phys.. 2020, 407: 109167

[43] Gaburro E, Dumbser M. A posteriori subcell finite volume limiter for general PNPM schemes: applications from gasdynamics to relativistic magnetohydrodynamics. J. Sci. Comput.. 2021, 86(3): 1-41

[44] Gaburro E, Öffner P, Ricchiuto M, Torlo D. High order entropy preserving ADER-DG scheme. Appl. Math. Comput.. 2023, 440 127644

[45] Gassner GJ, Winters AR, Kopriva DA. A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Appl. Math. Comput.. 2016, 272: 291-308

[46] Glaubitz J, Öffner P. Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points. Appl. Numer. Math.. 2020, 151: 98-118

[47] Gómez-Bueno I, Boscarino S, Castro MJ, Parés C, Russo G. Implicit and semi-implicit well-balanced finite-volume methods for systems of balance laws. Appl. Numer. Math.. 2023, 184: 18-48

[48] Gottlieb S, Shu C-W. Total variation diminishing Runge-Kutta schemes. Math. Comput. Am. Math. Soc.. 1998, 67(221): 73-85

[49] Hajduk H. Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws. Comput. Math. Appl.. 2021, 87: 120-138

[50] Han Veiga M, Öffner P, Torlo D. DeC and ADER: similarities, differences and a unified framework. J. Sci. Comput.. 2021, 87(1): 1-35

[51] Han Veiga M, Velasco-Romero DA, Abgrall R, Teyssier R. Capturing near-equilibrium solutions: a comparison between high-order discontinuous Galerkin methods and well-balanced schemes. Commun. Comput. Phys.. 2019, 26(1): 1-34

[52] Huang, D.Z., Avery, P., Farhat, C., Rabinovitch, J., Derkevorkian, A., Peterson, L.D.: Modeling, simulation and validation of supersonic parachute inflation dynamics during mars landing. In: AIAA Scitech 2020 Forum, pp. 0313 (2020)

[53] Huang J, Shu C-W. Positivity-preserving time discretizations for production-destruction equations with applications to non-equilibrium flows. J. Sci. Comput.. 2019, 78(3): 1811-1839

[54] Jund S, Salmon S. Arbitrary high-order finite element schemes and high-order mass lumping. Int. J. Appl. Math. Comput. Sci.. 2007, 17(3): 375-393

[55] Krais N, Beck A, Bolemann T, Frank H, Flad D, Gassner G, Hindenlang F, Hoffmann M, Kuhn T, Sonntag M, et al. . FLEXI: a high order discontinuous Galerkin framework for hyperbolic-parabolic conservation laws. Comput. Math. Appl.. 2021, 81: 186-219

[56] Kuzmin D. Entropy stabilization and property-preserving limiters for $\mathbb{P} _1$ discontinuous Galerkin discretizations of scalar hyperbolic problems. J. Numer. Math.. 2021, 29(4): 307-322

[57] Kuzmin D, de Luna MQ. Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws. Comput. Fluids. 2020, 213: 104742

[58] Lukáčová-Medvid’ová, M., Öffner, P.: Convergence of discontinuous Galerkin schemes for the Euler equations via dissipative weak solutions. Appl. Math. Comput. 436, 127508 (2023)

[59] Mantri Y, Noelle S. Well-balanced discontinuous Galerkin scheme for 2$\times$2 hyperbolic balance law. J. Comput. Phys.. 2021, 429: 110011

[60] Meister A, Ortleb S. On unconditionally positive implicit time integration for the DG scheme applied to shallow water flows. Int. J. Numer. Meth. Fluids. 2014, 76(2): 69-94

[61] Micalizzi, L., Torlo, D.: A new efficient explicit Deferred Correction framework: analysis and applications to hyperbolic PDEs and adaptivity (2022). https://doi.org/10.48550/arxiv.2210.02976

[62] Millington, R., Toro, E., Nejad, L.: Arbitrary high order methods for conservation laws I: the one dimensional scalar case. PhD thesis, Manchester Metropolitan University, Department of Computing and Mathematics (June 1999)

[63] Minion ML. Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci.. 2003, 1(3): 471-500

[64] Noelle S, Xing Y, Shu C-W. High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys.. 2007, 226(1): 29-58

[65] Öffner, P., Glaubitz, J., Ranocha, H.: Stability of correction procedure via reconstruction with summation-by-parts operators for Burgers’ equation using a polynomial chaos approach. ESAIM: Mathematical Modelling and Numerical Analysis 52(6), 2215–2245 (2018)

[66] Öffner P, Torlo D. Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes. Appl. Numer. Math.. 2020, 153: 15-34

[67] Perthame B, Shu C-W. On positivity preserving finite volume schemes for Euler equations. Numer. Math.. 1996, 73(1): 119-130

[68] Rabenseifner, R., Hager, G., Jost, G.: Hybrid MPI/OpenMP parallel programming on clusters of multi-core SMP nodes. In: 2009 17th Euromicro International Conference on Parallel, Distributed and Network-Based Processing, pp. 427–436 (2009)

[69] 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.. 2020, 42(2): 612-638

[70] Ricchiuto M. An explicit residual based approach for shallow water flows. J. Comput. Phys.. 2015, 280: 306-344

[71] 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

[72] Ricchiuto, M., Torlo, D.: Analytical travelling vortex solutions of hyperbolic equations for validating very high order schemes (2021). https://doi.org/10.48550/ARXIV.2109.10183

[73] Río-Martín L, Busto S, Dumbser M. A massively parallel hybrid finite volume/finite element scheme for computational fluid dynamics. Mathematics. 2021, 9(18): 2316

[74] Rusanov VV. Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR. 1961, 1: 267-279

[75] Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Quarteroni, A. (ed) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Berlin, Heidelberg (1998)

[76] Shu C-W, Osher S. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys.. 1988, 77: 439-471

[77] Spiegel, S.C., Huynh, H., DeBonis, J.R.: A survey of the isentropic Euler vortex problem using high-order methods. In: 22nd AIAA Computational Fluid Dynamics Conference, AIAA 2015-2444 (2015)

[78] Titarev V, Toro E. ADER: arbitrary high order Godunov approach. J. Sci. Comput.. 2002, 17(1/2/3/4): 609-618

[79] Titarev VA, Toro EF. ADER schemes for three-dimensional nonlinear hyperbolic systems. J. Comput. Phys.. 2005, 204: 715-736

[80] Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction. Springer, Berlin, Heidelberg (2009). https://books.google.com/books?id=SqEjX0um8o0C

[81] Tsoutsanis P, Antoniadis AF, Jenkins KW. Improvement of the computational performance of a parallel unstructured WENO finite volume CFD code for Implicit Large Eddy Simulation. Comput. Fluids. 2018, 173: 157-170

[82] Tsoutsanis, P., Drikakis, D.: Addressing the challenges of implementation of high-order finite-volume schemes for atmospheric dynamics on unstructured meshes. In: Papadrakakis, M., Papadopoulos, V., Stefanou, G., Plevris, V. (eds.) ECCOMAS Congress 2016-Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering, vol. 1, pp. 684–708. National Technical University of Athens, GRC (2016)

[83] Wintermeyer, N., Winters, A.R., Gassner, G.J., Kopriva, D.A.: An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. J. Comput. Phys. 340, 200–242 (2017)

[84] Winters AR, Gassner GJ. A comparison of two entropy stable discontinuous Galerkin spectral element approximations for the shallow water equations with non-constant topography. J. Comput. Phys.. 2015, 301: 357-376

[85] Xing Y, Shu C-W. High-order well-balanced finite difference WENO schemes for a class of hyperbolic systems with source terms. J. Sci. Comput.. 2006, 27(1): 477-494

[86] Zhang X, Shu C-W. Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments. Proc. R. Soci. A Math. Phys. Eng. Sci.. 2011, 467(2134): 2752-2776

Funding

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung(200020_204917)

Università degli Studi di Ferrara

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