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). https:// doi. org/ 10. 1007/ s10915- 017- 0498-4
3. 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)
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. 8, 125–160 (2022). https:// doi. org/ 10. 5802/ smai- jcm. 82
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. 453, 110955 (2022)
8. 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)
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, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160(2), 405–452 (2000)
11. Becker, R.: Stosswelle und detonation. Zeitschrift für Physik 8(1), 321–362 (1922)
12. Berberich, J.P., Chandrashekar, P., Klingenberg, C.: High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws. Comput. Fluids 219, 104858 (2021)
13. Boscheri, W., Balsara, D.S.: High order direct Arbitrary-Lagrangian-Eulerian (ALE) PNPM schemes with WENO Adaptive-Order reconstruction on unstructured meshes. J. Comput. Phys. 398, 108899 (2019)
14. Boscheri, W., Dumbser, M.: Arbitrary-Lagrangian-Eulerian one-step WENO finite volume schemes on unstructured triangular meshes. Commun. Comput. Phys. 14, 1174–1206 (2013)
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. 346, 449–479 (2017)
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. 32, 259–298 (2022)
17. Boscheri, W., Loubère, R.: High order accurate direct Arbitrary-Lagrangian-Eulerian ADERMOOD finite volume schemes for non-conservative hyperbolic systems with stiff source terms. Commun. Comput. Phys. 21, 271–312 (2017)
18. Boscheri, W., Loubère, R., Dumbser, M.: Direct arbitrary-Lagrangian-Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws. J. Comput. Phys. 292, 56–87 (2015)
19. Busto, S., Chiocchetti, S., Dumbser, M., Gaburro, E., Peshkov, I.: High order ADER schemes for continuum mechanics. Front. Phys. 8, 32 (2020). https:// doi. org/ 10. 3389/ fphy. 2020. 00032
20. Busto, S., Dumbser, M., Río-Martín, L.: Staggered semi-implicit hybrid finite volume/finite element schemes for turbulent and non-Newtonian flows. Mathematics 9(22), 2972 (2021)
21. Busto, S., Toro, E.F., Vázquez-Cendón, M.E.: Design and analysis of ADER-type schemes for model advection-diffusion-reaction equations. J. Comput. Phys. 327, 553–575 (2016)
22. Castro, M.J., Parés, C.: Well-balanced high-order finite volume methods for systems of balance laws. J. Sci. Comput. (2020). https:// doi. org/ 10. 1007/ s10915- 020- 01149-5
23. Chen, T., Shu, C.-W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427–461 (2017)
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. 1, 1–52 (2020)
25. Cheng, Y., Chertock, A., Herty, M., Kurganov, A., Wu, T.: A new approach for designing movingwater equilibria preserving schemes for the shallow water equations. J. Sci. Comput. 80(1), 538– 554 (2019)
26. Chertock, A., Cui, S., Kurganov, A., Özcan, C.N., Tadmor, E.: Well-balanced schemes for the Euler equations with gravitation: conservative formulation using global fluxes. J. Comput. Phys. 358, 36–52 (2018). https:// doi. org/ 10. 1016/j. jcp. 2017. 12. 026
27. 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). https:// doi. org/ 10. 1016/j. compfluid. 2022. 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 lawsMulti-dimensional Optimal Order Detection (MOOD). J. Comput. Phys. 230(10), 4028–4050 (2011)
30. Colonius, T., Lele, S., Moin, P.: Sound generation in a mixing layer. J. Fluid Mech. 330, 375–409 (1997)
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 64, 43–63 (2012)
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 73, 362–392 (2013)
33. Dumbser, M.: Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations. Comput. Fluids 39(1), 60–76 (2010)
34. Dumbser, M., Balsara, D.S., Toro, E.F., 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)
35. Dumbser, M., Munz, C.-D.: ADER discontinuous Galerkin schemes for aeroacoustics. Comptes Rendus Mécanique 333(9), 683–687 (2005)
36. Dumbser, M., Toro, E.F.: A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. J. Sci. Comput. 48, 70–88 (2011)
37. Dumbser, M., Zanotti, O.: Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations. J. Comput. Phys. 228(18), 6991–7006 (2009)
38. Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT 40(2), 241–266 (2000). https:// doi. org/ 10. 1023/A: 10223 38906 936
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. 102(1), 61–88 (1993). https:// doi. org/ 10. 1016/ 0045- 7825(93) 90141-J
40. Friedrich, L., Winters, A.R., Fernández, D.C.D.R., Gassner, G.J., Parsani, M., Carpenter, M.H.: An entropy stable h/p non-conforming discontinuous Galerkin method with the summation-by-parts property. J. Sci. Comput. 77(2), 689–725 (2018)
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. 407, 109167 (2020)
43. Gaburro, E., Dumbser, M.: A posteriori subcell finite volume limiter for general PNPM schemes: applications from gasdynamics to relativistic magnetohydrodynamics. J. Sci. Comput. 86(3), 1–41 (2021)
44. Gaburro, E., Öffner, P., Ricchiuto, M., Torlo, D.: High order entropy preserving ADER-DG scheme. Appl. Math. Comput. 440, 127644 (2023). https:// doi. org/ 10. 1016/j. amc. 2022. 127644
45. Gassner, G.J., Winters, A.R., Kopriva, D.A.: A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Appl. Math. Comput. 272, 291–308 (2016)
46. Glaubitz, J., Öffner, P.: Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points. Appl. Numer. Math. 151, 98–118 (2020)
47. Gómez-Bueno, I., Boscarino, S., Castro, M.J., Parés, C., Russo, G.: Implicit and semi-implicit wellbalanced finite-volume methods for systems of balance laws. Appl. Numer. Math. 184, 18–48 (2023)
48. Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. Am. Math. Soc. 67(221), 73–85 (1998)
49. Hajduk, H.: Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws. Comput. Math. Appl. 87, 120–138 (2021)
50. Han Veiga, M., Öffner, P., Torlo, D.: DeC and ADER: similarities, differences and a unified framework. J. Sci. Comput. 87(1), 1–35 (2021)
51. Han Veiga, M., Velasco-Romero, D.A., Abgrall, R., Teyssier, R.: Capturing near-equilibrium solutions: a comparison between high-order discontinuous Galerkin methods and well-balanced schemes. Commun. Comput. Phys. 26(1), 1–34 (2019). https:// doi. org/ 10. 4208/ cicp. oa- 2018- 0071
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. 78(3), 1811–1839 (2019)
54. 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)
55. Krais, N., Beck, A., Bolemann, T., Frank, H., Flad, D., Gassner, G., Hindenlang, F., Hoffmann, M., Kuhn, T., Sonntag, M., Munz, C.-D.: FLEXI: a high order discontinuous Galerkin framework for hyperbolic-parabolic conservation laws. Comput. Math. Appl. 81, 186–219 (2021)
56. Kuzmin, D.: Entropy stabilization and property-preserving limiters for P1 discontinuous Galerkin discretizations of scalar hyperbolic problems. J. Numer. Math. 29(4), 307–322 (2021)
57. Kuzmin, D., de Luna, M.Q.: Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws. Comput. Fluids 213, 104742 (2020)
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 2×2 hyperbolic balance law. J. Comput. Phys. 429, 110011 (2021)
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 76(2), 69–94 (2014)
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, M.L.: Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1(3), 471–500 (2003)
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. 226(1), 29–58 (2007)
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. 153, 15–34 (2020)
67. Perthame, B., Shu, C.-W.: On positivity preserving finite volume schemes for Euler equations. Numer. Math. 73(1), 119–130 (1996)
68. Rabenseifner, R., Hager, G., Jost, G.: Hybrid MPI/OpenMP parallel programming on clusters of multicore 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, D.I.: Relaxation Runge-Kutta methods: fully discrete explicit entropy-stable schemes for the compressible Euler and Navier-Stokes equations. SIAM J. Sci. Comput. 42(2), 612–638 (2020)
70. Ricchiuto, M.: An explicit residual based approach for shallow water flows. J. Comput. Phys. 280, 306–344 (2015)
71. 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)
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 9(18), 2316 (2021)
74. Rusanov, V.V.: Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279 (1961)
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. 77, 439–471 (1988)
77. Spiegel, S.C., Huynh, H., DeBonis, J.R.: A survey of the isentropic Euler vortex problem using highorder methods. In: 22nd AIAA Computational Fluid Dynamics Conference, AIAA 2015-2444 (2015)
78. Titarev, V.A., Toro, E.F.: ADER: arbitrary high order Godunov approach. J. Sci. Comput. 17(1/2/3/4), 609–618 (2002)
79. Titarev, V.A., Toro, E.F.: ADER schemes for three-dimensional nonlinear hyperbolic systems. J. Comput. Phys. 204, 715–736 (2005)
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= SqEjX 0um8o 0C
81. Tsoutsanis, P., Antoniadis, A.F., Jenkins, K.W.: Improvement of the computational performance of a parallel unstructured WENO finite volume CFD code for Implicit Large Eddy Simulation. Comput. Fluids 173, 157–170 (2018)
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, A.R., Gassner, G.J.: A comparison of two entropy stable discontinuous Galerkin spectral element approximations for the shallow water equations with non-constant topography. J. Comput. Phys. 301, 357–376 (2015)
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. 27(1), 477–494 (2006)
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. 467(2134), 2752–2776 (2011)
