1. Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25(6), 2050- 2065 (2004)
2. Barré de Saint-Venant, A.J.C.: Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et á l’introduction des marées dans leur lit. C. R. Acad. Sci. Paris Sér. I Math. 53, 147-154 (1871)
3. Barsukow, W.: The active flux scheme for nonlinear problems. J. Sci. Comput. 86(1), 1-34 (2021)
4. Barsukow, W., Berberich, J.P., Klingenberg, C.: On the active flux scheme for hyperbolic PDEs with source terms. SIAM J. Sci. Comput. 43(6), A4015-A4042 (2021)
5. Barsukow, W., Hohm, J., Klingenberg, C., Roe, P.L.: The active flux scheme on Cartesian grids and its low Mach number limit. J. Sci. Comput. 81(1), 594-622 (2019)
6. Barsukow, W., Klingenberg, C.: Exact solution and a truly multidimensional Godunov scheme for the acoustic equations. ESAIM: M2AN 56(1), 317-347 (2022)
7. 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)
8. Bermudez, A., Vázquez, E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23(8), 1049-1071 (1994)
9. Bollermann, A., Chen, G., Kurganov, A., Noelle, S.: A well-balanced reconstruction of wet/dry fronts for the shallow water equations. J. Sci. Comput. 56(2), 267-290 (2013)
10. Bollermann, A., Noelle, S., Lukáčová-Medvid’ová, M.: Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys. 10(2), 371-404 (2011)
11. Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Springer Science & Business Media, Berlin (2004)
12. Carrier, G.F., Wu, T.T., Yeh, H.: Tsunami run-up and draw-down on a plane beach. J. Fluid Mech. 475, 79-99 (2003)
13. Castro, M.J., González-Vida, J.M., Parés, C.: Numerical treatment of wet/dry fronts in shallow flows with a modified Roe scheme. Math. Models Methods Appl. Sci. 16(06), 897-931 (2006)
14. Castro, M.J., Semplice, M.: Third- and fourth-order well-balanced schemes for the shallow water equations based on the CWENO reconstruction. Int. J. Numer. Methods Fluids 89(8), 304-325 (2019)
15. 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)
16. Chinnayya, A., LeRoux, A.-Y., Seguin, N.: A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon. Int. J. Finite 1(1), 1-33 (2004)
17. Cozzolino, L., Della Morte, R., Del Giudice, G., Palumbo, A., Pianese, D.: A well-balanced spectral volume scheme with the wetting-drying property for the shallow-water equations. J. Hydroinform. 14(3), 745-760 (2012)
18. Eymann, T.A.: Active flux schemes. PhD thesis. University of Michigan, MI, USA (2013)
19. Eymann, T.A., Roe, P.L.: Active flux schemes for systems. In: Proceedings of the 20th AIAA Computational Fluid Dynamics Conference, AIAA 2011-3840, AIAA (2011)
20. Eymann, T.A., Roe, P.L.: Multidimensional active flux schemes. In: Proceedings of the 21st AIAA Computational Fluid Dynamics Conference, AIAA 2013-2940, AIAA (2013)
21. Fan, D.: On the acoustic component of active flux schemes for nonlinear hyperbolic conservation laws. PhD thesis. University of Michigan, MI, USA (2017)
22. Gallardo, J.M., Parés, C., Castro, M.: On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys. 227(1), 574-601 (2007)
23. Gallouët, T., Hérard, J.-M., Seguin, N.: Some approximate Godunov schemes to compute shallowwater equations with topography. Comput. Fluids 32(4), 479-513 (2003)
24. Gosse, L.: A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math. Models Methods Appl. Sci. 11(02), 339-365 (2001)
25. Helzel, C., Kerkmann, D., Scandurra, L.: A new ADER method inspired by the active flux method. J. Sci. Comput. 80(3), 1463-1497 (2019)
26. Kurganov, A.: Finite-volume schemes for shallow-water equations. Acta Numer. 27, 289-351 (2018)
27. Kurganov, A., Levy, D.: Central-upwind schemes for the Saint-Venant system. ESAIM: Math. Model. Numer. Anal. 36(3), 397-425 (2002)
28. LeVeque, R.J.: Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146(1), 346-365 (1998)29. Parés, C., Castro, M.: On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: Math. Model. Numer. Anal. 38(5), 821-852 (2004)
30. Pareschi, L., Rey, T.: Residual equilibrium schemes for time dependent partial differential equations. Comput. Fluids 156, 329-342 (2017)
31. Ricchiuto, M., Bollermann, A.: Stabilized residual distribution for shallow water simulations. J. Comput. Phys. 228(4), 1071-1115 (2009)
32. Roe, P.: Designing CFD methods for bandwidth—a physical approach. Comput. Fluids 214, 104774
(2021)
33. Roe, P.L.: Upwind differencing schemes for hyperbolic conservation laws with source terms. In: Nonlinear Hyperbolic Problems, pp. 41-51. Springer, Berlin (1987)
34. Roe, P.L., Lung, T., Maeng, J.: New approaches to limiting. In: Proceedings of the 22nd AIAA Computational Fluid Dynamics Conference, AIAA 2015-2913, AIAA (2015)
35. Rogers, B.D., Borthwick, A.G.L., Taylor, P.H.: Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver. J. Comput. Phys. 192(2), 422-451 (2003)
36. Thacker, W.C.: Some exact solutions to the nonlinear shallow-water wave equations. J. Fluid Mech. 107, 499-508 (1981)
37. Van Leer, B.: Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection. J. Comput. Phys. 23(3), 276-299 (1977)
38. Vater, S., Behrens, J.: Well-balanced inundation modeling for shallow-water flows with discontinuous Galerkin schemes. In: Finite Volumes for Complex Applications VII—Elliptic, Parabolic and Hyperbolic Problems, pp. 965-973. Springer, Berlin (2014)
39. Xing, Y., Shu, C.-W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208(1), 206-227 (2005)
40. Xing, Y., Shu, C.-W.: High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys. 214(2), 567- 598 (2006)
41. Xing, Y., Shu, C.-W.: High-order finite volume WENO schemes for the shallow water equations with dry states. Adv. Water Resour. 34(8), 1026-1038 (2011)
42. Xing, Y., Shu, C.-W.: A survey of high order schemes for the shallow water equations. J. Math. Study 47(3), 221-249 (2014)
43. Xing, Y., Zhang, X., Shu, C.-W.: Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations. Adv. Water Resour. 33(12), 1476-1493 (2010)
44. Xu, K.: A well-balanced gas-kinetic scheme for the shallow-water equations with source terms. J. Comput. Phys. 178(2), 533-562 (2002)
45. Zhang, X., Shu, C.-W.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229(23), 8918-8934 (2010)
46. Zhou, J.G., Causon, D.M., Mingham, C.G., Ingram, D.M.: The surface gradient method for the treatment of source terms in the shallow-water equations. J. Comput. Phys. 168(1), 1-25 (2001)
