1. Ahusborde, E., Azaiez, M., Caltagirone, J.P.:A primal formulation for the Helmholtz decomposition. J. Comput. Phys. 225(1), 13-19 (2007) 2. Ahusborde, E., Azaïez, M., Caltagirone, J.P., Gerritsma, M., Lemoine, A.:Discrete Hodge Helmholtz decomposition. Monografías Matemáticas García de Galdeano 39, 1-10 (2014) 3. Amrouche, C., Bernardi, C., Dauge, M., Girault, V.:Vector potentials in three-dimensional nonsmooth domains. Math. Methods Appl. Sci. 21(9), 823-864 (1998). https://doi.org/10.1002/(SICI)1099-1476(199806).21:9<823::AID-MMA976>3.0.CO;2-B 4. Angot, P., Caltagirone, J.P., Fabrie, P.:Fast discrete Helmholtz-Hodge decompositions in bounded domains. Appl. Math. Lett. 26(4), 445-451 (2013). https://doi.org/10.1016/j.aml.2012.11.006 5. Beresnyak, A., Lazarian, A.:Turbulence in magnetohydrodynamics. Stud. Math. Phys. 12. Walter de Gruyter GmbH & Co KG (2019) 6. Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.:Julia:a fresh approach to numerical computing. SIAM Rev. 59(1), 65-98 (2017). https://doi.org/10.1137/141000671.arxiv:1411.1607[cs.MS] 7. Bhatia, H., Norgard, G., Pascucci, V., Bremer, P.T.:The Helmholtz-Hodge decomposition-A survey. IEEE Trans. Visual Comput. Graphics 19(8), 1386-1404 (2012). https://doi.org/10.1109/TVCG.2012.316 8. Chan, J.:On discretely entropy conservative and entropy stable discontinuous Galerkin methods. J. Comput. Phys. 362, 346-374 (2018). https://doi.org/10.1016/j.jcp.2018.02.033 9. Estrin, R., Orban, D., Saunders, M.A.:LSLQ:an iterative method for linear least-squares with an error minimization property. SIAM J. Matrix Anal. Appl. 40(1), 254-275 (2019). https://doi.org/10.1137/17M1113552 10. Fernandes, P., Gilardi, G.:Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7(07), 957-991 (1997). https://doi.org/10.1142/S0218202597000487 11. Fernández, D.C.D.R., Boom, P.D., Carpenter, M.H., Zingg, D.W.:Extension of tensor-product generalized and dense-norm summation-by-parts operators to curvilinear coordinates. J. Sci. Comput. 80(3), 1957-1996 (2019). https://doi.org/10.1007/s10915-019-01011-3 12. Fernández, D.C.D.R., Boom, P.D., Zingg, D.W.:A generalized framework for nodal frst derivative summation-by-parts operators. J. Comput. Phys. 266, 214-239 (2014). https://doi.org/10.1016/j.jcp.2014.01.038 13. Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.:Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial diferential equations. Comput. Fluids 95, 171-196 (2014). https://doi.org/10.1016/j.compfuid.2014.02.016 14. Fisher, T.C., Carpenter, M.H.:High-order entropy stable fnite diference schemes for nonlinear conservation laws:fnite domains. J. Comput. Phys. 252, 518-557 (2013). https://doi.org/10.1016/j.jcp.2013.06.014 15. Fong, D.C.L., Saunders, M.A.:LSMR:an iterative algorithm for sparse least-squares problems. SIAM J. Sci. Comput. 33(5), 2950-2971 (2011). https://doi.org/10.1137/10079687X 16. Gao, L., Fernández, D.C.D.R., Carpenter, M., Keyes, D.:SBP-SAT fnite diference discretization of acoustic wave equations on staggered block-wise uniform grids. J. Comput. Appl. Math. 348, 421-444 (2019). https://doi.org/10.1016/j.cam.2018.08.040 17. Gassner, G.J.:A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT fnite diference methods. SIAM J. Sci. Comput. 35(3), A1233-A1253 (2013). https://doi.org/10.1137/120890144 18. Girault, V., Raviart, P.A.:Finite Element Methods for Navier-Stokes Equations:Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5. Springer, Berlin Heidelberg (2012). https://doi.org/10.1007/978-3-642-61623-5 19. Glaßmeier, K.H.:Refection of MHD-waves in the pc4-5 period range at ionospheres with nonuniform conductivity distributions. Geophys. Res. Lett. 10(8), 678-681 (1983). https://doi.org/10.1029/GL010i008p00678 20. Glaßmeier, K.H.:On the infuence of ionospheres with non-uniform conductivity distribution on hydromagnetic waves. J. Geophys. 54, 125-137 (1984) 21. Glaßmeier, K.H.:Reconstruction of the ionospheric infuence on ground-based observations of a short-duration ULF pulsation event. Planet. Space Sci. 36(8), 801-817 (1988). https://doi.org/10.1016/0032-0633(88)90086-4 22. Glaßmeier, K.H., Othmer, C., Cramm, R., Stellmacher, M., Engebretson, M.:Magnetospheric feld line resonances:a comparative planetology approach. Surv. Geophys. 20(1), 61-109 (1999). https://doi.org/10.1023/A:1006659717963 23. Hicken, J.E., Zingg, D.W.:Summation-by-parts operators and high-order quadrature. J. Comput. Appl. Math. 237(1), 111-125 (2013). https://doi.org/10.1016/j.cam.2012.07.015 24. Huynh, H.T.:A fux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In:18th AIAA Computational Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics (2007). https://doi.org/10.2514/6.2007-4079 25. Hyman, J.M., Shashkov, M.:Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl. 33(4), 81-104 (1997). https://doi.org/10.1016/S0898-1221(97)00009-6 26. Hyman, J.M., Shashkov, M.:The orthogonal decomposition theorems for mimetic fnite diference methods. SIAM J. Numer. Anal. 36(3), 788-818 (1999). https://doi.org/10.1137/S0036142996314044 27. Jikov, V.V., Kozlov, S.M., Oleinik, O.A.:Homogenization of Diferential Operators and Integral Functionals. Springer, Berlin Heidelberg (1994). https://doi.org/10.1007/978-3-642-84659-5 28. Kowal, G., Lazarian, A.:Velocity feld of compressible magnetohydrodynamic turbulence:wavelet decomposition and mode scalings. Astrophys J. 720(1), 742-756 (2010). https://doi.org/10.1088/0004-637X/720/1/742 29. Kreiss, H.O., Scherer, G.:Finite element and fnite diference methods for hyperbolic partial differential equations. In:de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195-212. Academic Press, New York (1974) 30. Lemoine, A., Caltagirone, J.P., Azaïez, M., Vincent, S.:Discrete Helmholtz-Hodge decomposition on polyhedral meshes using compatible discrete operators. J. Sci. Comput. 65(1), 34-53 (2015). https://doi.org/10.1007/s10915-014-9952-8 31. Linders, V., Lundquist, T., Nordström, J.:On the order of accuracy of fnite diference operators on diagonal norm based summation-by-parts form. SIAM J. Numer. Anal. 56(2), 1048-1063 (2018). https://doi.org/10.1137/17M1139333 32. Linders, V., Nordström, J., Frankel, S.H.:Convergence and Stability Properties of Summation-byParts in Time. Technical Report LiTH-MAT-R, ISSN 0348-2960; 2019:4, Linköping University, Linköping, Sweden (2019) 33. Lipnikov, K., Manzini, G., Shashkov, M.:Mimetic fnite diference method. J. Comput. Phys. 257, 1163-1227 (2014). https://doi.org/10.1016/j.jcp.2013.07.031 34. Mattsson, K., Almquist, M., Carpenter, M.H.:Optimal diagonal-norm SBP operators. J. Comput. Phys. 264, 91-111 (2014). https://doi.org/10.1016/j.jcp.2013.12.041 35. Mattsson, K., Almquist, M., van der Weide, E.:Boundary optimized diagonal-norm SBP operators. J. Comput. Phys. 374, 1261-1266 (2018). https://doi.org/10.1016/j.jcp.2018.06.010 36. Mattsson, K., Nordström, J.:Summation by parts operators for fnite diference approximations of second derivatives. J. Comput. Phys. 199(2), 503-540 (2004). https://doi.org/10.1016/j.jcp.2004.03.001 37. Mattsson, K., O'Reilly, O.:Compatible diagonal-norm staggered and upwind SBP operators. J. Comput. Phys. 352, 52-75 (2018). https://doi.org/10.1016/j.jcp.2017.09.044 38. Nordström, J., Björck, M.:Finite volume approximations and strict stability for hyperbolic problems. Appl. Numer. Math. 38(3), 237-255 (2001). https://doi.org/10.1016/S0168-9274(01)00027-7 39. Nordström, J., Forsberg, K., Adamsson, C., Eliasson, P.:Finite volume methods, unstructured meshes and strict stability for hyperbolic problems. Appl. Numer. Math. 45(4), 453-473 (2003). https://doi.org/10.1016/S0168-9274(02)00239-8 40. O'Reilly, O., Lundquist, T., Dunham, E.M., Nordström, J.:Energy stable and high-order-accurate fnite diference methods on staggered grids. J. Comput. Phys. 346, 572-589 (2017). https://doi. org/10.1016/j.jcp.2017.06.030 41. Paige, C.C., Saunders, M.A.:Algorithm 583 LSQR:sparse linear equations and least squares problems. ACM Trans. Math. Softw. (TOMS) 8(2), 195-209 (1982). https://doi.org/10.1145/355993.356000 42. Paige, C.C., Saunders, M.A.:LSQR:an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. (TOMS) 8(1), 43-71 (1982). https://doi.org/10.1145/355984.355989 43. Ranocha, H.:Shallow water equations:split-form, entropy stable, well-balanced, and positivity preserving numerical methods. GEM Int. J. Geomath. 8(1), 85-133 (2017). https://doi.org/10.1007/s13137-016-0089-9.arxiv:1609.08029[math.NA] 44. Ranocha, H.:Mimetic properties of diference operators:product and chain rules as for functions of bounded variation and entropy stability of second derivatives. BIT Numer. Math. 59(2), 547-563 (2019). https://doi.org/10.1007/s10543-018-0736-7.arxiv:1805.09126[math.NA] 45. Ranocha, H.:Some notes on summation by parts time integration methods. Results Appl. Math. 1, 100004 (2019). https://doi.org/10.1016/j.rinam.2019.100004.arxiv:1901.08377[math.NA] 46. Ranocha, H., Öfner, P., Sonar, T.:Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. 311, 299-328 (2016). https://doi.org/10.1016/j.jcp.2016.02.009.arxiv:1511.02052[math.NA] 47. Ranocha, H., Öfner, P., Sonar, T.:Extended skew-symmetric form for summation-by-parts operators and varying Jacobians. J. Comput. Phys. 342, 13-28 (2017). https://doi.org/10.1016/j.jcp.2017.04.044.arxiv:1511.08408[math.NA] 48. Ranocha, H., Ostaszewski, K., Heinisch, P.:Numerical methods for the magnetic induction equation with hall efect and projections onto divergence-free vector felds (2018). Submitted. arxiv:1810.01397[math.NA] 49. Ranocha, H., Ostaszewski, K., Heinisch, P.:2019_SBP_vector_calculus_REPRO. Discrete vector calculus and Helmholtz Hodge decomposition for classical fnite diference summation by parts operators. https://github.com/IANW-Projects/2019_SBP_vector_calculus_REPRO (2019). https://doi.org/10.5281/zenodo.3375170 50. Schnack, D.D.:Lectures in Magnetohydrodynamics with an Appendix on Extended MHD. Springer, Berlin Heidelberg (2009). https://doi.org/10.1007/978-3-642-00688-3 51. Schweizer, B.:On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. In:E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin (eds.) Trends in Applications of Mathematics to Mechanics, Springer INdAM Series, vol. 27, pp. 65-79. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-75940-1_4 52. Silberman, Z.J., Adams, T.R., Faber, J.A., Etienne, Z.B., Ruchlin, I.:Numerical generation of vector potentials from specifed magnetic felds. J. Comput. Phys. 379, 421-437 (2019). https://doi.org/10.1016/j.jcp.2018.12.006 53. Sims, J., Giorgi, M., Oliveira, M., Meneghetti, J., Gutierrez, M.:Directional analysis of cardiac motion feld from gated fuorodeoxyglucose PET images using the discrete Helmholtz Hodge decomposition. Comput. Med. Imaging Graph. 65, 69-78 (2018). https://doi.org/10.1016/j.compmedimag.2017.06.004 54. Sjögreen, B., Yee, H.C., Kotov, D.:Skew-symmetric splitting and stability of high order central schemes. J. Phys. Conf. Ser. vol. 837, p. 012019. IOP Publishing (2017). https://doi.org/10.1088/1742-6596/837/1/012019 55. Strand, B.:Summation by parts for fnite diference approximations for d/dx. J. Comput. Phys. 110(1), 47-67 (1994). https://doi.org/10.1006/jcph.1994.1005 56. Svärd, M.:On coordinate transformations for summation-by-parts operators. J. Sci. Comput. 20(1), 29-42 (2004). https://doi.org/10.1023/A:1025881528802 57. Svärd, M.:A note on L∞ bounds and convergence rates of summation-by-parts schemes. BIT Numer. Math. 54(3), 823-830 (2014). https://doi.org/10.1007/s10543-014-0471-7 58. Svärd, M., Nordström, J.:On the order of accuracy for diference approximations of initial-boundary value problems. J. Comput. Phys. 218(1), 333-352 (2006). https://doi.org/10.1016/j.jcp.2006.02.014 59. Svärd, M., Nordström, J.:Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17-38 (2014). https://doi.org/10.1016/j.jcp.2014.02.031 60. Svärd, M., Nordström, J.:On the Convergence Rates of Energy-Stable Finite-Diference Schemes. Technical Report LiTH-MAT-R-2017/14-SE, Linköping University, Linköping, Sweden (2017) |