A divergence-free hybrid finite volume / finite element scheme for the incompressible MHD equations based on compatible finite element spaces with a posteriori limiting
Authors: 
E. Zampa, S. Busto, M. DumbserAuthors Info & Claims
E. Zampa, S. Busto, M. DumbserAuthors Info & ClaimsReferences
[1]
J. Brackbill, D. Barnes, The effect of nonzero ∇ ⋅ B on the numerical solutions of the magnetohydrodynamics equation, J. Comput. Phys. 430 (1980) 426–430.
[2]
C.D. Munz, P. Omnes, R. Schneider, E. Sonnendrücker, U. Voss, Divergence correction techniques for Maxwell solvers based on a hyperbolic model, J. Comput. Phys. 161 (2000) 484–511.
[3]
A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, M. Wesenberg, Hyperbolic divergence cleaning for the MHD equations, J. Comput. Phys. 175 (2002) 645–673.
[4]
K. Powell, An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension), Tech. Rep. ICASE-Report 94-24 (NASA CR-194902) NASA Langley Research Center, Hampton, VA, 1994.
[5]
K. Powell, P. Roe, T. Linde, T. Gombosi, D.D. Zeeuw, A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. Comput. Phys. 154 (1999) 284–309.
[6]
S. Godunov, Symmetric form of the magnetohydrodynamic equation, Numer. Methods Mech. Contin. Media 3 (1) (1972) 26–34.
[7]
K. Yee, Numerical solution of initial voundary value problems involving Maxwell equation in isotropic media, IEEE Trans. Antennas Propag. 14 (1966) 302–307.
[8]
C. DeVore, Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics, J. Comput. Phys. 92 (1991) 142–160.
[9]
T. Gardiner, J. Stone, An unsplit Godunov method for ideal MHD via constrained transport, J. Comput. Phys. 205 (2) (2005) 509–539.
[10]
D. Balsara, D. Spicer, A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations, J. Comput. Phys. 149 (1999) 270–292.
[11]
D. Balsara, D. Spicer, Maintaining pressure positivity in magnetohydrodynamic simulations, J. Comput. Phys. 148 (1999) 133–148.
[12]
D. Balsara, Divergence-free adaptive mesh refinement for magnetohydrodynamics, J. Comput. Phys. 174 (2) (2001) 614–648.
[13]
D. Balsara, Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser. 151 (2004) 149–184.
[14]
D. Balsara, Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows, J. Comput. Phys. 229 (2010) 1970–1993.
[15]
D. Balsara, M. Dumbser, Divergence-free MHD on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers, J. Comput. Phys. 299 (2015) 687–715.
[16]
M. Dumbser, D.S. Balsara, M. Tavelli, F. Fambri, A divergence-free semi-implicit finite volume scheme for ideal, viscous and resistive magnetohydrodynamics, Int. J. Numer. Methods Fluids 89 (2019) 16–42.
[17]
F. Fambri, A novel structure preserving semi-implicit finite volume method for viscous and resistive magnetohydrodynamics, Int. J. Numer. Methods Fluids 93 (12) (2021) 3447–3489.
[18]
F. Fambri, E. Zampa, S. Busto, L. Río-Martín, F. Hindenlang, E. Sonnendrücker, M. Dumbser, A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equations, J. Comput. Phys. 493 (2023).
[19]
K. Hu, Y. Ma, X. J, Stable finite element methods preserving ∇ ⋅ B = 0 exactly for MHD models, Numer. Math. 135 (2017).
[20]
T. Cheng, C. Shu, Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws, J. Comput. Phys. 345 (2017) 427–461.
[21]
Y. Liu, C. Shu, M. Zhang, Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes, J. Comput. Phys. 354 (2018) 163–178.
[22]
D. Derigs, A.R. Winters, G. Gassner, S. Walch, M. Bohm, Ideal GLM-MHD: about the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations, J. Comput. Phys. 364 (2018) 420–467.
[23]
P. Chandrashekar, C. Klingenberg, Entropy stable finite volume scheme for ideal compressible MHD on 2-D Cartesian meshes, SIAM J. Numer. Anal. 54 (2) (2016) 1313–1340.
[24]
D. Ray, P. Chandrashekar, An entropy stable finite volume scheme for the two dimensional Navier–Stokes equations on triangular grids, Appl. Math. Comput. 314 (2017) 257–286.
[25]
S. Busto, M. Dumbser, A new thermodynamically compatible finite volume scheme for magnetohydrodynamics, SIAM J. Numer. Anal. 61 (2023) 343–364.
[26]
S. Busto, M. Dumbser, A new class of general, efficient and simple finite volume schemes for overdetermined thermodynamically compatible hyperbolic systems, Commun. Appl. Math. Comput. Sci. (2023).
[27]
D. Arnold, R. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006) 1–155.
[28]
D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Am. Math. Soc. (N. S.) 47 (2) (2010) 281–354.
[29]
R. Hiptmair, L. Li, S. Mao, W. Zheng, A fully divergence-free finite element method for magnetohydrodynamic equations, Math. Models Methods Appl. Sci. 28 (4) (2018) 659–695.
[30]
K. Hu, J. Lee, J. Xu, Helicity-conservative finite element discretization for incompressible MHD systems, J. Comput. Phys. 436 (2021).
[31]
F. Laakmann, K. Hu, P.E. Farrell, Structure-preserving and helicity-conserving finite element approximations and preconditioning for the Hall MHD equations, J. Comput. Phys. 492 (2023).
[32]
E. Gawlik, F. Gay-Balmaz, A structure-preserving finite element method for compressible ideal and resistive magnetohydrodynamics, J. Plasma Phys. 87 (2021).
[33]
E. Gawlik, F. Gay-Balmaz, A finite element method for MHD that preserves energy, cross-helicity, magnetic helicity, incompressibility, and div b, J. Comput. Phys. 450 (2022).
[34]
F. Li, L. Xu, S. Yakovlev, Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field, J. Comput. Phys. 230 (12) (2011) 4828–4847.
[35]
F. Li, L. Xu, Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations, J. Comput. Phys. 231 (6) (2012) 2655–2675.
[36]
Z. Xu, Y. Liu, New central and central discontinuous Galerkin schemes on overlapping cells of unstructured grids for solving ideal magnetohydrodynamic equations with globally divergence-free magnetic field, J. Comput. Phys. 327 (2016) 203–224.
[37]
G. Fu, An explicit divergence-free DG method for incompressible magnetohydrodynamics, J. Sci. Comput. 79 (3) (2019) 1737–1752.
[38]
G. Wimmer, X. Tang, Structure preserving transport stabilized compatible finite element methods for magnetohydrodynamics, J. Comput. Phys. 501 (2024) 112777.
[39]
G. Cohen, P. Joly, J.E. Roberts, N. Tordjman, Higher order triangular finite elements with mass lumping for the wave equation, SIAM J. Numer. Anal. 38 (6) (2001) 2047–2078.
[40]
J.-L. Guermond, R. Pasquetti, A correction technique for the dispersive effects of mass lumping for transport problems, Comput. Methods Appl. Mech. Eng. 253 (2013) 186–198.
[41]
R. Abgrall, High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices, J. Sci. Comput. 73 (2–3) (2017) 461–494.
[42]
A. Bermúdez, S. Busto, M. Dumbser, J. Ferrín, L. Saavedra, M. Vázquez-Cendón, A staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows, J. Comput. Phys. 421 (2020).
[43]
S. Busto, L.D. Rio, M. Vázquez-Cendón, M. Dumbser, A semi-implicit hybrid finite volume / finite element scheme for all Mach number flows on staggered unstructured meshes, Appl. Math. Comput. 402 (2021).
[44]
S. Busto, M. Dumbser, L. Río-Martín, An arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations, Appl. Math. Comput. 437 (2023).
[45]
A. Lucca, S. Busto, M. Dumbser, An implicit staggered hybrid finite volume/finite element solver for the incompressible Navier-Stokes equations, East Asian J. Appl. Math. 13 (2023) 671–716.
[46]
L. Río-Martín, S. Busto, M. Dumbser, A massively parallel hybrid finite volume/finite element scheme for computational fluid dynamics, Mathematics 9 (2021) 2316.
[47]
S. Busto, M. Dumbser, L. Río-Martín, Staggered semi-implicit hybrid finite volume/finite element schemes for turbulent and non-Newtonian flows, Mathematics 9 (2021) 2972.
[48]
S. Busto, M. Dumbser, A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers, Appl. Numer. Math. 175 (2022) 108–132.
[49]
R. Hiptmair, C. Pagliantini, Splitting-based structure preserving discretization of magnetohydrodynamics, SMAI J. Comput. Math. 4 (2018) 225–257.
[50]
F. Rapetti, A. Bossavit, Whitney forms of higher degree, SIAM J. Numer. Anal. 47 (3) (2009) 2369–2386.
[51]
S.H. Christiansen, F. Rapetti, On high order finite element spaces of differential forms, Math. Comput. 85 (2013) 517–548.
[52]
A. Alonso Rodríguez, L. Bruni Bruno, F. Rapetti, Minimal sets of unisolvent weights for high order Whitney forms on simplices, in: Numerical Mathematics and Advanced Applications—Enumath 2019, in: Lect. Notes Comput. Sci. Eng., Springer, Cham, 2020, pp. 195–203.
[53]
E. Zampa, A.A. Rodríguez, F. Rapetti, Using the FES framework to derive new physical degrees of freedom for finite element spaces of differential forms, Adv. Comput. Math. 49 (17) (2023).
[54]
L. Bruni Bruno, E. Zampa, Unisolvent and minimal physical degrees of freedom for the second family of polynomial differential forms, ESAIM Math. Model. Numer. Anal. 56 (6) (2022) 2239–2253.
[55]
S. Clain, S. Diot, R. Loubère, A high-order finite volume method for systems of conservation lawsmulti-dimensional optimal order detection (MOOD), J. Comput. Phys. 230 (2011) 4028–4050.
[56]
S. Diot, S. Clain, R. Loubère, Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials, Comput. Fluids 64 (2012) 43–63.
[57]
S. Diot, R. Loubère, S. Clain, The MOOD method in the three-dimensional case: very-high-order finite volume method for hyperbolic systems, Int. J. Numer. Methods Fluids 73 (2013) 362–392.
[58]
M. Tavelli, M. Dumbser, A pressure-based semi-implicit space-time discontinuous Galerkin method on staggered unstructured meshes for the solution of the compressible Navier-Stokes equations at all Mach numbers, J. Comput. Phys. 341 (2017) 341–376.
[59]
S. Klainermann, A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluid, Commun. Pure Appl. Math. 34 (1981) 481–524.
[60]
S. Klainermann, A. Majda, Compressible and incompressible fluids, Commun. Pure Appl. Math. 35 (1982) 629–651.
[61]
C. Munz, R. Klein, S. Roller, K. Geratz, The extension of incompressible flow solvers to the weakly compressible regime, Comput. Fluids 32 (2003) 173–196.
[62]
R. Klein, N. Botta, T. Schneider, C. Munz, S. Roller, A. Meister, L. Hoffmann, T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics, J. Eng. Math. 39 (2001) 261–343.
[63]
H. Heumann, R. Hiptmair, Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms, Discrete Contin. Dyn. Syst. 29 (4) (2011) 1471–1495.
[64]
W. Boscheri, A. Chiozzi, M.G. Carlino, G. Bertaglia, A new family of semi-implicit finite volume / virtual element methods for incompressible flows on unstructured meshes, Comput. Methods Appl. Mech. Eng. 414 (2023).
[65]
S. Busto, J.L. Ferrín, E.F. Toro, M.E. Vázquez-Cendón, A projection hybrid high order finite volume/finite element method for incompressible turbulent flows, J. Comput. Phys. 353 (2018) 169–192.
[66]
A. Bermúdez, J.L. Ferrín, L. Saavedra, M.E. Vázquez-Cendón, A projection hybrid finite volume/element method for low-Mach number flows, J. Comput. Phys. 271 (2014) 360–378.
[67]
E.T. Chung, C.S. Lee, A staggered discontinuous Galerkin method for the convection–diffusion equation, J. Numer. Math. 20 (2012) 1–31.
[68]
E.T. Chung, C.S. Lee, A staggered discontinuous Galerkin method for the curl-curl operator, IMA J. Numer. Anal. 32 (3) (2012) 1241–1265.
[69]
H.H. Kim, E.T. Chung, C.S. Lee, A staggered discontinuous Galerkin method for the Stokes system, SIAM J. Numer. Anal. 51 (6) (2013) 3327–3350.
[70]
M. Tavelli, M. Dumbser, A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier-Stokes equations, Appl. Math. Comput. 248 (2014) 70–92.
[71]
M. Tavelli, M. Dumbser, A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes, J. Comput. Phys. 319 (2016) 294–323.
[72]
S. Busto, M. Tavelli, W. Boscheri, M. Dumbser, Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems, Comput. Fluids 198 (2020).
[73]
M. Tavelli, M. Dumbser, Arbitrary high order accurate space-time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity, J. Comput. Phys. 366 (2018) 386–414.
[74]
M.E. Vázquez-Cendón, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. Comput. Phys. 148 (2) (1999) 497–526.
[75]
S. Delcourte, K. Domelevo, P. Omnes, A discrete duality finite volume approach to Hodge decomposition and div-curl problems on almost arbitrary two-dimensional meshes, SIAM J. Numer. Anal. 45 (2007) 1142–1174.
[76]
S. Delcourte, P. Omnes, A discrete duality finite volume discretization of the vorticity-velocity-pressure Stokes problem on almost arbitrary two-dimensional grids, Numer. Methods Partial Differ. Equ. 31 (2015) 1–30.
[77]
M. Dumbser, A. Hidalgo, M. Castro, C. Parés, E.F. Toro, FORCE schemes on unstructured meshes II: non-conservative hyperbolic systems, Comput. Methods Appl. Mech. Eng. 199 (2010) 625–647.
[78]
R. Costa, S. Clain, G. Machado, A sixth-order finite volume scheme for the steady-state incompressible Stokes equations on staggered unstructured meshes, J. Comput. Phys. 349 (2017) 501–527.
[79]
E.F. Toro, R.C. Millington, L.A.M. Nejad, Godunov Methods, Springer, 2001, Ch. Towards very high order Godunov schemes.
[80]
E. Toro, Riemann Solvers and Numerical Methods for Fluids Dynamics, Springer, 2009.
[81]
D. Balsara, M. Dumbser, R. Abgrall, Multidimensional HLLC Riemann solver for unstructured meshes - with application to Euler and MHD flows, J. Comput. Phys. 261 (2014) 172–208.
[82]
J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, 2nd edition, Springer, 2012.
[83]
J.-C. Nédélec, A new family of mixed finite elements in R 3, Numer. Math. 50 (1) (1986) 57–81.
[84]
J.-C. Nédélec, Mixed finite elements in R 3, Numer. Math. 35 (3) (1980) 315–341.
[85]
M. Licht, Averaging based local projections in finite element exterior calculus, 2023.
[86]
J. Lohi, Systematic implementation of higher order Whitney forms in methods based on discrete exterior calculus, Numer. Algorithms 91 (2022) 1261–1285.
[87]
M. Bonazzoli, E. Gaburro, V. Dolean, F. Rapetti, High order edge finite element approximations for the time-harmonic Maxwell's equations, in: 2014 IEEE Conference on Antenna Measurements & Applications (CAMA), 2014, pp. 1–4.
[88]
L. Kettunen, J. Lohi, J. Räbinä, S. Mönkölä, T. Rossi, Generalized finite difference schemes with higher order Whitney forms, ESAIM Math. Model. Numer. Anal. 55 (4) (2021) 1439–1459.
[89]
W. Tonnon, R. Hiptmair, Semi-Lagrangian Finite-Element Exterior Calculus for incompressible flows, 2023.
[90]
A. Ern, J.-L. Guermond, Theory and Practice of Finite Elements, vol. 159, Springer, 2004.
[91]
H.-G. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, vol. 24, second edition, Springer-Verlag, Berlin, 2008.
[92]
J.-L. Guermond, B. Popov, Y. Yang, The effect of the consistent mass matrix on the maximum-principle for scalar conservation equations, J. Sci. Comput. 70 (2017) 1358–1366.
[93]
P. Raviart, The use of numerical integration in finite element methods for solving parabolic equations, in: Topics in Numerical Analysis (Proc. Roy. Irish Acad. Conf., University Coll., Dublin, Ireland, 1972), Academic Press, London, UK, 1973, pp. 233-2.
[94]
M. Tabata, A finite element approximation corresponding to the upwind differencing, Mem. Numer. Math. 1 (1977) 47–63.
[95]
P. Lax, B. Wendroff, Systems of conservation laws, Commun. Pure Appl. Math. 13 (2) (1960) 217–237.
[96]
J. Qiu, M. Dumbser, C.-W. Shu, The discontinuous Galerkin method with Lax–Wendroff type time discretizations, Comput. Methods Appl. Mech. Eng. 194 (2005) 4528–4543.
[97]
M. Dumbser, T. Schwartzkopff, C.-D. Munz, Arbitrary high order finite volume schemes for linear wave propagation, in: E. Krause, Y. Shokin, M. Resch, N. Shokina (Eds.), Computational Science and High Performance Computing II, Springer Berlin Heidelberg, Berlin, Heidelberg, 2006, pp. 129–144.
[98]
J. Donea, A Taylor–Galerkin method for convective transport problems, Int. J. Numer. Methods Eng. 20 (1) (1984) 101–119.
[99]
J. Donea, S. Giuliani, H. Laval, L. Quartapelle, Time-accurate solution of advection-diffusion problems by finite elements, Comput. Methods Appl. Mech. Eng. 45 (1) (1984) 123–145.
[100]
T. Tarhasaari, L. Kettunen, A. Bossavit, Some realizations of a discrete Hodge operator: a reinterpretation of finite element techniques, IEEE Trans. Magn. 35 (3) (1999) 1494–1497.
[101]
R. Hiptmair, Discrete Hodge operators, Numer. Math. 90 (2001) 265–289.
[102]
M. Campos Pinto, E. Sonnendrücker, Gauss-compatible Galerkin schemes for time-dependent Maxwell equations, Math. Comput. 85 (2016) 2651–2685.
[103]
M. Dumbser, O. Zanotti, R. Loubère, S. Diot, A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws, J. Comput. Phys. 278 (2014) 47–75.
[104]
O. Zanotti, F. Fambri, M. Dumbser, A. Hidalgo, Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting, Comput. Fluids 118 (2015) 204–224.
[105]
M. Dumbser, R. Loubère, A simple robust and accurate a posteriori sub–cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes, J. Comput. Phys. 319 (2016) 163–199.
[106]
C. Pagliantini, Computational magnetohydrodynamics with discrete differential forms, Ph.D. thesis ETH, Zürich, 2016.
[107]
J.-L. Guermond, R. Pasquetti, Entropy-based nonlinear viscosity for Fourier approximations of conservation laws, C. R. Math. Acad. Sci. Paris 346 (13–14) (2008) 801–806.
[108]
J.-L. Guermond, R. Pasquetti, B. Popov, Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys. 230 (11) (2011) 4248–4267.
[109]
V. Zingan, J.-L. Guermond, J. Morel, B. Popov, Implementation of the entropy viscosity method with the discontinuous Galerkin method, Comput. Methods Appl. Mech. Eng. 253 (2013) 479–490.
[110]
G. Carré, S.D. Pino, B. Després, E. Labourasse, A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension, J. Comput. Phys. 228 (2009) 5160–5183.
[111]
P. Maire, R. Abgrall, J. Breil, J. Ovadia, A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comput. 29 (2007) 1781–1824.
[112]
P. Maire, A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J. Comput. Phys. 228 (2009) 2391–2425.
[113]
P. Maire, B. Nkonga, Multi–scale Godunov–type method for cell–centered discrete Lagrangian hydrodynamics, J. Comput. Phys. 228 (2009) 799–821.
[114]
W. Boscheri, R. Loubère, P. Maire, A 3D cell-centered ADER MOOD finite volume method for solving updated Lagrangian hyperelasticity on unstructured grids, J. Comput. Phys. 449 (2022).
[115]
D. Balsara, A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows, J. Comput. Phys. 231 (2012) 7476–7503.
[116]
D. Balsara, Three dimensional HLL Riemann solver for conservation laws on structured meshes; Application to Euler and magnetohydrodynamic flows, J. Comput. Phys. 295 (2015) 1–23.
[117]
D. Balsara, Multidimensional Riemann problem with self-similar internal structure – part I – application to hyperbolic conservation laws on structured meshes, J. Comput. Phys. 277 (2014) 163–200.
[118]
D. Balsara, M. Dumbser, Multidimensional Riemann problem with self-similar internal structure – part II – application to hyperbolic conservation laws on unstructured meshes, J. Comput. Phys. 287 (2015) 269–292.
[119]
D. Arnold, R. Falk, R. Winther, Geometric decompositions and local bases for spaces of finite element differential forms, Comput. Methods Appl. Mech. Eng. 198 (21) (2009) 1660–1672.
[120]
J.A. Rossmanith, An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows, SIAM J. Sci. Comput. 28 (5) (2006) 1766–1797.
[121]
S. Komissarov, Multidimensional numerical scheme for resistive relativistic magnetohydrodynamics, Mon. Not. R. Astron. Soc. 382 (3) (2007) 995–1004.
[122]
H. Schlichting, K. Gersten, Boundarylayer Theory, Springer, 2016.
[123]
J. De Loera, J. Rambau, F. Santos, Triangulations: Structures for Algorithms and Applications, Algorithms and Computation in Mathematics, Springer Berlin Heidelberg, 2010.
[124]
L. Pareschi, G. Russo, Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations, Adv. Theory Comput. Math. 3 (2000) 269–288.
[125]
G. Dimarco, R. Loubère, V. Michel-Dansac, M.H. Vignal, Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime, J. Comput. Phys. 372 (2018) 178–201.
[126]
W. Boscheri, G. Dimarco, R. Loubère, M. Tavelli, M.H. Vignal, A second order all Mach number IMEX finite volume solver for the three dimensional Euler equations, J. Comput. Phys. 415 (2020).
[127]
A. Thomann, G. Puppo, C. Klingenberg, An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity, J. Comput. Phys. 4201 (2020).
[128]
A. Thomann, M. Zenk, G. Puppo, C. Klingenberg, An all speed second order IMEX relaxation scheme for the Euler equations, Commun. Comput. Phys. 28 (2020) 591–620.
[129]
V. Michael-Dansac, A. Thomann, TVD–MOOD schemes based on implicit–explicit time integration, Appl. Math. Comput. 433 (2022).
[130]
H. Heumann, Eulerian and semi-Lagrangian methods for advection-diffusion of differential forms, Ph.D. thesis ETH Zürich, 2011.
[131]
R. Abgrall, S. Busto, M. Dumbser, A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics, Appl. Math. Comput. 440 (2023).
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