Second Order, Unconditionally Stable, Linear Ensemble Algorithms for the Magnetohydrodynamics Equations

Mohebujjaman, M., Rebholz, L.G.: An efficient algorithm for computation of MHD flow ensembles. Comput. Methods Appl. Math. 17(1), 121–137 (2017). https://doi.org/10.1515/cmam-2016-0033 Wei, D., Zhang, Z.: Global well-posedness of the MHD equations in a homogeneous magnetic field. Anal. PDE 10(6), 1361–1406 (2017). https://doi.org/10.2140/apde.2017.10.1361 Trenchea, C.: Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows. Appl. Math. Lett. 27, 97–100 (2014). https://doi.org/10.1016/j.aml.2013.06.017 Mohebujjaman, M., Wang, H., Rebholz, L.G., Mahbub, M.A.A.: An efficient algorithm for simulating ensembles of parameterized MHD flow problems (2021). arXiv:2108.05110 Jiang, N., Layton, W.: An algorithm for fast calculation of flow ensembles. Int. J. Uncertain. Quantif. 4(4), 273–301 (2014). https://doi.org/10.1615/Int.J.UncertaintyQuantification.2014007691 Jiang, N., Layton, W.: Numerical analysis of two ensemble eddy viscosity numerical regularizations of fluid motion. Numer. Methods Partial Differ. Equ. 31(3), 630–651 (2015). https://doi.org/10.1002/num.21908 Jiang, N.: A higher order ensemble simulation algorithm for fluid flows. J. Sci. Comput. 64(1), 264–288 (2015). https://doi.org/10.1007/s10915-014-9932-z Jiang, N.: A second-order ensemble method based on a blended backward differentiation formula timestepping scheme for time-dependent Navier–Stokes equations. Numer. Methods Partial Differ. Equ. 33(1), 34–61 (2017). https://doi.org/10.1002/num.22070 Gunzburger, M., Jiang, N., Schneier, M.: An ensemble-proper orthogonal decomposition method for the nonstationary Navier–Stokes equations. SIAM J. Numer. Anal. 55(1), 286–304 (2017). https://doi.org/10.1137/16M1056444 Fiordilino, J.A.: A second order ensemble time stepping algorithm for natural convection. SIAM J. Numer. Anal. 56(2), 816–837 (2018). https://doi.org/10.1137/17M1135104 Jiang, N.: A pressure-correction ensemble scheme for computing evolutionary Boussinesq equations. J. Sci. Comput. 80(1), 315–350 (2019). https://doi.org/10.1007/s10915-019-00939-w Gunzburger, M., Jiang, N., Wang, Z.: An efficient algorithm for simulating ensembles of parameterized flow problems. IMA J. Numer. Anal. 39(3), 1180–1205 (2019). https://doi.org/10.1093/imanum/dry029 Jiang, N., Qiu, C.: An efficient ensemble algorithm for numerical approximation of stochastic Stokes-Darcy equations. Comput. Methods Appl. Mech. Eng. 343, 249–275 (2019). https://doi.org/10.1016/j.cma.2018.08.020 Jiang, N., Li, Y., Yang, H.: An artificial compressibility Crank–Nicolson leap-frog method for the Stokes-Darcy model and application in ensemble simulations. SIAM J. Numer. Anal. 59(1), 401–428 (2021). https://doi.org/10.1137/20M1321644 Jiang, N., Schneier, M.: An efficient, partitioned ensemble algorithm for simulating ensembles of evolutionary MHD flows at low magnetic Reynolds number. Numer. Methods Partial Differ. Equ. 34(6), 2129–2152 (2018). https://doi.org/10.1002/num.22281 Carter, J., Jiang, N.: Numerical analysis of a second order ensemble method for evolutionary magnetohydrodynamics equations at small magnetic Reynolds number. Numer. Methods Partial Differ. Equ. (2022). https://doi.org/10.1002/num.22843 Elsasser, W.M.: The hydromagnetic equations. Phys. Rev. 79, 183–183 (1950). https://doi.org/10.1103/PhysRev.79.183 Guillén-González, F., Tierra, G.: On linear schemes for a Cahn–Hilliard diffuse interface model. J. Comput. Phys. 234, 140–171 (2013). https://doi.org/10.1016/j.jcp.2012.09.020 Yang, X., Zhao, J., Wang, Q.: Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J. Comput. Phys. 333, 104–127 (2017). https://doi.org/10.1016/j.jcp.2016.12.025 Yang, X., Ju, L.: Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model. Comput. Methods Appl. Mech. Eng. 318, 1005–1029 (2017). https://doi.org/10.1016/j.cma.2017.02.011 Gong, Y., Zhao, J., Wang, Q.: Arbitrarily high-order linear energy stable schemes for gradient flow models. J. Comput. Phys. 419, 109610–20 (2020). https://doi.org/10.1016/j.jcp.2020.109610 Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018) Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019). https://doi.org/10.1137/17M1150153 Yang, Z., Dong, S.: An unconditionally energy-stable scheme based on an implicit auxiliary energy variable for incompressible two-phase flows with different densities involving only precomputable coefficient matrices. J. Comput. Phys. 393, 229–257 (2019). https://doi.org/10.1016/j.jcp.2019.05.018 Yang, Z., Dong, S.: A roadmap for discretely energy-stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity. J. Comput. Phys. 404, 109121–46 (2020). https://doi.org/10.1016/j.jcp.2019.109121 Yang, X.: A novel fully-decoupled, second-order and energy stable numerical scheme of the conserved Allen–Cahn type flow-coupled binary surfactant model. Comput. Methods Appl. Mech. Eng. 373, 113502 (2021). https://doi.org/10.1016/j.cma.2020.113502 Li, S.J. Xiaoli, Liu, Z.: New sav-pressure correction methods for the Navier–Stokes equations: stability and error analysis. Math. Comput. https://doi.org/10.1090/mcom/3651 (2021) Jiang, N., Yang, H.: SAV decoupled ensemble algorithms for fast computation of Stokes–Darcy flow ensembles. Comput. Methods Appl. Mech. Eng. 387, 34–114150 (2021). https://doi.org/10.1016/j.cma.2021.114150 Labovsky, A., Layton, W.J., Manica, C.C., Neda, M., Rebholz, L.G.: The stabilized extrapolated trapezoidal finite-element method for the Navier–Stokes equations. Comput. Methods Appl. Mech. Engrg. 198(9–12), 958–974 (2009). https://doi.org/10.1016/j.cma.2008.11.004 Li, X., Wang, W., Shen, J.: Stability and error analysis of IMEX SAV schemes for the magneto-hydrodynamic equations. SIAM J. Numer. Anal. 60(3), 1026–1054 (2022). https://doi.org/10.1137/21M1430376 Zhang, C., Ouyang, J., Wang, C., Wise, S.M.: Numerical comparison of modified-energy stable SAV-type schemes and classical BDF methods on benchmark problems for the functionalized Cahn–Hilliard equation. J. Comput. Phys. 423, 35–109772 (2020). https://doi.org/10.1016/j.jcp.2020.109772 Jiang, M., Zhang, Z., Zhao, J.: Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation. J. Comput. Phys. 456, 110954 (2022). https://doi.org/10.1016/j.jcp.2022.110954 Foias, C., Holm, D.D., Titi, E.S.: The Navier–Stokes-alpha model of fluid turbulence. vol. 152/153, pp. 505–519 (2001). https://doi.org/10.1016/S0167-2789(01)00191-9. Advances in nonlinear mathematics and science. https://doi-org.libproxy.mst.edu/10.1016/S0167-2789(01)00191-9 Chen, S., Holm, D.D., Margolin, L.G., Zhang, R.: Direct numerical simulations of the Navier-Stokes alpha model. 133, 66–83 (1999) Predictability: quantifying uncertainty in models of complex phenomena (Los Alamos, NM, 1998). https://doi.org/10.1016/S0167-2789(99)00099-8 Wang, C., Wang, J., Xia, Z., Xu, L.: Optimal error estimates of a Crank–Nicolson finite element projection method for magnetohydrodynamic equations. ESAIM Math. Model. Numer. Anal. 56(3), 767–789 (2022). https://doi.org/10.1051/m2an/2022020 Li, B., Wang, J., Xu, L.: A convergent linearized Lagrange finite element method for the magneto-hydrodynamic equations in two-dimensional nonsmooth and nonconvex domains. SIAM J. Numer. Anal. 58(1), 430–459 (2020). https://doi.org/10.1137/18M1205649 Li, B., Ma, S., Ueda, Y.: Analysis of fully discrete finite element methods for 2D Navier–Stokes equations with critical initial data. ESAIM Math. Model. Numer. Anal. 56(6), 2105–2139 (2022). https://doi.org/10.1051/m2an/2022073 Li, B., Ma, S., Wang, N.: Second-order convergence of the linearly extrapolated Crank–Nicolson method for the Navier–Stokes equations with \(H^1\) initial data. J. Sci. Comput. 88(3), 20–70 (2021). https://doi.org/10.1007/s10915-021-01588-8 Gunzburger, M.D.: Finite element methods for viscous incompressible flows. Computer Science and Scientific Computing. Academic Press, San Diego (1989). https://doi.org/10.1016/B978-0-12-307350-1.50002-8. https://www.sciencedirect.com/science/article/pii/B9780123073501500028 Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics. Springer (2007). https://books.google.com/books?id=ci4c_R0WKYYC Kraichnan, R.H.: Inertial ranges in two-dimensional turbulence. Phys. Fluids 10(7), 1417–1423 (1967). https://doi.org/10.1063/1.1762301 Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M., Wells, G.: The fenics project version 1.5 3 (2015). https://doi.org/10.11588/ans.2015.100.20553 Zhang, G., He, X., Yang, X.: Fully decoupled, linear and unconditionally energy stable time discretization scheme for solving the magneto-hydrodynamic equations. J. Comput. Appl. Math. 369, 112636 (2019). https://doi.org/10.1016/j.cam.2019.112636