Time-Implicit High-Order Accurate Positivity-Preserving Discretizations for the Navier–Stokes and Navier–Stokes–Korteweg Equations

Abstract

We extend the framework of the Karush–Kuhn–Tucker (KKT) limiter from second-order nonlinear scalar equations to complex systems of equations and construct time-implicit high-order accurate positivity-preserving local discontinuous Galerkin methods for the Navier-Stokes(NS) and Navier-Stokes-Korteweg(NSK) equations, which are second-order and third-order nonlinear systems, respectively. For the NS and NSK equations, a meaningful numerical approximation must ensure that the density and pressure are non-negative. Borrowing from the idea of the KKT limiter, we use Lagrange multipliers to couple the positivity constraints of density and pressure with local discontinuous Galerkin discretizations combined with a diagonally implicit Runge–Kutta time integration method and obtain high-order accurate positivity-preserving schemes for the NS and NSK equations. The use of diagonally implicit Runge–Kutta time integration methods greatly enlarges the time step in the numerical simulation of the NS and NSK equations. Different from the original application of the KKT limiter, the nonlinearity of pressure with respect to the conservative variables increases the difficulty in implementation. Numerical examples validate the efficiency of these methods.

Keywords

Karush–Kuhn–Tucker limiter / Navier–Stokes equations / Navier–Stokes–Korteweg equations / Positivity-preserving / Local discontinuous Galerkin methods / 65M60 / 35B09 / 76M10 / 76N06 / 76T10

References

[1] Alexander RDiagonally implicit Runge–Kutta methods for stiff ODE’s. SIAM J. Numer. Anal., 1977, 14(6): 1006-1021.

[2] Cheng Q, Shen JA new Lagrange multiplier approach for constructing structure preserving schemes, I. Positivity preserving. Comput. Methods Appl. Mech. Eng., 2022, 391: 114585.

[3] Cheng Q, Shen JA new Lagrange multiplier approach for constructing structure preserving schemes, II. Bound preserving. SIAM J. Numer. Anal., 2022, 60(3): 970-998.

[4] Christlieb AJ, Liu Y, Tang Q, Xu ZHigh order parametrized maximum-principle-preserving and positivity-preserving WENO schemes on unstructured meshes. J. Comput. Phys., 2015, 281334-351.

[5] Christlieb AJ, Liu Y, Tang Q, Xu ZPositivity-preserving finite difference weighted ENO schemes with constrained transport for ideal magnetohydrodynamic equations. SIAM J. Sci. Comput., 2015, 37(4): A1825-A1845.

[6] Cockburn B, Hou S, Shu C-WThe Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput., 1990, 54(190): 545-581

[7] Cockburn B, Lin S-Y, Shu C-WTVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys., 1989, 84(1): 90-113.

[8] Cockburn B, Shu C-WTVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput., 1989, 52(186): 411-435

[9] Cockburn B, Shu C-WThe Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys., 1998, 141(2): 199-224.

[10] Diehl, D.: Higher order schemes for simulation of compressible liquid vapor flows with phase change. PhD thesis, Universität Freiburg (2007)

[11] Facchinei F, Pang J-SFinite-Dimensional Variational Inequalities and Complementarity Problems, 2003, Berlin: Springer

[12] Fan C, Zhang X, Qiu JPositivity-preserving high order finite difference WENO schemes for compressible Navier–Stokes equations. J. Comput. Phys., 2022, 467: 111446.

[13] Han S-P, Pang J-S, Rangaraj NGlobally convergent Newton methods for nonsmooth equations. Math. Oper. Res., 1992, 17(3): 586-607.

[14] Hsu, C.-J.: Numerical Heat Transfer and Fluid Flow (1981)

[15] Hu, J., Tian,T., Pan,S., Wen, Z.: On the local convergence of the semismooth Newton method for composite optimization. arXiv:2211.01127 (2022)

[16] Huang F, Shen JBound/positivity preserving and energy stable scalar auxiliary variable schemes for dissipative systems: applications to Keller–Segel and Poisson–Nernst–Planck equations. SIAM J. Sci. Comput., 2021, 43(3): A1832-A1857.

[17] Ito K, Kunisch KLagrange Multiplier Approach to Variational Problems and Applications, 2008, New Delhi: SIAM

[18] Ito K, Kunisch KOn a semi-smooth Newton method and its globalization. Math. Program., 2009, 118(2): 347-370.

[19] Li Y, Wen Z, Yang C, Yuan Y-XA semismooth Newton method for semidefinite programs and its applications in electronic structure calculations. SIAM J. Sci. Comput., 2018, 40(6): A4131-A4157.

[20] Liang C, Xu ZParametrized maximum principle preserving flux limiters for high order schemes solving multi-dimensional scalar hyperbolic conservation laws. J. Sci. Comput., 2014, 5841-60.

[21] Liu Y, Wen Z, Yin WA multiscale semi-smooth newton method for optimal transport. J. Sci. Comput., 2022, 91(2): 39.

[22] Meister A, Ortleb SOn unconditionally positive implicit time integration for the DG scheme applied to shallow water flows. Int. J. Numer. Methods Fluids, 2014, 76(2):69-94.

[23] Meng X, Xu YAdaptive local discontinuous Galerkin methods with semi-implicit time discretizations for the Navier–Stokes equations. Adv. Aerodyn., 2022, 4(1): 22.

[24] Milzarek A, Xiao X, Cen S, Wen Z, Ulbrich MA stochastic semismooth Newton method for nonsmooth nonconvex optimization. SIAM J. Optim., 2019, 29(4):2916-2948.

[25] Qin T, Shu C-WImplicit positivity-preserving high-order discontinuous Galerkin methods for conservation laws. SIAM J. Sci. Comput., 2018, 40(1): A81-A107.

[26] Skvortsov LMDiagonally implicit Runge–Kutta methods for stiff problems. Comput. Math. Math. Phys., 2006, 46(12): 2110.

[27] Tian L, Xu Y, Kuerten J, van der Vegt JJA local discontinuous Galerkin method for the (non)-isothermal Navier–Stokes–Korteweg equations. J. Comput. Phys., 2015, 295: 685-714.

[28] Toro EFRiemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 2013, Berlin: Springer

[29] van der Vegt JJ, Xia Y, Xu YPositivity preserving limiters for time-implicit higher order accurate discontinuous Galerkin discretizations. SIAM J. Sci. Comput., 2019, 41(3): A2037-A2063.

[30] Xiong T, Qiu J-M, Xu ZA parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows. J. Comput. Phys., 2013, 252310-331.

[31] Xiong T, Qiu J-M, Xu ZParametrized positivity preserving flux limiters for the high order finite difference WENO scheme solving compressible Euler equations. J. Sci. Comput., 2016, 67: 1066-1088.

[32] Xu ZParametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem. Math. Comput., 2014, 83(289): 2213-2238.

[33] Yan F, Van der Vegt JJW, Xia Y, Xu YEntropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations. J. Comput. Appl. Math., 2024, 441(22): 115674.

[34] Yan, F., Van der Vegt,J. J. W., Xia,Y., Xu, Y.: Higher order accurate bounds preserving time-implicit discretizations for the chemically reactive Euler equations. To appear in Communications in Computational Physics, (2024)

[35] Zhang XOn positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations. J. Comput. Phys., 2017, 328: 301-343.

[36] Zhang X, Shu C-WOn maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys., 2010, 229(9): 3091-3120.

[37] Zhang X, Shu C-WOn positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys., 2010, 229(23): 8918-8934.

[38] Zhang X, Shu C-WPositivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms. J. Comput. Phys., 2011, 230(4): 1238-1248.

[39] Zhang X, Xia Y, Shu C-WMaximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes. J. Sci. Comput., 2012, 50(1): 29-62.

[40] Zhang Y, Zhang X, Shu C-WMaximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes. J. Comput. Phys., 2013, 234295-316.

Funding

China Scholarship Council(201806340215)

Rights & permissions

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature