Abstract
Accurate simulations of the Stokes–Darcy system face many difficulties including the coupling of flows in two different subdomains via interface conditions, the incompressibility constraint for the free flow and uncertainties in model parameters. In this report, we propose and study efficient, decoupled, artificial compressibility (AC) ensemble schemes based on a recently developed SAV approach for fast computation of Stokes–Darcy flow ensembles. The proposed algorithms (1) do not require any time step condition and (2) decouple the computation of the velocity and pressure in the free flow region, and (3) result in a common coefficient matrix for all realizations after spatial discretization for which efficient iterative linear solvers such as block CG or block GMRES can be used to greatly reduce the computational cost. We prove the long time stability under two parameter conditions, without any timestep constraints. In particular, for one single simulation, they are unconditionally stable schemes. Several numerical tests are presented to demonstrate the efficiency of the algorithms and illustrate their applications in realistic flow problems.
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References
Babus̆ka, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45, 1005–1034 (2007)
Barth, A., Lang, A.: Multilevel Monte Carlo method with applications to stochastic partial differential equations. Int. J. Comput. Math. 89, 2479–2498 (2012)
Bear, J.: Hydraulics of Groundwater. McGraw-Hill, New York (1979)
Beavers, G., Joseph, D.: Boundary conditions at a naturally impermeable wall. J. Fluid Mech. 30, 197–207 (1967)
Calandra, H., Gratton, S., Langou, J., Pinel, X., Vasseur, X.: Flexible variants of block restarted GMRES methods with application to geophysics. SIAM J. Sci. Comput. 34(2), 714–736 (2012)
Cao, Y., Gunzburger, M., He, X., Wang, X.: Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes–Darcy systems. Math. Comput. 83, 1617–1644 (2014)
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. 38, 1407–1436 (2022)
Chen, W., Gunzburger, M., Sun, D., Wang, X.: Efficient and long-time accurate second-order methods for the Stokes–Darcy system. SIAM J. Numer. Anal. 51, 2563–2584 (2013)
Chen, R.M., Layton, W., McLaughlin, M.: Analysis of variable-step/non-autonomous artificial compression methods. J. Math. Fluid Mech. 21, 30 (2019)
Chorin, A.J.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26 (1967)
Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968)
Connors, J.: An ensemble-based conventional turbulence model for fluid-fluid interactions. Int. J. Numer. Anal. Model. 15, 492–519 (2018)
DeCaria, V., Layton, W., McLaughlin, M.: A conservative, second order, unconditionally stable artificial compression method. Comput. Methods Appl. Mech. Eng. 325, 733–747 (2017)
DeCaria, V., Illiescu, T., Layton, W., McLaughlin, M., Schneier, M.: An artificial compression reduced order model. SIAM J. Numer. Anal. 58, 565–589 (2020)
Discacciati, M., Quarteroni, A.: Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations. Comput. Vis. Sci. 6, 93–103 (2004)
Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43, 57–74 (2002)
Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd edn. Oxford University Press, New York (2014)
Ervin, V., Jenkins, E., Sun, S.: Coupling nonlinear Stokes and Darcy flow using mortar finite elements. Appl. Numer. Math. 61, 1198–1222 (2011)
Feng, Y.T., Owen, D.R.J., Peric, D.: A block conjugate gradient method applied to linear systems with multiple right hand sides. Comput. Methods Appl. Mech. 127, 1–4 (1995)
Fiordilino, J.: A second order ensemble timestepping algorithm for natural convection. SIAM J. Numer. Anal. 56, 816–837 (2018)
Fiordilino, J., Khankan, S.: Ensemble timestepping algorithms for natural convection. Int. J. Numer. Anal. Model. 15, 524–551 (2018)
Gallopulos, E., Simoncini, V.: Convergence of BLOCK GMRES and matrix polynomials. Linear Algebra Appl. 247, 97–119 (1996)
Ganis, B., Klie, H., Wheeler, M., Wildey, T., Yotov, I., Zhang, D.: Stochastic collocation and mixed finite elements for flow in porous media. Comput. Methods Appl. Mech. Eng. 197, 3547–3559 (2008)
Girault, V., Vassilev, D., Yotov, I.: Mortar multiscale finite element methods for Stokes–Darcy flows. Numer. Math. 127, 93–165 (2014)
Goda, K.: A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows. J. Comput. Phys. 30, 76–95 (1979)
Guermond, J., Minev, P.: High-order time stepping for the incompressible Navier–Stokes equations. SIAM J. Sci. Comput. 37, A2656–A2681 (2015)
Guermond, J., Minev, P.: High-order adaptive time stepping for the incompressible Navier–Stokes equations. SIAM J. Sci. Comput. 41, A770–A788 (2019)
Guermond, J.L., Shen, J.: Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41, 112–134 (2003)
Guermond, J.L., Shen, J.: On the error estimates for the rotational pressure-correction projection methods. Math. Comput. 73, 1719–1737 (2004)
Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011–6045 (2006)
Gunzburger, M., Jiang, N., Schneier, M.: An ensemble-proper orthogonal decomposition method for the nonstationary Navier–Stokes equations. SIAM J. Numer. Anal. 55, 286–304 (2017)
Gunzburger, M., Jiang, N., Schneier, M.: A higher-order ensemble/proper orthogonal decomposition method for the nonstationary Navier–Stokes equations. Int. J. Numer. Anal. Model. 15, 608–627 (2018)
Gunzburger, M., Jiang, N., Wang, Z.: An efficient algorithm for simulating ensembles of parameterized flow problems. IMA J. Numer. Anal. 39, 1180–1205 (2019)
Gunzburger, M., Jiang, N., Wang, Z.: A second-order time-stepping scheme for simulating ensembles of parameterized flow problems. Comput. Methods Appl. Math. 19, 681–701 (2019)
Gunzburger, M., Iliescu, T., Schneier, M.: A Leray regularized ensemble-proper orthogonal decomposition method for parameterized convection-dominated flows. IMA J. Numer. Anal. 40, 886–913 (2020)
He, X., Jiang, N., Qiu, C.: An artificial compressibility ensemble algorithm for a stochastic Stokes–Darcy model with random hydraulic conductivity and interface conditions. Int. J. Numer. Methods Eng. 121, 712–739 (2020)
Helton, J.C., Davis, F.J.: Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliab. Eng. Syst. Saf. 81, 23–69 (2003)
Hosder, S., Walters, R., Perez, R.: A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations. In: AIAA-Paper 2006-891, 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 2006, CD-ROM
Jager, W., Mikelic, A.: On the boundary condition at the interface between a porous medium and a free fluid. SIAM J. Appl. Math. 60, 1111–1127 (2000)
Ji, H., Li, Y.: A breakdown-free block conjugate gradient method. BIT Numer. Math. 57(2), 379–403 (2017)
Jiang, N.: A higher order ensemble simulation algorithm for fluid flows. J. Sci. Comput. 64, 264–288 (2015)
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, 34–61 (2017)
Jiang, N.: A pressure-correction ensemble scheme for computing evolutionary Boussinesq equations. J. Sci. Comput. 80, 315–350 (2019)
Jiang, N., Layton, W.: An algorithm for fast calculation of flow ensembles. Int. J. Uncertain. Quantif. 4, 273–301 (2014)
Jiang, N., Layton, W.: Numerical analysis of two ensemble eddy viscosity numerical regularizations of fluid motion. Numer. Methods Partial Differ. Equ. 31, 630–651 (2015)
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)
Jiang, N., Qiu, C.: Numerical analysis of a second order ensemble algorithm for numerical approximation of stochastic Stokes–Darcy equations. J. Comput. Appl. Math. 406, 113934 (2022)
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, 2129–2152 (2018)
Jiang, N., Yang, H.: Stabilized scalar auxiliary variable ensemble algorithms for parameterized flow problems. SIAM J. Sci. Comput. 43(4), A2869–A2896 (2021)
Jiang, N., Yang, H.: SAV decoupled ensemble algorithms for fast computation of Stokes–Darcy flow ensembles. Comput. Methods Appl. Mech. Eng. 387, 114150 (2021)
Jiang, N., Kaya, S., Layton, W.: Analysis of model variance for ensemble based turbulence modeling. Comput. Methods Appl. Math. 15, 173–188 (2015)
Jiang, N., Kubacki, M., Layton, W., Moraiti, M., Tran, H.: A Crank–Nicolson Leapfrog stabilization: unconditional stability and two applications. J. Comput. Appl. Math. 281, 263–276 (2015)
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, 401–428 (2021)
Jiang, N., Li, Y., Yang, H.: A second order ensemble method with different subdomain time steps for simulating coupled surface-groundwater flows. Accepted in Numerical Methods for Partial Differential Equations, in press, (2022)
Jiang, N., Takhirov, A., Waters, J.: Robust SAV-ensemble algorithms for parametrized flow problems with energy stable open boundary conditions. Comput. Methods Appl. Mech. Eng. 392, 114709 (2022)
Kubacki, M., Moraiti, M.: Analysis of a second-order, unconditionally stable, partitioned method for the evolutionary Stokes–Darcy model. Int. J. Numer. Anal. Model. 12, 704–730 (2015)
Kuo, F., Schwab, C., Sloan, I.: Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50, 3351–3374 (2012)
Kuznetsov, B., Vladimirova, N., Yanenko, N.: Numerical Calculation of the Symmetrical Flow of Viscous Incompressible Liquid around a Plate (in Russian). Studies in Mathematics and its Applications, Nauka, Moscow (1966)
Layton, W., McLaughlin, M.: Doubly-adaptive artificial compression methods for incompressible flow. J. Numer. Math. 28, 175–192 (2020)
Layton, W., Xu, S.: Conditioning of linear systems arising from penalty methods. arXiv:2206.06971 (2022)
Layton, W., Tran, H., Xiong, X.: Long time stability of four methods for splitting the evolutionary Stokes–Darcy problem into Stokes and Darcy subproblems. J. Comput. Appl. Math. 236, 3198–3217 (2012)
Layton, W., Tran, H., Trenchea, C.: Analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater-surface water flows. SIAM J. Numer. Anal. 51, 248–272 (2013)
Li, Y., Hou, Y., Rong, Y.: A second-order artificial compression method for the evolutionary Stokes–Darcy system. Numer. Algorithms 84, 1019–1048 (2020)
Li, X., Shen, J., Liu, Z.: New SAV-pressure correction methods for the Navier–Stokes equations: stability and error analysis. Math. Comput. 91, 141–167 (2022)
McCarthy, J.F.: Block-conjugate-gradient method. Phys. Rev. D 40, 2149 (1989)
Mohebujjaman, M., Rebholz, L.: An efficient algorithm for computation of MHD flow ensembles. Comput. Methods Appl. Math. 17, 121–137 (2017)
Mu, M., Zhu, X.: Decoupled schemes for a non-stationary mixed Stokes–Darcy model. Math. Comput. 79, 707–731 (2010)
Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2309–2345 (2008)
O’Leary, D.P.: The block conjugate gradient algorithm and related methods. Linear Algebra Appl. 29, 293–322 (1980)
Reagan, M., Najm, H.N., Ghanem, R.G., Knio, O.M.: Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection. Combust. Flame 132, 545–555 (2003)
Romero, V., Burkardt, J., Gunzburger, M., Peterson, J.: Comparison of pure and “Latinized’’ centroidal Voronoi tessellation against various other statistical sampling methods. Reliab. Eng. Syst. Saf. 91, 1266–1280 (2006)
Rong, Y., Layton, W., Zhao, H.: Numerical analysis of an artificial compression method for magnetohydrodynamic flows at low magnetic Reynolds numbers. J. Sci. Comput. 76, 1458–1483 (2018)
Saffman, P.: On the boundary condition at the interface of a porous medium. Stud. Appl. Math. 1, 93–101 (1971)
Shan, L., Zheng, H., Layton, W.: A decoupling method with different subdomain time steps for the nonstationary Stokes–Darcy model. Numer. Methods for Partial Differ. Equ. 29, 549–583 (2013)
Shen, J.: On error estimates of projection methods for Navier–Stokes equations: first-order schemes. SIAM J. Numer. Anal. 29, 57–77 (1992)
Shen, J.: On error estimates of projection methods for Navier–Stokes equations: second-order schemes. Math. Comput. 65, 1039–1065 (1996)
Takhirov, A., Waters, J.: Ensemble algorithm for parametrized flow problems with energy stable open boundary conditions. Comput. Methods Appl. Math. 20, 531–554 (2020)
Takhirov, A., Neda, M., Waters, J.: Time relaxation algorithm for flow ensembles. Numer. Methods Partial Differ. Equ. 32, 757–777 (2016)
Temam, R.: Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I). Arch. Rational. Mech. Anal. 33, 135–153 (1969)
Temam, R.: Sur l’approximation de la solution des equations de Navier–Stokes par la m\(\acute{e}\)thode des fractionnarires II. Arch. Ration. Mech. Anal. 33, 377–385 (1969)
van Kan, J.: A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Stat. Comput. 7, 870–891 (1987)
Weinan, E., Liu, J.-G.: Projection method I: convergence and numerical boundary layers. SIAM J. Numer. Anal. 32, 1017–1057 (1995)
Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005)
Funding
Nan Jiang was partially supported by the US National Science Foundation grants DMS-1720001 and DMS-2120413. Huanhuan Yang was supported in part by the National Natural Science Foundation of China under Grant 11801348, the key research projects of general universities in Guangdong Province (Grant No. 2019KZDXM034), and the basic research and applied basic research projects in Guangdong Province (Projects of Guangdong, Hong Kong and Macao Center for Applied Mathematics, Grant No. 2020B1515310018).
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Jiang, N., Yang, H. Fast and Accurate Artificial Compressibility Ensemble Algorithms for Computing Parameterized Stokes–Darcy Flow Ensembles. J Sci Comput 94, 17 (2023). https://doi.org/10.1007/s10915-022-02069-2
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DOI: https://doi.org/10.1007/s10915-022-02069-2