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doi: 10.3934/era.2021031

A multigrid based finite difference method for solving parabolic interface problem

Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA

* Corresponding author: Shan Zhao

Received  November 2020 Revised  February 2021 Published  April 2021

Fund Project: The authors are supported by NSF grant DMS-1812930

In this paper, a new Cartesian grid finite difference method is introduced to solve two-dimensional parabolic interface problems with second order accuracy achieved in both temporal and spatial discretization. Corrected central difference and the Matched Interface and Boundary (MIB) method are adopted to restore second order spatial accuracy across the interface, while the standard Crank-Nicolson scheme is employed for the implicit time stepping. In the proposed augmented MIB (AMIB) method, an augmented system is formulated with auxiliary variables introduced so that the central difference discretization of the Laplacian could be disassociated with the interface corrections. A simple geometric multigrid method is constructed to efficiently invert the discrete Laplacian in the Schur complement solution of the augmented system. This leads a significant improvement in computational efficiency in comparing with the original MIB method. Being free of a stability constraint, the implicit AMIB method could be asymptotically faster than explicit schemes. Extensive numerical results are carried out to validate the accuracy, efficiency, and stability of the proposed AMIB method.

Citation: Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem. Electronic Research Archive, doi: 10.3934/era.2021031
References:
[1]

L. Adams and T. P. Chartier, New geometric immersed interface multigrid solvers, SIAM J. Sci. Comput., 25 (2004), 1516-1533.  doi: 10.1137/S1064827503421707.  Google Scholar

[2]

L. Adams and Z. Li, The immersed interface/multigrid methods for interface problems, SIAM J. Sci. Comput., 24 (2002), 463-479.  doi: 10.1137/S1064827501389849.  Google Scholar

[3]

C. Attanayake and D. Senaratne, Convergence of an immersed finite element method for semilinear parabolic interface problems, Appl. Math. Sci. (Ruse), 5 (2011), 135-147.   Google Scholar

[4]

P. A. Berthelsen, A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions, J. Comput. Phys., 197 (2004), 364-386.  doi: 10.1016/j.jcp.2003.12.003.  Google Scholar

[5]

F. Bouchon and G. H. Peichl, An immersed interface technique for the numerical solution of the heat equation on a moving domain, Numerical Mathematics and Advanced Applications 2009, Springer Berlin Heidelberg, (2010), 181–189. doi: 10.1007/978-3-642-11795-4_18.  Google Scholar

[6]

F. Bouchon and G. H. Peichl, The immersed interface technique for parabolic problems with mixed boundary conditions, SIAM J. Numer. Anal., 48 (2010), 2247-2266.  doi: 10.1137/09075384X.  Google Scholar

[7]

T. F. Chan and W. L. Wan, Robust multigrid methods for nonsmooth coefficient elliptic linear systems, J. Comput. Appl. Math., 123 (2000), 323-352.  doi: 10.1016/S0377-0427(00)00411-8.  Google Scholar

[8]

Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175-202.  doi: 10.1007/s002110050336.  Google Scholar

[9]

A. Coco and G. Russo, Finite-difference ghost-point multigrid methods on Cartesian grids for elliptic problems in arbitrary domains, J. Comput. Phys., 241 (2013), 464-501.  doi: 10.1016/j.jcp.2012.11.047.  Google Scholar

[10]

A. Coco and G. Russo, Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface, J. Comput. Phys., 361 (2018), 299-330.  doi: 10.1016/j.jcp.2018.01.016.  Google Scholar

[11]

J. Douglas Jr., On the numerical integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by implicit methods, J. Soc. Indust. Appl. Math., 3 (1955), 42-65.   Google Scholar

[12]

J. Douglas and D. Peaceman, Numerical solution of two-dimensional heat-flow problems, AIChEJ., 1 (1955), 505-512.   Google Scholar

[13]

W. FengX. HeY. Lin and X. Zhang, Immersed finite element method for interface problems with algebraic multigrid solver, Commun. Comput. Phys., 15 (2014), 1045-1067.  doi: 10.4208/cicp.150313.171013s.  Google Scholar

[14]

H. FengG. Long and S. Zhao, An augmented matched interface and boundary (MIB) method for solving elliptic interface problem, J. Comput. Appl. Math., 361 (2019), 426-443.  doi: 10.1016/j.cam.2019.05.004.  Google Scholar

[15]

H. Feng and S. Zhao, FFT-based high order central difference schemes for the three-dimensional Poisson's equation with various types of boundary conditions, J. Comput. Phys., 410 (2020), 109391, 24 pp. doi: 10.1016/j.jcp.2020.109391.  Google Scholar

[16]

H. Feng and S. Zhao, A fourth order finite difference method for solving elliptic interface problems with the FFT acceleration, J. Comput. Phys., 419 (2020), 109677, 25 pp. doi: 10.1016/j.jcp.2020.109677.  Google Scholar

[17]

F. Gibou and R. Fedkiw, A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. Comput. Phys., 202 (2005), 577-601.  doi: 10.1016/j.jcp.2004.07.018.  Google Scholar

[18]

J. D. Kandilarov and L. G. Vulkov, The immersed interface method for a nonlinear chemical diffusion equation with local sites of reactions, Numer. Algorithms, 36 (2004), 285-307.  doi: 10.1007/s11075-004-2170-y.  Google Scholar

[19]

J. D. Kandilarov and L. G. Vulkov, The immersed interface method for two-dimensional heat-diffusion equations with singular own sources, Appl. Numer. Math., 57 (2007), 486-497.  doi: 10.1016/j.apnum.2006.07.002.  Google Scholar

[20]

Z. LiX. Chen and Z. Zhang, On multiscale ADI methods for parabolic PDEs with a discontinuous coefficient, Multiscale Model. Simul., 16 (2018), 1623-1647.  doi: 10.1137/17M1151985.  Google Scholar

[21]

Z. LiH. Ji and X. Chen, Accurate solution and gradient computation for elliptic interface problems with variable coefficients, SIAM J. Numer. Anal., 55 (2017), 570-597.  doi: 10.1137/15M1040244.  Google Scholar

[22]

Z. Li and A. Mayo, ADI methods for heat quations with discontinuous along an arbitrary interface, Proc. Sympos. Appl. Math., 48 (1993), 311-315.  doi: 10.1090/psapm/048/1314863.  Google Scholar

[23]

C. LiZ. WeiG. LongC. CampbellS. Ashlyn and S. Zhao, Alternating direction ghost-fluid methods for solving the heat equation with interfaces, Comput. Math. Appl., 80 (2020), 714-732.  doi: 10.1016/j.camwa.2020.04.027.  Google Scholar

[24]

C. Li and S. Zhao, A matched Peaceman–Achford ADI method for solving parabolic interface problems, Appl. Math. Comput., 299 (2017), 28-44.  doi: 10.1016/j.amc.2016.11.033.  Google Scholar

[25]

T. LinQ. Yang and X. Zhang, Partially penalized immersed finite element methods for parabolic interface problems, Numer. Methods Partial Differential Equations, 31 (2015), 1925-1947.  doi: 10.1002/num.21973.  Google Scholar

[26]

J. Liu and Z. Zheng, IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces, Appl. Math. Model., 37 (2013), 1196-1207.  doi: 10.1016/j.apm.2012.03.047.  Google Scholar

[27]

J. Liu and Z. Zheng, A dimension by dimension splitting immersed interface method for heat conduction equation with interfaces, J. Comput. Appl. Math., 261 (2014), 221-231.  doi: 10.1016/j.cam.2013.10.051.  Google Scholar

[28]

S. F. McCormick, Multigrid Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1987. doi: 10.1137/1.9781611971057.  Google Scholar

[29]

D. W. Paeceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic equations, J. Soc. Indust. Appl. Math., 3 (1955), 28-41.  doi: 10.1137/0103003.  Google Scholar

[30]

J. PapacF. Gibou and C. Ratsch, Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions, J. Comput. Phys., 229 (2010), 875-889.  doi: 10.1016/j.jcp.2009.10.017.  Google Scholar

[31]

W. H. Press and S. A. Teukolsky, Numerical Recipes in FORTRAN, the Art of Scientific Computing, 2$^{nd}$ edition, Cambridge University Press, New York, 1992. Google Scholar

[32]

R. K. Sinha and B. Deka, Optimal error estimates for linear parabolic problems with discontinuous coefficients, SIAM J. Numer. Anal., 43 (2005), 733-749.  doi: 10.1137/040605357.  Google Scholar

[33]

R. K. Sinha and B. Deka, Finite element methods for semilinear elliptic and parabolic interface problems, Appl. Numer. Math., 59 (2009), 1870-1883.  doi: 10.1016/j.apnum.2009.02.001.  Google Scholar

[34]

L. Song and S. Zhao, Symmetric interior penalty Galerkin approaches for two-dimensional parabolic interface problems with low regularity solutions, J. Comput. Appl. Math., 330 (2018), 356-379.  doi: 10.1016/j.cam.2017.09.018.  Google Scholar

[35]

J. W. L. Wan and X.-D. Liu, A boundary condition-capturing multigrid approach to irregular boundary problems, SIAM J. Sci. Comput., 25 (2004), 1982-2003.  doi: 10.1137/S1064827503428540.  Google Scholar

[36]

Z. WeiC. Li and S. Zhao, A spatially second order alternating direction implicit (ADI) method for three dimensional parabolic interface problems, Comput. Math. Appl., 75 (2018), 2173-2192.  doi: 10.1016/j.camwa.2017.06.037.  Google Scholar

[37]

A. Wiegmann and K. P. Bube, The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal., 37 (2000), 827-862.  doi: 10.1137/S0036142997328664.  Google Scholar

[38]

K. XiaM. Zhan and G.-W. Wei, MIB method for elliptic equations with multi-material interfaces, J. Comput. Phys., 230 (2011), 4588-4615.  doi: 10.1016/j.jcp.2011.02.037.  Google Scholar

[39]

Q. Yang and X. Zhang, Discontinuous Galerkin immersed finite element methods for parabolic interface problems, J. Comput. Appl. Math., 299 (2016), 127-139.   Google Scholar

[40]

S. Zhao, A matched alternating direction implicit (ADI) method for solving the heat equation with interfaces, J. Sci. Comput., 63 (2015), 118-137.  doi: 10.1007/s10915-014-9887-0.  Google Scholar

[41]

S. Zhao and G. W. Wei, High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces, J. Comput. Phys., 200 (2004), 60-103.  doi: 10.1016/j.jcp.2004.03.008.  Google Scholar

[42]

Y. C. ZhouS. ZhaoM. Feig and G. W. Wei, High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular source, J. Comput. Phys., 213 (2006), 1-30.  doi: 10.1016/j.jcp.2005.07.022.  Google Scholar

show all references

References:
[1]

L. Adams and T. P. Chartier, New geometric immersed interface multigrid solvers, SIAM J. Sci. Comput., 25 (2004), 1516-1533.  doi: 10.1137/S1064827503421707.  Google Scholar

[2]

L. Adams and Z. Li, The immersed interface/multigrid methods for interface problems, SIAM J. Sci. Comput., 24 (2002), 463-479.  doi: 10.1137/S1064827501389849.  Google Scholar

[3]

C. Attanayake and D. Senaratne, Convergence of an immersed finite element method for semilinear parabolic interface problems, Appl. Math. Sci. (Ruse), 5 (2011), 135-147.   Google Scholar

[4]

P. A. Berthelsen, A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions, J. Comput. Phys., 197 (2004), 364-386.  doi: 10.1016/j.jcp.2003.12.003.  Google Scholar

[5]

F. Bouchon and G. H. Peichl, An immersed interface technique for the numerical solution of the heat equation on a moving domain, Numerical Mathematics and Advanced Applications 2009, Springer Berlin Heidelberg, (2010), 181–189. doi: 10.1007/978-3-642-11795-4_18.  Google Scholar

[6]

F. Bouchon and G. H. Peichl, The immersed interface technique for parabolic problems with mixed boundary conditions, SIAM J. Numer. Anal., 48 (2010), 2247-2266.  doi: 10.1137/09075384X.  Google Scholar

[7]

T. F. Chan and W. L. Wan, Robust multigrid methods for nonsmooth coefficient elliptic linear systems, J. Comput. Appl. Math., 123 (2000), 323-352.  doi: 10.1016/S0377-0427(00)00411-8.  Google Scholar

[8]

Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), 175-202.  doi: 10.1007/s002110050336.  Google Scholar

[9]

A. Coco and G. Russo, Finite-difference ghost-point multigrid methods on Cartesian grids for elliptic problems in arbitrary domains, J. Comput. Phys., 241 (2013), 464-501.  doi: 10.1016/j.jcp.2012.11.047.  Google Scholar

[10]

A. Coco and G. Russo, Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface, J. Comput. Phys., 361 (2018), 299-330.  doi: 10.1016/j.jcp.2018.01.016.  Google Scholar

[11]

J. Douglas Jr., On the numerical integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by implicit methods, J. Soc. Indust. Appl. Math., 3 (1955), 42-65.   Google Scholar

[12]

J. Douglas and D. Peaceman, Numerical solution of two-dimensional heat-flow problems, AIChEJ., 1 (1955), 505-512.   Google Scholar

[13]

W. FengX. HeY. Lin and X. Zhang, Immersed finite element method for interface problems with algebraic multigrid solver, Commun. Comput. Phys., 15 (2014), 1045-1067.  doi: 10.4208/cicp.150313.171013s.  Google Scholar

[14]

H. FengG. Long and S. Zhao, An augmented matched interface and boundary (MIB) method for solving elliptic interface problem, J. Comput. Appl. Math., 361 (2019), 426-443.  doi: 10.1016/j.cam.2019.05.004.  Google Scholar

[15]

H. Feng and S. Zhao, FFT-based high order central difference schemes for the three-dimensional Poisson's equation with various types of boundary conditions, J. Comput. Phys., 410 (2020), 109391, 24 pp. doi: 10.1016/j.jcp.2020.109391.  Google Scholar

[16]

H. Feng and S. Zhao, A fourth order finite difference method for solving elliptic interface problems with the FFT acceleration, J. Comput. Phys., 419 (2020), 109677, 25 pp. doi: 10.1016/j.jcp.2020.109677.  Google Scholar

[17]

F. Gibou and R. Fedkiw, A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. Comput. Phys., 202 (2005), 577-601.  doi: 10.1016/j.jcp.2004.07.018.  Google Scholar

[18]

J. D. Kandilarov and L. G. Vulkov, The immersed interface method for a nonlinear chemical diffusion equation with local sites of reactions, Numer. Algorithms, 36 (2004), 285-307.  doi: 10.1007/s11075-004-2170-y.  Google Scholar

[19]

J. D. Kandilarov and L. G. Vulkov, The immersed interface method for two-dimensional heat-diffusion equations with singular own sources, Appl. Numer. Math., 57 (2007), 486-497.  doi: 10.1016/j.apnum.2006.07.002.  Google Scholar

[20]

Z. LiX. Chen and Z. Zhang, On multiscale ADI methods for parabolic PDEs with a discontinuous coefficient, Multiscale Model. Simul., 16 (2018), 1623-1647.  doi: 10.1137/17M1151985.  Google Scholar

[21]

Z. LiH. Ji and X. Chen, Accurate solution and gradient computation for elliptic interface problems with variable coefficients, SIAM J. Numer. Anal., 55 (2017), 570-597.  doi: 10.1137/15M1040244.  Google Scholar

[22]

Z. Li and A. Mayo, ADI methods for heat quations with discontinuous along an arbitrary interface, Proc. Sympos. Appl. Math., 48 (1993), 311-315.  doi: 10.1090/psapm/048/1314863.  Google Scholar

[23]

C. LiZ. WeiG. LongC. CampbellS. Ashlyn and S. Zhao, Alternating direction ghost-fluid methods for solving the heat equation with interfaces, Comput. Math. Appl., 80 (2020), 714-732.  doi: 10.1016/j.camwa.2020.04.027.  Google Scholar

[24]

C. Li and S. Zhao, A matched Peaceman–Achford ADI method for solving parabolic interface problems, Appl. Math. Comput., 299 (2017), 28-44.  doi: 10.1016/j.amc.2016.11.033.  Google Scholar

[25]

T. LinQ. Yang and X. Zhang, Partially penalized immersed finite element methods for parabolic interface problems, Numer. Methods Partial Differential Equations, 31 (2015), 1925-1947.  doi: 10.1002/num.21973.  Google Scholar

[26]

J. Liu and Z. Zheng, IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces, Appl. Math. Model., 37 (2013), 1196-1207.  doi: 10.1016/j.apm.2012.03.047.  Google Scholar

[27]

J. Liu and Z. Zheng, A dimension by dimension splitting immersed interface method for heat conduction equation with interfaces, J. Comput. Appl. Math., 261 (2014), 221-231.  doi: 10.1016/j.cam.2013.10.051.  Google Scholar

[28]

S. F. McCormick, Multigrid Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1987. doi: 10.1137/1.9781611971057.  Google Scholar

[29]

D. W. Paeceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic equations, J. Soc. Indust. Appl. Math., 3 (1955), 28-41.  doi: 10.1137/0103003.  Google Scholar

[30]

J. PapacF. Gibou and C. Ratsch, Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions, J. Comput. Phys., 229 (2010), 875-889.  doi: 10.1016/j.jcp.2009.10.017.  Google Scholar

[31]

W. H. Press and S. A. Teukolsky, Numerical Recipes in FORTRAN, the Art of Scientific Computing, 2$^{nd}$ edition, Cambridge University Press, New York, 1992. Google Scholar

[32]

R. K. Sinha and B. Deka, Optimal error estimates for linear parabolic problems with discontinuous coefficients, SIAM J. Numer. Anal., 43 (2005), 733-749.  doi: 10.1137/040605357.  Google Scholar

[33]

R. K. Sinha and B. Deka, Finite element methods for semilinear elliptic and parabolic interface problems, Appl. Numer. Math., 59 (2009), 1870-1883.  doi: 10.1016/j.apnum.2009.02.001.  Google Scholar

[34]

L. Song and S. Zhao, Symmetric interior penalty Galerkin approaches for two-dimensional parabolic interface problems with low regularity solutions, J. Comput. Appl. Math., 330 (2018), 356-379.  doi: 10.1016/j.cam.2017.09.018.  Google Scholar

[35]

J. W. L. Wan and X.-D. Liu, A boundary condition-capturing multigrid approach to irregular boundary problems, SIAM J. Sci. Comput., 25 (2004), 1982-2003.  doi: 10.1137/S1064827503428540.  Google Scholar

[36]

Z. WeiC. Li and S. Zhao, A spatially second order alternating direction implicit (ADI) method for three dimensional parabolic interface problems, Comput. Math. Appl., 75 (2018), 2173-2192.  doi: 10.1016/j.camwa.2017.06.037.  Google Scholar

[37]

A. Wiegmann and K. P. Bube, The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal., 37 (2000), 827-862.  doi: 10.1137/S0036142997328664.  Google Scholar

[38]

K. XiaM. Zhan and G.-W. Wei, MIB method for elliptic equations with multi-material interfaces, J. Comput. Phys., 230 (2011), 4588-4615.  doi: 10.1016/j.jcp.2011.02.037.  Google Scholar

[39]

Q. Yang and X. Zhang, Discontinuous Galerkin immersed finite element methods for parabolic interface problems, J. Comput. Appl. Math., 299 (2016), 127-139.   Google Scholar

[40]

S. Zhao, A matched alternating direction implicit (ADI) method for solving the heat equation with interfaces, J. Sci. Comput., 63 (2015), 118-137.  doi: 10.1007/s10915-014-9887-0.  Google Scholar

[41]

S. Zhao and G. W. Wei, High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces, J. Comput. Phys., 200 (2004), 60-103.  doi: 10.1016/j.jcp.2004.03.008.  Google Scholar

[42]

Y. C. ZhouS. ZhaoM. Feig and G. W. Wei, High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular source, J. Comput. Phys., 213 (2006), 1-30.  doi: 10.1016/j.jcp.2005.07.022.  Google Scholar

Figure 1.  Black dots denote real function values, while empty circles indicate fictitious values. They are used in the polynomials for approximating derivative jump across the interface at $ x = \alpha $
Figure 2.  Derivative jump approximation in two scenarios. Empty circles denote the fictitious values at the grid points, while black dots indicate real function values
Figure 3.  Consider the case where the interface intersects $ y = y_j $ at $ (x_{o}, y_j). $ At the irregular points $ P_1(i, j) $ and $ P_2(i+1, j) $, fictitious values (in box) can be constructed. Here $ \theta $ is the angle between positive $ x- $ direction and the normal vector $ \vec{n} $.
Figure 4.  The needed nine points on fine mesh for restriction or interpolation in the multigrid approach
Figure 5.  The temporal convergence for example 2. The left is for AMIB-GMRES, and the right is for AMIB-LU
Figure 6.  Stability test for the two AMIB methods. The left one is for AMIB-GMRES, and the right one is for AMIB-LU. All the numerical errors are bounded with $ n_x = 128 $, and $ T = 10^4\Delta t. $
Figure 7.  The numerical errors from two AMIB methods
Figure 8.  The solution of heat equation evolves within time equal to 1. The rapid change during initial period indicates quick temperature drop
Table 1.  Spatial convergence for example 1
$ [n_x, n_y] $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 2.08E-4 8.82E-4 0.94
$ [64, 64] $ 3.21E-5 2.70 1.69E-4 2.38 5.48
$ [128, 128] $ 7.52E-6 2.09 3.13E-5 2.43 26.6
$ [256, 256] $ 1.94E-6 1.95 9.07E-6 1.79 175
$ [512, 512] $ 5.66E-7 1.95 2.97E-6 1.79 1143
AMIB-LU
$ L_{2} $ $ L_{\infty} $
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 2.08E-4 8.82E-4 0.30
$ [64, 64] $ 3.19E-5 2.70 1.68E-4 2.38 1.49
$ [128, 128] $ 7.52E-6 2.09 3.13E-5 2.43 6.50
$ [256, 256] $ 1.94E-6 1.95 9.08E-6 1.79 29.8
$ [512, 512] $ 5.71E-7 1.95 2.99E-6 1.79 200
MIB
$ L_{2} $ $ L_{\infty} $
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 2.07E-4 8.75E-4 0.62
$ [64, 64] $ 3.19E-5 2.70 1.68E-4 2.38 4.76
$ [128, 128] $ 7.50E-6 2.08 3.13E-5 2.42 35.4
$ [256, 256] $ 1.94E-6 1.94 9.09E-6 1.79 336
$ [512, 512] $ 5.66E-7 1.95 2.97E-6 1.79 2935
$ [n_x, n_y] $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 2.08E-4 8.82E-4 0.94
$ [64, 64] $ 3.21E-5 2.70 1.69E-4 2.38 5.48
$ [128, 128] $ 7.52E-6 2.09 3.13E-5 2.43 26.6
$ [256, 256] $ 1.94E-6 1.95 9.07E-6 1.79 175
$ [512, 512] $ 5.66E-7 1.95 2.97E-6 1.79 1143
AMIB-LU
$ L_{2} $ $ L_{\infty} $
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 2.08E-4 8.82E-4 0.30
$ [64, 64] $ 3.19E-5 2.70 1.68E-4 2.38 1.49
$ [128, 128] $ 7.52E-6 2.09 3.13E-5 2.43 6.50
$ [256, 256] $ 1.94E-6 1.95 9.08E-6 1.79 29.8
$ [512, 512] $ 5.71E-7 1.95 2.99E-6 1.79 200
MIB
$ L_{2} $ $ L_{\infty} $
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 2.07E-4 8.75E-4 0.62
$ [64, 64] $ 3.19E-5 2.70 1.68E-4 2.38 4.76
$ [128, 128] $ 7.50E-6 2.08 3.13E-5 2.42 35.4
$ [256, 256] $ 1.94E-6 1.94 9.09E-6 1.79 336
$ [512, 512] $ 5.66E-7 1.95 2.97E-6 1.79 2935
Table 2.  Temporal convergence for example 1
$ n_t $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 1.32E-4 5.46E-4 2.76
$ 4 $ 8.98E-5 0.56 3.25E-4 0.75 5.28
$ 8 $ 2.48E-5 1.86 7.42E-5 2.13 9.73
$ 16 $ 6.21E-6 2.00 1.87E-5 1.99 19.36
$ 32 $ 1.56E-6 1.99 4.88E-6 2.02 37.88
$ 64 $ 6.26E-7 1.32 3.21E-6 0.60 76.9
$ 128 $ 5.60E-7 0.16 2.99E-6 0.10 148.4
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 1.32E-4 5.46E-4 58
$ 4 $ 8.98E-5 0.56 3.25E-4 0.75 59.4
$ 8 $ 2.48E-5 1.86 7.42E-5 2.13 61.5
$ 16 $ 6.21E-6 2.00 1.87E-5 1.99 63.2
$ 32 $ 1.56E-6 1.99 4.88E-6 1.94 64.7
$ 64 $ 6.34E-7 1.30 3.24E-6 0.59 66.8
$ 128 $ 5.76E-7 0.13 3.04E-6 0.09 74
MIB
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 1.32E-4 5.46E-4 21.97
$ 4 $ 8.98E-5 0.56 3.25E-4 0.75 42.34
$ 8 $ 2.48E-5 1.86 7.42E-5 2.14 81.72
$ 16 $ 6.21E-6 2.00 1.87E-5 1.98 153.34
$ 32 $ 1.56E-6 1.99 4.88E-6 1.94 287.03
$ 64 $ 6.27E-7 1.31 3.22E-6 0.60 533
$ 128 $ 5.69E-7 0.14 3.02E-6 0.09 863
$ n_t $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 1.32E-4 5.46E-4 2.76
$ 4 $ 8.98E-5 0.56 3.25E-4 0.75 5.28
$ 8 $ 2.48E-5 1.86 7.42E-5 2.13 9.73
$ 16 $ 6.21E-6 2.00 1.87E-5 1.99 19.36
$ 32 $ 1.56E-6 1.99 4.88E-6 2.02 37.88
$ 64 $ 6.26E-7 1.32 3.21E-6 0.60 76.9
$ 128 $ 5.60E-7 0.16 2.99E-6 0.10 148.4
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 1.32E-4 5.46E-4 58
$ 4 $ 8.98E-5 0.56 3.25E-4 0.75 59.4
$ 8 $ 2.48E-5 1.86 7.42E-5 2.13 61.5
$ 16 $ 6.21E-6 2.00 1.87E-5 1.99 63.2
$ 32 $ 1.56E-6 1.99 4.88E-6 1.94 64.7
$ 64 $ 6.34E-7 1.30 3.24E-6 0.59 66.8
$ 128 $ 5.76E-7 0.13 3.04E-6 0.09 74
MIB
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 1.32E-4 5.46E-4 21.97
$ 4 $ 8.98E-5 0.56 3.25E-4 0.75 42.34
$ 8 $ 2.48E-5 1.86 7.42E-5 2.14 81.72
$ 16 $ 6.21E-6 2.00 1.87E-5 1.98 153.34
$ 32 $ 1.56E-6 1.99 4.88E-6 1.94 287.03
$ 64 $ 6.27E-7 1.31 3.22E-6 0.60 533
$ 128 $ 5.69E-7 0.14 3.02E-6 0.09 863
Table 3.  Temporal convergence for example 2
$ n_t $ AMIB-GMRES-$ f^{n+\frac{1}{2}} $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU $ L_{2} $ $ L_{\infty} $ CPU
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 3.22E-2 7.55E-2 3.04 1.39E-3 3.43E-3 2.81
$ 4 $ 5.56E-3 2.53 1.57E-2 2.27 5.9 3.14E-4 2.15 7.77E-4 2.14 5.69
$ 8 $ 9.47E-4 2.55 2.94E-3 2.42 12.5 7.59E-5 2.05 1.88E-4 2.05 11.2
$ 16 $ 1.67E-4 2.50 4.35E-4 2.76 23 1.89E-5 2.01 4.68E-5 2.01 22
$ 32 $ 3.82E-5 2.13 8.32E-5 2.39 45 5.00E-6 1.92 1.21E-5 1.95 45.4
$ 64 $ 9.68E-6 1.99 2.13E-5 1.98 87 1.44E-6 1.80 3.30E-6 1.87 96.3
$ 128 $ 2.579E-6 1.91 6.05E-6 1.82 168 6.60E-7 1.13 2.84E-6 0.22 170
$ 256 $ 9.03E-7 1.53 3.25E-6 0.91 334
AMIB-LU-$ f^{n+\frac{1}{2}} $ AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU $ L_{2} $ $ L_{\infty} $ CPU
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 3.22E-2 7.55E-2 65 1.40E-3 3.44E-3 57
$ 4 $ 5.56E-3 2.53 1.57E-2 2.27 69 3.15E-4 2.15 7.79E-4 2.15 59.6
$ 8 $ 9.47E-4 2.55 2.94E-3 2.42 62 7.64E-5 2.04 1.89E-4 2.04 58.9
$ 16 $ 1.67E-4 2.50 4.35E-4 2.76 62 1.90E-5 2.01 4.70E-5 2.01 57.7
$ 32 $ 3.82E-5 2.13 8.32E-5 2.39 60.6 4.82E-6 1.98 1.17E-5 2.01 60.7
$ 64 $ 9.65E-6 1.99 2.11E-5 1.98 66 1.34E-6 1.85 3.05E-6 1.94 64.6
$ 128 $ 2.57E-6 1.91 5.98E-6 1.82 75.1 6.16E-7 1.12 2.70E-6 0.18 75.9
$ 256 $ 8.88E-7 1.53 3.18E-6 0.91 91
$ 512 $ 5.69E-7 0.64 2.76E-6 0.20 129
$ n_t $ AMIB-GMRES-$ f^{n+\frac{1}{2}} $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU $ L_{2} $ $ L_{\infty} $ CPU
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 3.22E-2 7.55E-2 3.04 1.39E-3 3.43E-3 2.81
$ 4 $ 5.56E-3 2.53 1.57E-2 2.27 5.9 3.14E-4 2.15 7.77E-4 2.14 5.69
$ 8 $ 9.47E-4 2.55 2.94E-3 2.42 12.5 7.59E-5 2.05 1.88E-4 2.05 11.2
$ 16 $ 1.67E-4 2.50 4.35E-4 2.76 23 1.89E-5 2.01 4.68E-5 2.01 22
$ 32 $ 3.82E-5 2.13 8.32E-5 2.39 45 5.00E-6 1.92 1.21E-5 1.95 45.4
$ 64 $ 9.68E-6 1.99 2.13E-5 1.98 87 1.44E-6 1.80 3.30E-6 1.87 96.3
$ 128 $ 2.579E-6 1.91 6.05E-6 1.82 168 6.60E-7 1.13 2.84E-6 0.22 170
$ 256 $ 9.03E-7 1.53 3.25E-6 0.91 334
AMIB-LU-$ f^{n+\frac{1}{2}} $ AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU $ L_{2} $ $ L_{\infty} $ CPU
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 3.22E-2 7.55E-2 65 1.40E-3 3.44E-3 57
$ 4 $ 5.56E-3 2.53 1.57E-2 2.27 69 3.15E-4 2.15 7.79E-4 2.15 59.6
$ 8 $ 9.47E-4 2.55 2.94E-3 2.42 62 7.64E-5 2.04 1.89E-4 2.04 58.9
$ 16 $ 1.67E-4 2.50 4.35E-4 2.76 62 1.90E-5 2.01 4.70E-5 2.01 57.7
$ 32 $ 3.82E-5 2.13 8.32E-5 2.39 60.6 4.82E-6 1.98 1.17E-5 2.01 60.7
$ 64 $ 9.65E-6 1.99 2.11E-5 1.98 66 1.34E-6 1.85 3.05E-6 1.94 64.6
$ 128 $ 2.57E-6 1.91 5.98E-6 1.82 75.1 6.16E-7 1.12 2.70E-6 0.18 75.9
$ 256 $ 8.88E-7 1.53 3.18E-6 0.91 91
$ 512 $ 5.69E-7 0.64 2.76E-6 0.20 129
Table 4.  Spatial convergence for example 2
$ [n_x, n_y] $ AMIB-GMRES-$ f^{n+\frac{1}{2}} $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU $ L_{2} $ $ L_{\infty} $ CPU
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 1.15E-4 5.16E-4 1.11 2.25E-4 5.16E-4 1.19
$ [64, 64] $ 1.10E-4 1.03 3.42E-4 0.59 7.39 1.10E-4 1.03 3.42E-4 0.59 7.86
$ [128, 128] $ 1.44E-5 2.93 6.40E-5 2.42 42 1.44E-5 2.93 6.40E-5 2.42 43
$ [256, 256] $ 2.06E-6 2.82 1.54E-5 2.61 264 2.04E-6 2.82 1.04E-5 2.61 258
$ [512, 512] $ 5.18E-7 2.01 2.65E-6 1.99 1457 5.05E-7 2.01 2.62E-6 1.99 1463
AMIB-LU-$ f^{n+\frac{1}{2}} $ AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU $ L_{2} $ $ L_{\infty} $ CPU
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 1.15E-4 5.16E-4 0.34 2.25E-4 5.16E-4 0.7
$ [64, 64] $ 1.10E-4 0.06 3.42E-4 0.59 1.66 1.10E-4 1.03 3.42E-4 0.59 1.78
$ [128, 128] $ 1.44E-5 2.93 6.40E-5 2.42 8.04 1.44E-5 2.93 6.40E-5 2.43 8.76
$ [256, 256] $ 2.06E-6 2.81 1.05E-5 2.61 39.4 2.05E-6 2.91 1.04E-5 1.79 41.6
$ [512, 512] $ 5.28E-7 1.96 2.68E-6 1.97 231 5.15E-7 1.99 2.65E-6 1.99 227
$ [n_x, n_y] $ AMIB-GMRES-$ f^{n+\frac{1}{2}} $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU $ L_{2} $ $ L_{\infty} $ CPU
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 1.15E-4 5.16E-4 1.11 2.25E-4 5.16E-4 1.19
$ [64, 64] $ 1.10E-4 1.03 3.42E-4 0.59 7.39 1.10E-4 1.03 3.42E-4 0.59 7.86
$ [128, 128] $ 1.44E-5 2.93 6.40E-5 2.42 42 1.44E-5 2.93 6.40E-5 2.42 43
$ [256, 256] $ 2.06E-6 2.82 1.54E-5 2.61 264 2.04E-6 2.82 1.04E-5 2.61 258
$ [512, 512] $ 5.18E-7 2.01 2.65E-6 1.99 1457 5.05E-7 2.01 2.62E-6 1.99 1463
AMIB-LU-$ f^{n+\frac{1}{2}} $ AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU $ L_{2} $ $ L_{\infty} $ CPU
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 1.15E-4 5.16E-4 0.34 2.25E-4 5.16E-4 0.7
$ [64, 64] $ 1.10E-4 0.06 3.42E-4 0.59 1.66 1.10E-4 1.03 3.42E-4 0.59 1.78
$ [128, 128] $ 1.44E-5 2.93 6.40E-5 2.42 8.04 1.44E-5 2.93 6.40E-5 2.43 8.76
$ [256, 256] $ 2.06E-6 2.81 1.05E-5 2.61 39.4 2.05E-6 2.91 1.04E-5 1.79 41.6
$ [512, 512] $ 5.28E-7 1.96 2.68E-6 1.97 231 5.15E-7 1.99 2.65E-6 1.99 227
Table 5.  Spatial convergence for example 2 with respect to different coefficient contrasts. $ n_x = 128 $, $ \Delta t = 0.002 $
$ [\beta^+, \beta^-] $ AMIB-GMRES
Small $ \beta $ side Large $ \beta $ side
$ L_{2} $ $ L_{\infty} $ CPU time(s) $ L_{2} $ $ L_{\infty} $ CPU time(s)
$ [10000, 1] $ 1.63E-5 5.11E-5 71.4 1.86E-5 6.70E-5 92.6
$ [1000, 1] $ 1.63E-5 5.10E-5 65.4 1.86E-5 6.69E-5 92.8
$ [100, 1] $ 1.61E-5 5.04E-5 52.4 1.84E-5 6.60E-5 69.3
$ [10, 1] $ 1.47E-5 4.43E-5 31.4 1.65E-5 5.74E-5 33.8
$ [1, 10] $ 9.48E-6 4.71E-5 36.2 7.01E-6 3.44E-5 36.2
$ [1, 100] $ 6.03E-5 1.56E-4 58.3 6.20E-5 1.45E-4 66.4
$ [1, 1000] $ 4.66E-4 9.00E-4 67.8 6.38E-4 1.19E-3 81.6
$ [1, 10000] $ 5.26E-4 8.52E-4 71.2 5.62E-3 1.02E-2 89.5
$ [\beta^+, \beta^-] $ AMIB-LU
Small $ \beta $ side Large $ \beta $ side
$ L_{2} $ $ L_{\infty} $ CPU time(s) $ L_{2} $ $ L_{\infty} $ CPU time(s)
$ [10000, 1] $ 1.63E-5 5.11E-5 10.9 1.86E-5 6.72E-5 10.9
$ [1000, 1] $ 1.63E-5 5.10E-5 10.58 1.86E-5 6.71E-5 10.46
$ [100, 1] $ 1.61E-5 5.04E-5 8.8 1.84E-5 6.61E-5 9.1
$ [10, 1] $ 1.47E-5 4.43E-5 6.8 1.66E-5 5.75E-5 6.9
$ [1, 10] $ 9.48E-6 4.71E-5 7.0 7.03E-6 3.43E-5 6.9
$ [1, 100] $ 6.03E-5 1.56E-4 9.2 6.20E-5 1.45E-4 8.5
$ [1, 1000] $ 4.66E-4 9.00E-4 8.6 6.38E-4 1.19E-3 8.6
$ [1, 10000] $ 5.29E-4 8.56E-4 9.0 5.63E-3 1.02E-2 8.7
$ [\beta^+, \beta^-] $ AMIB-GMRES
Small $ \beta $ side Large $ \beta $ side
$ L_{2} $ $ L_{\infty} $ CPU time(s) $ L_{2} $ $ L_{\infty} $ CPU time(s)
$ [10000, 1] $ 1.63E-5 5.11E-5 71.4 1.86E-5 6.70E-5 92.6
$ [1000, 1] $ 1.63E-5 5.10E-5 65.4 1.86E-5 6.69E-5 92.8
$ [100, 1] $ 1.61E-5 5.04E-5 52.4 1.84E-5 6.60E-5 69.3
$ [10, 1] $ 1.47E-5 4.43E-5 31.4 1.65E-5 5.74E-5 33.8
$ [1, 10] $ 9.48E-6 4.71E-5 36.2 7.01E-6 3.44E-5 36.2
$ [1, 100] $ 6.03E-5 1.56E-4 58.3 6.20E-5 1.45E-4 66.4
$ [1, 1000] $ 4.66E-4 9.00E-4 67.8 6.38E-4 1.19E-3 81.6
$ [1, 10000] $ 5.26E-4 8.52E-4 71.2 5.62E-3 1.02E-2 89.5
$ [\beta^+, \beta^-] $ AMIB-LU
Small $ \beta $ side Large $ \beta $ side
$ L_{2} $ $ L_{\infty} $ CPU time(s) $ L_{2} $ $ L_{\infty} $ CPU time(s)
$ [10000, 1] $ 1.63E-5 5.11E-5 10.9 1.86E-5 6.72E-5 10.9
$ [1000, 1] $ 1.63E-5 5.10E-5 10.58 1.86E-5 6.71E-5 10.46
$ [100, 1] $ 1.61E-5 5.04E-5 8.8 1.84E-5 6.61E-5 9.1
$ [10, 1] $ 1.47E-5 4.43E-5 6.8 1.66E-5 5.75E-5 6.9
$ [1, 10] $ 9.48E-6 4.71E-5 7.0 7.03E-6 3.43E-5 6.9
$ [1, 100] $ 6.03E-5 1.56E-4 9.2 6.20E-5 1.45E-4 8.5
$ [1, 1000] $ 4.66E-4 9.00E-4 8.6 6.38E-4 1.19E-3 8.6
$ [1, 10000] $ 5.29E-4 8.56E-4 9.0 5.63E-3 1.02E-2 8.7
Table 6.  Spatial convergence for example 3
$ [n_x, n_y] $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 6.70E-4 2.17E-3 1.67
$ [64, 64] $ 1.52E-4 2.14 5.05E-4 2.10 9.82
$ [128, 128] $ 3.63E-5 2.07 1.39E-4 1.86 53
$ [256, 256] $ 8.62E-6 2.07 3.52E-5 1.98 399
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 6.70E-4 2.17E-3 0.42
$ [64, 64] $ 1.52E-4 2.14 5.05E-4 2.10 1.94
$ [128, 128] $ 3.63E-5 2.07 1.39E-4 1.86 8.54
$ [256, 256] $ 8.62E-6 2.07 3.52E-5 1.98 45.5
$ [n_x, n_y] $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 6.70E-4 2.17E-3 1.67
$ [64, 64] $ 1.52E-4 2.14 5.05E-4 2.10 9.82
$ [128, 128] $ 3.63E-5 2.07 1.39E-4 1.86 53
$ [256, 256] $ 8.62E-6 2.07 3.52E-5 1.98 399
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 6.70E-4 2.17E-3 0.42
$ [64, 64] $ 1.52E-4 2.14 5.05E-4 2.10 1.94
$ [128, 128] $ 3.63E-5 2.07 1.39E-4 1.86 8.54
$ [256, 256] $ 8.62E-6 2.07 3.52E-5 1.98 45.5
Table 7.  Temporal convergence for example 3
$ n_t $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 4.83E-5 2.18E-4 4.1
$ 4 $ 1.39E-5 1.80 5.61E-5 1.96 7.9
$ 8 $ 5.22E-6 1.41 1.92E-5 1.55 15.9
$ 16 $ 2.68E-6 0.96 9.40E-6 1.03 31.5
$ 32 $ 2.15E-6 0.32 8.56E-6 0.14 66.6
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 4.62E-5 2.13E-4 62.7
$ 4 $ 1.23E-5 1.91 5.46E-5 1.96 63.2
$ 8 $ 4.25E-6 1.53 1.60E-5 1.77 63.5
$ 16 $ 2.57E-6 0.73 9.15E-6 0.81 67.6
$ 32 $ 2.23E-6 0.20 8.88E-6 0.04 68.1
$ n_t $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 4.83E-5 2.18E-4 4.1
$ 4 $ 1.39E-5 1.80 5.61E-5 1.96 7.9
$ 8 $ 5.22E-6 1.41 1.92E-5 1.55 15.9
$ 16 $ 2.68E-6 0.96 9.40E-6 1.03 31.5
$ 32 $ 2.15E-6 0.32 8.56E-6 0.14 66.6
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 4.62E-5 2.13E-4 62.7
$ 4 $ 1.23E-5 1.91 5.46E-5 1.96 63.2
$ 8 $ 4.25E-6 1.53 1.60E-5 1.77 63.5
$ 16 $ 2.57E-6 0.73 9.15E-6 0.81 67.6
$ 32 $ 2.23E-6 0.20 8.88E-6 0.04 68.1
Table 8.  Efficiency test for example 3
$ (n_x, n_t) $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio ANMG
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 1.53E-4 5.08E-4 0.25 31.3
$ (129, 40) $ 3.63E-5 2.08 1.39E-4 1.87 2.22 8.9 35.2
$ (257, 80) $ 8.63E-6 2.07 3.52E-5 1.98 26.7 12.0 48.1
$ (513, 160) $ 2.16E-6 2.00 8.88E-6 1.99 342.9 12.8 57.3
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 1.53E-4 5.08E-4 0.15 15.2
$ (129, 40) $ 3.63E-5 2.08 1.39E-4 1.87 1.15 7.7 15.2
$ (257, 80) $ 8.63E-6 2.07 3.52E-5 1.98 10.1 8.8 15.2
$ (513, 160) $ 2.14E-6 2.01 8.79E-6 2.00 100.8 10 15.2
MIB
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 1.47E-4 5.05E-4 0.43
$ (129, 40) $ 3.62E-5 2.08 1.42E-4 1.87 5.94 13.8
$ (257, 80) $ 8.53E-6 2.07 3.56E-5 1.98 113.4 19.1
$ (513, 160) $ NA NA NA NA NA NA
$ (n_x, n_t) $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio ANMG
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 1.53E-4 5.08E-4 0.25 31.3
$ (129, 40) $ 3.63E-5 2.08 1.39E-4 1.87 2.22 8.9 35.2
$ (257, 80) $ 8.63E-6 2.07 3.52E-5 1.98 26.7 12.0 48.1
$ (513, 160) $ 2.16E-6 2.00 8.88E-6 1.99 342.9 12.8 57.3
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 1.53E-4 5.08E-4 0.15 15.2
$ (129, 40) $ 3.63E-5 2.08 1.39E-4 1.87 1.15 7.7 15.2
$ (257, 80) $ 8.63E-6 2.07 3.52E-5 1.98 10.1 8.8 15.2
$ (513, 160) $ 2.14E-6 2.01 8.79E-6 2.00 100.8 10 15.2
MIB
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 1.47E-4 5.05E-4 0.43
$ (129, 40) $ 3.62E-5 2.08 1.42E-4 1.87 5.94 13.8
$ (257, 80) $ 8.53E-6 2.07 3.56E-5 1.98 113.4 19.1
$ (513, 160) $ NA NA NA NA NA NA
Table 9.  Spatial convergence for example 4
$ [n_x, n_y] $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 4.08E-3 6.81E-3 1.67
$ [64, 64] $ 6.26E-4 2.70 1.12E-3 2.60 12.5
$ [128, 128] $ 2.26E-4 1.47 3.92E-4 1.51 89
$ [256, 256] $ 9.01E-5 1.33 1.54E-4 1.35 524
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 4.08E-3 6.81E-3 0.48
$ [64, 64] $ 6.26E-4 2.70 1.12E-3 2.60 2.50
$ [128, 128] $ 2.25E-4 1.47 3.91E-4 1.52 12.4
$ [256, 256] $ 8.83E-5 1.33 1.51E-4 1.37 57
$ [n_x, n_y] $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 4.08E-3 6.81E-3 1.67
$ [64, 64] $ 6.26E-4 2.70 1.12E-3 2.60 12.5
$ [128, 128] $ 2.26E-4 1.47 3.92E-4 1.51 89
$ [256, 256] $ 9.01E-5 1.33 1.54E-4 1.35 524
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 4.08E-3 6.81E-3 0.48
$ [64, 64] $ 6.26E-4 2.70 1.12E-3 2.60 2.50
$ [128, 128] $ 2.25E-4 1.47 3.91E-4 1.52 12.4
$ [256, 256] $ 8.83E-5 1.33 1.51E-4 1.37 57
Table 10.  Temporal convergence for example 4
$ n_t $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 3.28E-3 5.60E-3 5.2
$ 4 $ 8.22E-4 2.00 1.38E-3 2.02 10.1
$ 8 $ 2.23E-4 1.88 3.65E-4 1.92 20.2
$ 16 $ 7.45E-5 1.58 1.16E-4 1.65 40.1
$ 32 $ 3.75E-5 0.99 6.05E-5 0.94 76.6
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 3.28E-3 5.60E-3 75.8
$ 4 $ 8.21E-4 2.00 1.38E-3 2.02 75.3
$ 8 $ 2.23E-4 1.88 3.64E-4 1.92 77.7
$ 16 $ 7.41E-5 1.59 1.15E-4 1.66 80.1
$ 32 $ 3.75E-5 0.98 6.10E-5 0.94 82.7
$ n_t $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 3.28E-3 5.60E-3 5.2
$ 4 $ 8.22E-4 2.00 1.38E-3 2.02 10.1
$ 8 $ 2.23E-4 1.88 3.65E-4 1.92 20.2
$ 16 $ 7.45E-5 1.58 1.16E-4 1.65 40.1
$ 32 $ 3.75E-5 0.99 6.05E-5 0.94 76.6
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ 2 $ 3.28E-3 5.60E-3 75.8
$ 4 $ 8.21E-4 2.00 1.38E-3 2.02 75.3
$ 8 $ 2.23E-4 1.88 3.64E-4 1.92 77.7
$ 16 $ 7.41E-5 1.59 1.15E-4 1.66 80.1
$ 32 $ 3.75E-5 0.98 6.10E-5 0.94 82.7
Table 11.  Efficiency test for example 5
$ (n_x, n_t) $ AMIB-GMRES
$\text{Solution}$
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 3.00E-2 7.43E-2 0.37
$ (129, 40) $ 8.96E-3 1.74 2.24E-2 1.73 3.96 10.7
$ (257, 80) $ 1.57E-3 2.51 4.02E-3 2.48 44.46 11.2
$ (513, 160) $ 3.23E-4 2.28 8.28E-4 2.28 562 12.6
$\text{Gradient}$
$ L_{2} $ $ L_{\infty} $
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 9.38E-2 0.37
$ (129, 40) $ 2.83E-2 1.732 0.12 1.62
$ (257, 80) $ 5.07E-3 2.48 2.15E-2 2.48
$ (513, 160) $ 1.21E-3 2.07 5.29E-3 2.02
AMIB-LU
$\text{Solution}$
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 3.00E-2 7.43E-2 0.15
$ (129, 40) $ 8.95E-3 1.75 2.23E-2 1.74 1.22 8.1
$ (257, 80) $ 1.58E-3 2.50 4.05E-3 2.46 11.7 9.6
$ (513, 160) $ 3.71E-4 2.09 9.54E-4 2.09 127 10.9
$\text{Gradient}$
$ L_{2} $ $ L_{\infty} $
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 9.38E-2 0.37
$ (129, 40) $ 2.82E-2 1.73 0.12 1.62
$ (257, 80) $ 5.11E-3 2.46 2.17E-2 2.47
$ (513, 160) $ 1.21E-3 2.08 5.29E-3 2.04
MIB
$\text{Solution}$
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 3.61E-2 9.21E-2 0.44
$ (129, 40) $ 1.10E-2 1.74 2.87E-2 1.68 7.38 16.8
$ (257, 80) $ 1.41E-3 2.46 3.90E-3 2.88 143 19.4
$ (513, 160) $ 2.17E-4 2.09 9.89E-4 1.98 2372 16.6
$\text{Gradient}$
$ L_{2} $ $ L_{\infty} $
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 0.12 1.93
$ (129, 40) $ 3.53E-2 1.76 0.44 2.13
$ (257, 80) $ 4.51E-3 2.97 0.13 1.76
$ (513, 160) $ 9.73E-4 2.21 2.42E-2 2.43
$ (n_x, n_t) $ AMIB-GMRES
$\text{Solution}$
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 3.00E-2 7.43E-2 0.37
$ (129, 40) $ 8.96E-3 1.74 2.24E-2 1.73 3.96 10.7
$ (257, 80) $ 1.57E-3 2.51 4.02E-3 2.48 44.46 11.2
$ (513, 160) $ 3.23E-4 2.28 8.28E-4 2.28 562 12.6
$\text{Gradient}$
$ L_{2} $ $ L_{\infty} $
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 9.38E-2 0.37
$ (129, 40) $ 2.83E-2 1.732 0.12 1.62
$ (257, 80) $ 5.07E-3 2.48 2.15E-2 2.48
$ (513, 160) $ 1.21E-3 2.07 5.29E-3 2.02
AMIB-LU
$\text{Solution}$
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 3.00E-2 7.43E-2 0.15
$ (129, 40) $ 8.95E-3 1.75 2.23E-2 1.74 1.22 8.1
$ (257, 80) $ 1.58E-3 2.50 4.05E-3 2.46 11.7 9.6
$ (513, 160) $ 3.71E-4 2.09 9.54E-4 2.09 127 10.9
$\text{Gradient}$
$ L_{2} $ $ L_{\infty} $
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 9.38E-2 0.37
$ (129, 40) $ 2.82E-2 1.73 0.12 1.62
$ (257, 80) $ 5.11E-3 2.46 2.17E-2 2.47
$ (513, 160) $ 1.21E-3 2.08 5.29E-3 2.04
MIB
$\text{Solution}$
$ L_{2} $ $ L_{\infty} $ CPU time(s) time ratio
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 3.61E-2 9.21E-2 0.44
$ (129, 40) $ 1.10E-2 1.74 2.87E-2 1.68 7.38 16.8
$ (257, 80) $ 1.41E-3 2.46 3.90E-3 2.88 143 19.4
$ (513, 160) $ 2.17E-4 2.09 9.89E-4 1.98 2372 16.6
$\text{Gradient}$
$ L_{2} $ $ L_{\infty} $
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ (65, 20) $ 0.12 1.93
$ (129, 40) $ 3.53E-2 1.76 0.44 2.13
$ (257, 80) $ 4.51E-3 2.97 0.13 1.76
$ (513, 160) $ 9.73E-4 2.21 2.42E-2 2.43
Table 12.  Spatial convergence for example 6
$ [n_x, n_y] $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 4.09E-5 6.66E-5 0.65
$ [64, 64] $ 2.24E-5 0.87 3.63E-5 0.88 4.02
$ [128, 128] $ 5.04E-6 2.15 8.19E-6 2.15 22.3
$ [256, 256] $ 9.47E-7 2.41 1.56E-6 2.39 151
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 4.09E-5 6.66E-5 0.25
$ [64, 64] $ 2.23E-5 0.88 3.62E-5 0.88 1.36
$ [128, 128] $ 5.06E-6 2.14 8.22E-6 2.24 7.06
$ [256, 256] $ 9.97E-7 2.34 1.63E-6 2.33 34.9
$ [n_x, n_y] $ AMIB-GMRES
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 4.09E-5 6.66E-5 0.65
$ [64, 64] $ 2.24E-5 0.87 3.63E-5 0.88 4.02
$ [128, 128] $ 5.04E-6 2.15 8.19E-6 2.15 22.3
$ [256, 256] $ 9.47E-7 2.41 1.56E-6 2.39 151
AMIB-LU
$ L_{2} $ $ L_{\infty} $ CPU time(s)
$\text{Error}$ $\text{Order}$ $\text{Error}$ $\text{Order}$
$ [32, 32] $ 4.09E-5 6.66E-5 0.25
$ [64, 64] $ 2.23E-5 0.88 3.62E-5 0.88 1.36
$ [128, 128] $ 5.06E-6 2.14 8.22E-6 2.24 7.06
$ [256, 256] $ 9.97E-7 2.34 1.63E-6 2.33 34.9
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