×

High-order numerical methods for 2D parabolic problems in single and composite domains. (English) Zbl 1395.65024

Summary: In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests–with respect to accuracy and convergence–for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations

Software:

IIMPACK; CutFEM
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abarbanel, S; Ditkowski, A, Asymptotically stable fourth-order accurate schemes for the diffusion equation on complex shapes, J. Comput. Phys., 133, 279-288, (1997) · Zbl 0891.65099 · doi:10.1006/jcph.1997.5653
[2] Adams, L; Li, Z, The immersed interface/multigrid methods for interface problems, SIAM J. Sci. Comput., 24, 463-479, (2002) · Zbl 1014.65099 · doi:10.1137/S1064827501389849
[3] Albright, J.: Numerical Methods Based on Difference Potentials for Models with Material Interfaces. Ph.D. thesis, University of Utah (2016) · Zbl 1408.65078
[4] Albright, J; Epshteyn, Y; Medvinsky, M; Xia, Q, High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces, Appl. Numer. Math., 111, 64-91, (2017) · Zbl 1353.65110 · doi:10.1016/j.apnum.2016.08.017
[5] Albright, J; Epshteyn, Y; Steffen, KR, High-order accurate difference potentials methods for parabolic problems, Appl. Numer. Math., 93, 87-106, (2015) · Zbl 1326.65103 · doi:10.1016/j.apnum.2014.08.002
[6] Albright, J; Epshteyn, Y; Xia, Q, High-order accurate methods based on difference potentials for 2D parabolic interface models, Commun. Math. Sci., 15, 985-1019, (2017) · Zbl 1516.65063 · doi:10.4310/CMS.2017.v15.n4.a4
[7] Almquist, M; Karasalo, I; Mattsson, K, Atmospheric sound propagation over large-scale irregular terrain, J. Sci. Comput., 61, 369-397, (2014) · Zbl 1299.76168 · doi:10.1007/s10915-014-9830-4
[8] Appelö, D; Petersson, NA, A stable finite difference method for the elastic wave equation on complex geometries with free surfaces, Commun. Comput. Phys., 5, 84-107, (2009) · Zbl 1364.74016
[9] Bedrossian, J; Brecht, JH; Zhu, S; Sifakis, E; Teran, JM, A second order virtual node method for elliptic problems with interfaces and irregular domains, J. Comput. Phys., 229, 6405-6426, (2010) · Zbl 1197.65168 · doi:10.1016/j.jcp.2010.05.002
[10] Berg, J; Nordström, J, Spectral analysis of the continuous and discretized heat and advection equation on single and multiple domains, Appl. Numer. Math., 62, 1620-1638, (2012) · Zbl 1251.65120 · doi:10.1016/j.apnum.2012.05.002
[11] Bouchon, M; Campillo, M; Gaffet, S, A boundary integral equation-discrete wavenumber representation method to study wave propagation in multilayered media having irregular interfaces, Geophysics, 54, 1134-1140, (1989) · doi:10.1190/1.1442748
[12] Britt, S; Tsynkov, S; Turkel, E, Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials, J. Comput. Phys., (2017) · Zbl 1380.65146 · doi:10.1016/j.jcp.2017.10.049
[13] Burman, E; Claus, S; Hansbo, P; Larson, MG; Massing, A, Cutfem: discretizing geometry and partial differential equations, Int. J. Numer. Methods Eng., 104, 472-501, (2015) · Zbl 1352.65604 · doi:10.1002/nme.4823
[14] Burman, E; Hansbo, P, Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method, Appl. Numer. Math., 62, 328-341, (2012) · Zbl 1316.65099 · doi:10.1016/j.apnum.2011.01.008
[15] Burman, E; Hansbo, P; Larson, MG; Zahedi, S, Cut finite element methods for coupled bulksurface problems, Numer. Math., 133, 203-231, (2016) · Zbl 1341.65044 · doi:10.1007/s00211-015-0744-3
[16] Carpenter, MH; Gottlieb, D; Abarbanel, S, Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, J. Comput. Phys., 111, 220-236, (1994) · Zbl 0832.65098 · doi:10.1006/jcph.1994.1057
[17] Carpenter, MH; Nordström, J; Gottlieb, D, Revisiting and extending interface penalties for multi-domain summation-by-parts operators, J. Sci. Comput., 45, 118-150, (2009) · Zbl 1203.65176 · doi:10.1007/s10915-009-9301-5
[18] Coco, A; Russo, G, Second order multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface, I: one dimensional problems, Numer. Math. Theory Methods Appl., 5, 19-42, (2012) · Zbl 1265.65251 · doi:10.4208/nmtma.2011.m12si02
[19] Crockett, R; Colella, P; Graves, DT, A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions, J. Comput. Phys., 230, 2451-2469, (2011) · Zbl 1220.65121 · doi:10.1016/j.jcp.2010.12.017
[20] Rey Fernández, DC; Boom, PD; Zingg, DW, A generalized framework for nodal first derivative summation-by-parts operators, J. Comput. Phys., 266, 214-239, (2014) · Zbl 1311.65002 · doi:10.1016/j.jcp.2014.01.038
[21] Rey Fernández, DC; Hicken, JE; Zingg, DW, Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations, Comput. Fluids, 95, 171-196, (2014) · Zbl 1390.65064 · doi:10.1016/j.compfluid.2014.02.016
[22] Demirdz̆ić, I; Muzaferija, S, Finite volume method for stress analysis in complex domains, Int. J. Numer. Methods. Eng., 37, 3751-3766, (1994) · Zbl 0814.73075 · doi:10.1002/nme.1620372110
[23] Ditkowski, A; Harness, Y, High-order embedded finite difference schemes for initial boundary value problems on time dependent irregular domains, J. Sci. Comput., 39, 394-440, (2009) · Zbl 1203.65133 · doi:10.1007/s10915-009-9277-1
[24] Duru, K; Virta, K, Stable and high order accurate difference methods for the elastic wave equation in discontinuous media, J. Comput. Phys., 279, 37-62, (2014) · Zbl 1351.74154 · doi:10.1016/j.jcp.2014.08.046
[25] Epshteyn, Y, Upwind-difference potentials method for patlak-Keller-Segel chemotaxis model, J. Sci. Comput., 53, 689-713, (2012) · Zbl 1317.92019 · doi:10.1007/s10915-012-9599-2
[26] Epshteyn, Y, Algorithms composition approach based on difference potentials method for parabolic problems, Commun. Math. Sci., 12, 723-755, (2014) · Zbl 1305.65184 · doi:10.4310/cms.2014.v12.n4.a7
[27] Epshteyn, Y., Medvinsky, M.: On the solution of the elliptic interface problems by difference potentials method. In: Kirby, R.M., Berzins, M., Hesthaven, J.S. (eds.) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014, pp. 197-205. Springer (2015). https://doi.org/10.1007/978-3-319-19800-2_16 · Zbl 1352.65418
[28] Epshteyn, Y; Phippen, S, High-order difference potentials methods for 1D elliptic type models, Appl. Numer. Math., 93, 69-86, (2015) · Zbl 1326.65142 · doi:10.1016/j.apnum.2014.02.005
[29] Epshteyn, Y; Sofronov, I; Tsynkov, S, Professor V. S. ryaben’kii. on the occasion of the 90-th birthday, Appl. Numer. Math., 93, 1-2, (2015) · Zbl 1315.00092 · doi:10.1016/j.apnum.2015.02.001
[30] Fadlun, E; Verzicco, R; Orlandi, P; Mohd-Yusof, J, Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. Comput. Phys., 161, 35-60, (2000) · Zbl 0972.76073 · doi:10.1006/jcph.2000.6484
[31] Fedkiw, RP; Aslam, T; Merriman, B; Osher, S, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152, 457-492, (1999) · Zbl 0957.76052 · doi:10.1006/jcph.1999.6236
[32] Gibou, F; Fedkiw, R, A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. Comput. Phys., 202, 577-601, (2005) · Zbl 1061.65079 · doi:10.1016/j.jcp.2004.07.018
[33] Godunov, SK; Zhukov, VT; Lazarev, MI; Sofronov, IL; Turchaninov, VI; Kholodov, AS; Tsynkov, SV; Chetverushkin, BN; Epshteyn, YY, Viktor solomonovich ryaben’kii and his school (on his 90th birthday), Russ. Math. Surv., 70, 1183, (2015) · Zbl 1342.01035 · doi:10.1070/RM2015v070n06ABEH004981
[34] Gong, J; Xuan, L; Ming, P; Zhang, W, An unstructured finite-volume method for transient heat conduction analysis of multilayer functionally graded materials with mixed grids, Numer. Heat Transfer, Part B, 63, 222-247, (2013) · doi:10.1080/10407790.2013.751251
[35] Guittet, A; Lepilliez, M; Tanguy, S; Gibou, F, Solving elliptic problems with discontinuities on irregular domains—the Voronoi interface method, J. Comput. Phys., 298, 747-765, (2015) · Zbl 1349.65579 · doi:10.1016/j.jcp.2015.06.026
[36] Guittet, A; Poignard, C; Gibou, F, A Voronoi interface approach to cell aggregate electropermeabilization, J. Comput. Phys., 332, 143-159, (2017) · Zbl 1378.92015 · doi:10.1016/j.jcp.2016.11.048
[37] Hansbo, A; Hansbo, P, An unfitted finite element method, based on nitsche’s method, for elliptic interface problems, Comput. Methods Appl. Mech. Eng., 191, 5537-5552, (2002) · Zbl 1035.65125 · doi:10.1016/S0045-7825(02)00524-8
[38] Hansbo, P; Larson, MG; Zahedi, S, A cut finite element method for a Stokes interface problem, Appl. Numer. Math., 85, 90-114, (2014) · Zbl 1299.76136 · doi:10.1016/j.apnum.2014.06.009
[39] Hellrung, JL; Wang, L; Sifakis, E; Teran, JM, A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions, J. Comput. Phys., 231, 2015-2048, (2012) · Zbl 1408.65078 · doi:10.1016/j.jcp.2011.11.023
[40] Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. Springer, Berlin (2008). https://doi.org/10.1007/978-0-387-72067-8 · Zbl 1134.65068 · doi:10.1007/978-0-387-72067-8
[41] Johansen, H; Colella, P, A Cartesian grid embedded boundary method for poisson’s equation on irregular domains, J. Comput. Phys., 147, 60-85, (1998) · Zbl 0923.65079 · doi:10.1006/jcph.1998.5965
[42] Kim, J; Kim, D; Choi, H, An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys., 171, 132-150, (2001) · Zbl 1057.76039 · doi:10.1006/jcph.2001.6778
[43] Knupp, P., Steinberg, S.: Fundamentals of Grid Generation. CRC Press, Boca Raton (1993) · Zbl 0855.65123
[44] Kozdon, JE; Wilcox, LC, Stable coupling of nonconforming, high-order finite difference methods, SIAM J. Sci. Comput., 38, a923-a952, (2016) · Zbl 1380.65160 · doi:10.1137/15M1022823
[45] Kreiss, H.O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical Aspects of Finite Elements in Partial Differential Equations, Symposium Proceedings pp. 195-212 (1974). https://doi.org/10.1016/B978-0-12-208350-1.50012-1
[46] Leveque, RJ; Li, Z, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31, 1019-1044, (1994) · Zbl 0811.65083 · doi:10.1137/0731054
[47] LeVeque, RJ; Li, Z, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18, 709-735, (1997) · Zbl 0879.76061 · doi:10.1137/S1064827595282532
[48] Li, Z; Ito, K, The immersed interface method: numerical solutions of PDEs involving interfaces and irregular domains, Front. Appl. Math. Soc. Ind. Appl. Math., (2006) · Zbl 1122.65096 · doi:10.1137/1.9780898717464
[49] Linnick, MN; Fasel, HF, A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains, J. Comput. Phys., 204, 157-192, (2005) · Zbl 1143.76538 · doi:10.1016/j.jcp.2004.09.017
[50] Liu, T; Khoo, B; Yeo, K, Ghost fluid method for strong shock impacting on material interface, J. Comput. Phys., 190, 651-681, (2003) · Zbl 1076.76592 · doi:10.1016/S0021-9991(03)00301-2
[51] Liu, XD; Sideris, T, Convergence of the ghost fluid method for elliptic equations with interfaces, Math. Comput., 72, 1731-1746, (2003) · Zbl 1027.65140 · doi:10.1090/S0025-5718-03-01525-4
[52] Magura, S; Petropavlovsky, S; Tsynkov, S; Turkel, E, High-order numerical solution of the Helmholtz equation for domains with reentrant corners, Appl. Numer. Math., 118, 87-116, (2017) · Zbl 1367.65155 · doi:10.1016/j.apnum.2017.02.013
[53] Massing, A; Larson, MG; Logg, A; Rognes, ME, A stabilized Nitsche fictitious domain method for the Stokes problem, J. Sci. Comput., 61, 604-628, (2014) · Zbl 1417.76028 · doi:10.1007/s10915-014-9838-9
[54] Mattsson, K, Summation by parts operators for finite difference approximations of second-derivatives with variable coefficient, J. Sci. Comput., 51, 650-682, (2012) · Zbl 1252.65055 · doi:10.1007/s10915-011-9525-z
[55] Mattsson, K; Carpenter, MH, Stable and accurate interpolation operators for high-order multiblock finite difference methods, SIAM J. Sci. Comput., 32, 2298-2320, (2010) · Zbl 1216.65107 · doi:10.1137/090750068
[56] Mayo, A, The fast solution of poisson’s and the biharmonic equations on irregular regions, SIAM J. Sci. Comput., 21, 285-299, (1984) · Zbl 1131.65303 · doi:10.1137/0721021
[57] McCorquodale, P; Colella, P; Johansen, H, A Cartesian grid embedded boundary method for the heat equation on irregular domains, J. Comput. Phys., 173, 620-635, (2001) · Zbl 0991.65099 · doi:10.1006/jcph.2001.6900
[58] Medvinsky, M; Tsynkov, S; Turkel, E, The method of difference potentials for the Helmholtz equation using compact high order schemes, J. Sci. Comput., 53, 150-193, (2012) · Zbl 1254.65118 · doi:10.1007/s10915-012-9602-y
[59] Medvinsky, M; Tsynkov, S; Turkel, E, Solving the Helmholtz equation for general smooth geometry using simple grids, Wave Motion, 62, 75-97, (2016) · doi:10.1016/j.wavemoti.2015.12.004
[60] Nitsche, J, Über ein variationsprinzip zur lösung von Dirichlet-problemen bei verwendung von teilrämen, die keinen randbedingungen unterworfen sind, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, 36, 9-15, (1971) · Zbl 0229.65079 · doi:10.1007/BF02995904
[61] Peskin, CS, The immersed boundary method, Acta Numer., 11, 479-517, (2002) · Zbl 1123.74309 · doi:10.1017/S0962492902000077
[62] Reznik, AA, Approximation of surface potentials of elliptic operators by difference potentials, Dokl. Akad. Nauk SSSR, 263, 1318-1321, (1982)
[63] Reznik, A.A.: Approximation of the Surface Potentials of Elliptic Operators by Difference Potentials and the Solution of Boundary Value Problems. Ph.D. thesis, Moscow Institute for Physics and Technology (1983) · Zbl 1305.65184
[64] Ryaben’kiĭ, VS, Boundary equations with projectors, Uspekhi Mat. Nauk, 40, 121-149, (1985)
[65] Ryaben’kii, V.S.: Method of Difference Potentials and Its Applications. Springer, Berlin (2002). https://doi.org/10.1007/978-3-642-56344-7 · Zbl 0994.65107 · doi:10.1007/978-3-642-56344-7
[66] Ryaben’kiĭ, VS, Difference potentials analogous to Cauchy integrals, Uspekhi Mat. Nauk, 67, 147-172, (2012)
[67] Ryaben’kii, VS; Turchaninov, VI; Epshteyn, YY, Algorithm composition scheme for problems in composite domains based on the difference potential method, Comp. Math. Math. Phys., 46, 1768-1784, (2006) · doi:10.1134/s0965542506100137
[68] Saye, RI, High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles, SIAM J. Sci. Comput., 37, a993-a1019, (2015) · Zbl 1328.65070 · doi:10.1137/140966290
[69] Sethian, JA; Wiegmann, A, Structural boundary design via level set and immersed interface methods, J. Comput. Phys., 163, 489-528, (2000) · Zbl 0994.74082 · doi:10.1006/jcph.2000.6581
[70] Sticko, S.: Towards Higher Order Immersed Finite Elements for the Wave Equation. Licentiate thesis, Uppsala University, Division of Scientific Computing (2016) · Zbl 1265.65251
[71] Sticko, S; Kreiss, G, A stabilized Nitsche cut element method for the wave equation, Comput. Methods Appl. Mech. Eng., 309, 364-387, (2016) · doi:10.1016/j.cma.2016.06.001
[72] Strand, B, Summation by parts for finite difference approximations for d/dx, J. Comput. Phys., 110, 47-67, (1994) · Zbl 0792.65011 · doi:10.1006/jcph.1994.1005
[73] Svärd, M; Nordström, J, Review of summation-by-parts schemes for initial-boundary-value problems, J. Comput. Phys., 268, 17-38, (2014) · Zbl 1349.65336 · doi:10.1016/j.jcp.2014.02.031
[74] Tseng, YH; Ferziger, JH, A ghost-cell immersed boundary method for flow in complex geometry, J. Comput. Phys., 192, 593-623, (2003) · Zbl 1047.76575 · doi:10.1016/j.jcp.2003.07.024
[75] Virta, K; Mattsson, K, Acoustic wave propagation in complicated geometries and heterogeneous media, J. Sci. Comput., 61, 90-118, (2014) · Zbl 1306.65247 · doi:10.1007/s10915-014-9817-1
[76] Wadbro, E; Zahedi, S; Kreiss, G; Berggren, M, A uniformly well-conditioned, unfitted Nitsche method for interface problems, BIT Numer. Math., 53, 791-820, (2013) · Zbl 1279.65134 · doi:10.1007/s10543-012-0417-x
[77] Wang, S; Kreiss, G, Convergence of summation-by-parts finite difference methods for the wave equation, J. Sci. Comput., 71, 219-245, (2017) · Zbl 1398.65222 · doi:10.1007/s10915-016-0297-3
[78] Wang, S; Virta, K; Kreiss, G, High order finite difference methods for the wave equation with non-conforming grid interfaces, J. Sci. Comput., 68, 1002-1028, (2016) · Zbl 1352.65274 · doi:10.1007/s10915-016-0165-1
[79] Wang, Y; Zhou, H; Yuan, S; Ye, Y, A fourth order accuracy summation-by-parts finite difference scheme for acoustic reverse time migration in boundary-conforming grids, J. Appl. Geophys., 136, 498-512, (2017) · doi:10.1016/j.jappgeo.2016.12.002
[80] Woodward, W.H.: On the Application of the Method of Difference Potentials to Linear Elastic Fracture Mechanics. Ph.D. thesis, The University of Manchester (2015) · Zbl 1352.74043
[81] Woodward, WH; Utyuzhnikov, S; Massin, P, On the application of the method of difference potentials to linear elastic fracture mechanics, Int. J. Numer. Methods Eng., 103, 703-736, (2015) · Zbl 1352.74043 · doi:10.1002/nme.4903
[82] Xia, K; Zhan, M; Wei, GW, MIB method for elliptic equations with multi-material interfaces, J. Comput. Phys., 230, 4588-4615, (2011) · Zbl 1220.65150 · doi:10.1016/j.jcp.2011.02.037
[83] Ye, T; Mittal, R; Udaykumar, H; Shyy, W, An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries, J. Comput. Phys., 156, 209-240, (1999) · Zbl 0957.76043 · doi:10.1006/jcph.1999.6356
[84] Yu, S; Wei, G, Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities, J. Comput. Phys., 227, 602-632, (2007) · Zbl 1128.65103 · doi:10.1016/j.jcp.2007.08.003
[85] Yu, S; Zhou, Y; Wei, GW, Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces, J. Comput. Phys., 224, 729-756, (2007) · Zbl 1120.65333 · doi:10.1016/j.jcp.2006.10.030
[86] Zhou, Y; Zhao, S; Feig, M; Wei, GW, High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, J. Comput. Phys., 213, 1-30, (2006) · Zbl 1089.65117 · doi:10.1016/j.jcp.2005.07.022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.