×

An integral equation-based numerical method for the forced heat equation on complex domains. (English) Zbl 1452.65272

Summary: Integral equation-based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M80 Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65D30 Numerical integration
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65R20 Numerical methods for integral equations
35K05 Heat equation

Software:

DLMF; Matlab; rbf_qr
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Kropinski, MC; Quaife, BD, Fast integral equation methods for Rothe’s method applied to the isotropic heat equation, Comput. Math. Appl., 61, 9, 2436-2446 (2011) · Zbl 1221.65279
[2] Chapko, R.; Kress, R., Rothe’s method for the heat equation and boundary integral equations, J. Integral. Equ. Appl., 9, 1, 47-69 (1997) · Zbl 0885.65101
[3] Chapko, R., On the combination of Rothe’s method and boundary integral equations for the nonstationary Stokes equation, J. Integral. Equ. Appl., 13, 2, 99-116 (2001) · Zbl 0995.65096
[4] Kropinski, MC; Quaife, BD, Fast integral equation methods for the modified Helmholtz equation, J. Comput. Phys., 230, 2, 425-434 (2011) · Zbl 1207.65144
[5] Kennedy, CA; Carpenter, MH, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44, 1, 139-181 (2003) · Zbl 1013.65103
[6] Dutt, A.; Greengard, L.; Rokhlin, V., Spectral deferred correction methods for ordinary differential equations, BIT, 40, 2, 241-266 (2000) · Zbl 0959.65084
[7] Jia, J.; Huang, J., Krylov deferred correction accelerated method of lines transpose for parabolic problems, J. Comput. Phys., 227, 3, 1739-1753 (2008) · Zbl 1134.65064
[8] Minion, ML, Semi-implicit spectral deferred correction methods for ordinary differential equations, Commun. Math. Sci., 1, 3, 471-500 (2003) · Zbl 1088.65556
[9] Fryklund, F.; Lehto, E.; Tornberg, A-K, Partition of unity extension of functions on complex domains, J. Comput. Phys., 375, 57-79 (2018) · Zbl 1416.65475
[10] Askham, T.; Cerfon, AJ, An adaptive fast multipole accelerated Poisson solver for complex geometries, J. Comput. Phys., 344, 1-22 (2017) · Zbl 1380.65413
[11] Bruno, OP; Lyon, M., High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements, J. Comput. Phys., 229, 6, 2009-2033 (2010) · Zbl 1185.65184
[12] Stein, DB; Guy, RD; Thomases, B., Immersed boundary smooth extension (IBSE): A high-order method for solving incompressible flows in arbitrary smooth domains, J. Comput. Phys., 335, 155-178 (2017) · Zbl 1375.76038
[13] Shirokoff, D.; Nave, J-C, A sharp-interface active penalty method for the incompressible Navier-Stokes equations, J. Sci. Comput., 62, 1, 53-77 (2015) · Zbl 1309.76144
[14] Hao, S.; Barnett, AH; Martinsson, PG; Young, P., High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane, Adv. Comput. Math., 40, 1, 245-272 (2014) · Zbl 1300.65093
[15] Helsing, J.; Holst, A., Variants of an explicit kernel-split panel-based Nyström discretization scheme for Helmholtz boundary value problems, Adv. Comput. Math., 41, 3, 691-708 (2015) · Zbl 1319.65117
[16] Helsing, J.; Ojala, R., On the evaluation of layer potentials close to their sources, J. Comput. Phys., 227, 5, 2899-2921 (2008) · Zbl 1135.65404
[17] Ojala, R.; Tornberg, A-K, An accurate integral equation method for simulating multi-phase stokes flow, J. Comput. Phys., 298, 145-160 (2015) · Zbl 1349.76635
[18] Klinteberg, L, Fryklund, F, Tornberg, A-K: An adaptive kernel-split quadrature method for parameter-dependent layer potentials. arXiv:1906.07713 (2019)
[19] Helsing, J., Integral equation methods for elliptic problems with boundary conditions of mixed type, J. Comput. Phys., 228, 23, 8892-8907 (2009) · Zbl 1177.65176
[20] af Klinteberg, L.; Askham, T.; Kropinski, MC, A fast integral equation method for the two-dimensional navier-stokes equations, J. Comput. Phys., 409, 109353 (2020) · Zbl 1435.76020
[21] Li, J.; Greengard, L., High order accurate methods for the evaluation of layer heat potentials, SIAM J. Sci. Comput., 31, 5, 3847-3860 (2009) · Zbl 1204.65117
[22] Wang, S.; Jiang, S.; Wang, J., Fast high-order integral equation methods for solving boundary value problems of two dimensional heat equation in complex geometry, J. Sci. Comput., 79, 2, 787-808 (2019) · Zbl 1464.65156
[23] Zhou, H-X; Pang, X., Electrostatic interactions in protein structure, folding, binding, and condensation, Chem. Rev., 118, 4, 1691-1741 (2018)
[24] Juffer, AH; Botta, EFF; van Keulen, BAM; van der Ploeg, A.; Berendsen, HJC, The electric potential of a macromolecule in a solvent: A fundamental approach, J. Comput. Phys., 97, 1, 144-171 (1991) · Zbl 0743.65094
[25] Chen, KH; Chen, JT, Adaptive dual boundary element method for solving oblique incident wave passing a submerged breakwater, Comput. Method. Appl. M., 196, 1, 551-565 (2006) · Zbl 1120.76343
[26] Vorobjev, YN, Modeling of electrostatic effects in macromolecules, 163-202 (2019), Cham: Springer International Publishing, Cham
[27] Liang, J.; Subramaniam, S., Computation of molecular electrostatics with boundary element methods, Biophys. J., 73, 4, 1830-1841 (1997)
[28] Kouibia, A.; Pasadas, M.; Reyah, L.; Akhrif, R., Approximation of surfaces by modified helmholtz splines, J. Comput. Appl. Math., 350, 262-273 (2019) · Zbl 1407.65262
[29] Chen, CS; Jiang, X.; Chen, W.; Yao, G., Fast solution for solving the modified Helmholtz equation with the method of fundamental solutions, Commun. Comput. Phys., 17, 3, 867-886 (2015) · Zbl 1388.65179
[30] Li, X., On solving boundary value problems of modified Helmholtz equations by plane wave functions, J. Comput. Appl. Math., 195, 1, 66-82 (2006) · Zbl 1099.65118
[31] Ascher, U.; Ruuth, S.; Wetton, B., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32, 3, 797-823 (1995) · Zbl 0841.65081
[32] Quaife, B: Fast integral equation methods for the modified helmholtz equation, Ph.D. Thesis, Simon Fraser University (2011) · Zbl 1207.65144
[33] Atkinson, KE, The numerical solution of integral equations of the second kind (1997), Cambridge: Cambridge University Press, Cambridge · Zbl 0899.65077
[34] Shepard, D: A two-dimensional interpolation function for irregularly-spaced data, vol 23 (1968)
[35] Fasshauer, G F: Meshfree Approximation Methods with MATLAB, World Scientific Publishing Co., Inc. River Edge, NJ, USA (2007) · Zbl 1123.65001
[36] Larsson, E.; Fornberg, B., Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions, Comput. Math. Appl., 49, 1, 103-130 (2005) · Zbl 1074.41012
[37] Larsson, E, Shcherbakov, V, Heryudono, A: A least squares radial basis function partition of unity method for solving PDEs. SIAM J. Sci. Comput. (2017) · Zbl 1377.65156
[38] Fornberg, B.; Larsson, E.; Flyer, N., Stable computations with Gaussian radial basis functions, SIAM J. Sci. Comput., 33, 2, 869-892 (2011) · Zbl 1227.65018
[39] Trefethen, L: Spectral methods in MATLAB, Society for Industrial and Applied Mathematics (2000) · Zbl 0953.68643
[40] Carrier, J.; Greengard, L.; Rokhlin, V., A fast adaptive multipole algorithm for particle simulations, SIAM J. Sci. Stat. Comp., 9, 4, 669-686 (1988) · Zbl 0656.65004
[41] Cheng, H.; Huang, J.; Leiterman, TJ, An adaptive fast solver for the modified helmholtz equation in two dimensions, J. Comput. Phys., 211, 2, 616-637 (2006) · Zbl 1117.65161
[42] Greengard, LF; Huang, J., A new version of the fast multipole method for screened Coulomb interactions in three dimensions, J. Comput. Phys., 180, 2, 642-658 (2002) · Zbl 1143.78372
[43] Verchota, G., Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal., 59, 3, 572-611 (1984) · Zbl 0589.31005
[44] Helsing, J: Solving integral equations on piecewise smooth boundaries using the RCIP method: a tutorial, ArXiv e-prints (2012) · Zbl 1328.65271
[45] NIST: Digital Library of Mathematical Functions, Release 1.0.16 of 2017-09-18 http://dlmf.nist.gov/
[46] Khatri, S.; Tornberg, A-K, An embedded boundary method for soluble surfactants with interface tracking for two-phase flows, J. Comput. Physics, 256, 768-790 (2014) · Zbl 1349.76059
[47] lsson, SP; Siegel, M.; Tornberg, A-K, Simulation and validation of surfactant-laden drops in two-dimensional Stokes flow, J. Comput. Phys., 386, 218-247 (2019) · Zbl 1452.76062
[48] Kropinski, MCA; Lushi, E., Efficient numerical methods for multiple surfactant-coated bubbles in a two-dimensional Stokes flow, J. Comput. Phys., 230, 12, 4466-4487 (2011) · Zbl 1416.76216
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.