×

A new mixed potential representation for unsteady, incompressible flow. (English) Zbl 1429.35055

Summary: We present a new integral representation for the unsteady, incompressible Stokes or Navier-Stokes equations, based on a linear combination of heat and harmonic potentials. For velocity boundary conditions, this leads to a coupled system of integral equations: one for the normal component of velocity and one for the tangential components. Each individual equation is well-conditioned, and we show that using them in predictor-corrector fashion, combined with spectral deferred correction, leads to high-order accuracy solvers. The fundamental unknowns in the mixed potential representation are densities supported on the boundary of the domain. We refer to one as the vortex source, the other as the pressure source, and to the coupled system as the combined source integral equation.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J08 Green’s functions for elliptic equations
35K05 Heat equation
35K08 Heat kernel
35Q30 Navier-Stokes equations

Software:

DLMF; pvfmm
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] B. K. Alpert, Hybrid Gauss-trapezoidal quadrature rules, SIAM J. Sci. Comput., 20 (1999), pp. 1551-1584, https://doi.org/10.1137/S1064827597325141. · Zbl 0933.41019
[2] C. R. Anderson, Vorticity boundary conditions and boundary vorticity generation for two-dimensional incompressible flows, J. Comput. Phys., 80 (1989), pp. 72-97. · Zbl 0656.76034
[3] R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1962. · Zbl 0123.41502
[4] M. Ben-Artzi, J.-P. Croisille, D. Fishelov, and S. Trachtenberg, A pure-compact scheme for the streamfunction formulation of the Navier-Stokes equations, J. Comput. Phys., 205 (2005), pp. 640-664. · Zbl 1087.76025
[5] M. Ben-Artzi, D. Fishelov, and S. Trachtenberg, Vorticity dynamics and numerical resolution of Navier-Stokes equations, Math. Model. Numer. Anal., 35 (2001), pp. 313-330. · Zbl 0987.35122
[6] K. Böhmer and e. H. J. Stetter, Defect Correction Methods, Theory and Applications, Springer-Verlag, New York, 1984. · Zbl 0545.00019
[7] A. Bourlioux, A. T. Layton, and M. L. Minion, High-order multi-implicit spectral deferred correction methods for problems of reactive flow, J. Comput. Phys., 189 (2003), pp. 651-675. · Zbl 1061.76053
[8] J. Bremer, On the Nyström discretization of integral equations on planar curves with corners, Appl. Comput. Harmon. Anal., 32 (2012), pp. 45-64. · Zbl 1269.65131
[9] J. Bremer and Z. Gimbutas, A Nyström method for weakly singular integral operators on surfaces, J. Comput. Phys., 231 (2012), pp. 4885-4903. · Zbl 1245.65177
[10] J. Bremer, V. Rokhlin, and I. Sammis, Universal quadratures for boundary integral equations on two-dimensional domains with corners, J. Comput. Phys., 229 (2010), pp. 8259-8280. · Zbl 1201.65213
[11] D. L. Brown, R. Cortez, and M. L. Minion, Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168 (2001), pp. 464-499. · Zbl 1153.76339
[12] T. F. Buttke, Velocity methods: Lagrangian numerical methods which preserve the Hamiltonian structure of incompressible fluid flow, in Vortex Flows and Related Numerical Methods, J. T. Beale, G.-H. Cottet, and S. Huberson, eds., Springer, 1993, pp. 39-57. · Zbl 0860.76064
[13] H. Cheng, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm in three dimensions, J. Comput. Phys., 155 (1999), pp. 468-498. · Zbl 0937.65126
[14] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), pp. 745-762. · Zbl 0198.50103
[15] A. Clebsch, Ueber die integration der hydrodynamischen gleichungen, J. Reine Angew. Math., 56 (1859), pp. 1-10. · ERAM 056.1468cj
[16] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, New York, 2012.
[17] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, SIAM, Philadelphia, 2013. · Zbl 1291.35003
[18] P. Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes equations, Comm. Math. Phys., 216 (2001), pp. 663-686. · Zbl 0988.76020
[19] R. Cortez, On the accuracy of impulse methods for fluid flow, SIAM J. Sci. Comput., 19 (1998), pp. 1290-1302, https://doi.org/10.1137/S1064827595293570. · Zbl 0956.76067
[20] E. J. Dean, R. Glowinski, and O. Pironneau, Iterative solution of the stream function-vorticity formulation of the Stokes problem. Applications to the numerical simulation of incompressible viscous flow, Comput. Method Appl. Mech. Engrg., 87 (1991), pp. 117-155. · Zbl 0760.76044
[21] A. Dutt, L. Greengard, and V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT, 40 (2000), pp. 241-266. · Zbl 0959.65084
[22] W. E and J.-G. Liu, Vorticity boundary condition and related issues for finite difference scheme, J. Comput. Phys., 124 (1996), pp. 368-382. · Zbl 0847.76050
[23] W. E and J.-G. Liu, Finite difference methods for \(3\)-D viscous incompressible flows in the vorticity-vector potential formulation on nonstaggered grids, J. Comput. Phys., 138 (1997), pp. 57-82. · Zbl 0901.76046
[24] W. E and J.-G. Liu, Gauge method for viscous incompressible flows, Commun. Math. Sci., 1 (2003), pp. 317-332. · Zbl 1160.76329
[25] F. Ethridge and L. Greengard, A new fast-multipole accelerated Poisson solver in two dimensions, SIAM J. Sci. Comput., 23 (2001), pp. 741-760, https://doi.org/10.1137/S1064827500369967. · Zbl 1002.65131
[26] E. Fabes, M. Jodeit, and N. Riviére, Potential theoretic techniques for boundary value problems on \({C}^1\) domains, Acta Math., 141 (1978), pp. 165-186. · Zbl 0402.31009
[27] W. Gautschi, Gauss-Radau formulae for Jacobi and Laguerre weight functions, Math. Comput. Simul., 54 (2000), pp. 403-412. · Zbl 0981.41017
[28] V. Girault and P. A. Raviart, Finite element methods for Navier-Stokes equations, Springer Ser. Comput. Math. 5, Springer-Verlag, Berlin, 1986. · Zbl 0585.65077
[29] L. Greengard and M. C. Kropinski, An integral equation approach to the incompressible Navier-Stokes equations in two dimensions, SIAM J. Sci. Comput., 20 (1998), pp. 318-336, https://doi.org/10.1137/S1064827597317648. · Zbl 0917.35094
[30] L. Greengard and J.-Y. Lee, A direct adaptive Poisson solver of arbitrary order accuracy, J. Comput. Phys., 125 (1996), pp. 415-424. · Zbl 0851.65090
[31] L. Greengard and P. Lin, Spectral approximation of the free-space heat kernel, Appl. Comput. Harmon. Anal., 9 (2000), pp. 83-97. · Zbl 0959.65111
[32] L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), pp. 325-348. · Zbl 0629.65005
[33] L. Greengard and J. Strain, A fast algorithm for the evaluation of heat potentials, Comm. Pure Appl. Math., 43 (1990), pp. 949-963. · Zbl 0719.65074
[34] R. B. Guenther and J. W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Prentice-Hall, 1988. · Zbl 0879.35001
[35] T. Hagstrom and R. Zhou, On the spectral deferred correction of splitting methods for initial value problems, Comm. Appl. Math. Comput. Sci., 1 (2006), pp. 169-205. · Zbl 1105.65076
[36] S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young, High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane, Adv. Comput. Math., 40 (2014), pp. 245-272. · Zbl 1300.65093
[37] J. Helsing, Solving integral equations on piecewise smooth boundaries using the RCIP method: A tutorial, Abstr. Appl. Anal., 2013 (2013), art. 938167. · Zbl 1328.65271
[38] J. Helsing and R. Ojala, Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning, J. Comput. Phys., 227 (2008), pp. 8820-8840. · Zbl 1152.65114
[39] W. D. Henshaw, A fourth-order accurate method for the incompressible Navier-Stokes equations on overlapping grids, J. Comput. Phys., 113 (1994), pp. 13-25. · Zbl 0808.76059
[40] T. Y. Hou and B. R. Wetton, Stable fourth-order stream-function methods for incompressible flows with boundaries, J. Comput. Math., 27 (2009), pp. 441-458. · Zbl 1212.65318
[41] S. Jiang, M. C. A. Kropinski, and B. Quaife, Second kind integral equation formulation for the modified biharmonic equation and its applications, J. Comput. Phys., 249 (2013), pp. 113-126. · Zbl 1427.76206
[42] S. Jiang, S. Veerapaneni, and L. Greengard, Integral equation methods for unsteady Stokes flow in two dimensions, SIAM J. Sci. Comput., 34 (2012), pp. A2197-A2219, https://doi.org/10.1137/110860537. · Zbl 1254.35179
[43] P. Kolm, S. Jiang, and V. Rokhlin, Quadruple and octuple layer potentials in two dimensions I: Analytical apparatus, Appl. Comput. Harmon. Anal., 14 (2003), pp. 47-74. · Zbl 1139.35397
[44] R. Kress, Linear Integral Equations, 3rd ed., Appl. Math. Sci. 82, Springer-Verlag, Berlin, 2014. · Zbl 1328.45001
[45] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York, 1969. · Zbl 0184.52603
[46] H. Lamb, Hydrodynamics, Cambridge University Press, 1974.
[47] H. M. Langston, L. Greengard, and D. Zorin, A free-space adaptive FMM-based PDE solver in three dimensions, Comm. Appl. Math. Comput. Sci., 6 (2011), pp. 79-122. · Zbl 1230.65131
[48] J.-R. Li and L. Greengard, High order accurate methods for the evaluation of layer heat potentials, SIAM J. Sci. Comput., 31 (2009), pp. 3847-3860, https://doi.org/10.1137/080732389. · Zbl 1204.65117
[49] J.-G. Liu, J. Liu, and R. L. Pego, Stable and accurate pressure approximation for unsteady incompressible viscous flow, J. Comput. Phys., 229 (2010), pp. 3428-3453. · Zbl 1307.76029
[50] C. Lubich and R. Schneider, Time discretization of parabolic boundary integral equations, Numer. Math., 63 (1992), pp. 455-481. · Zbl 0795.65061
[51] D. Malhotra and G. Biros, PVFMM: A parallel kernel independent FMM for particle and volume potentials, Commun. Comput. Phys., 18 (2015), pp. 808-830. · Zbl 1388.65169
[52] S. G. Mikhlin and S. Prossdorf, Singular integral operators, Springer-Verlag, Berlin, 1986. · Zbl 0612.47024
[53] M. L. Minion, Semi-implicit projection methods for incompressible flow based on spectral deferred corrections, Appl. Numer. Math., 48 (2004), pp. 369-387. · Zbl 1035.76040
[54] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, Cambridge University Press, 2010. · Zbl 1198.00002
[55] R. L. Pantom, Incompressible Flow, John Wiley & Sons, New York, 1996.
[56] W. Pogorzelski, Integral Equations and Their Applications, Pergamon Press, 1966. · Zbl 0137.30502
[57] L. Quartapelle, Numerical Solution of the Incompressible Navier-Stokes Equations, Birkhäuser Verlag, Basel, 1993. · Zbl 0784.76020
[58] K. B. Ranger, Parametrization of general solutions for the Navier-Stokes equations, Quart. Appl. Math., 2 (1994), pp. 335-341. · Zbl 0807.76018
[59] R. Saye, Interfacial gauge methods for incompressible fluid dynamics, Sci. Adv., 2 (2016), art. e150869.
[60] M. Scholle, P. H. Gaskell, and F. Marner, Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications, J. Math. Phys., 59 (2018), p. 043101. · Zbl 1391.76094
[61] M. Scholle, A. Haas, and P. H. Gaskell, A first integral of Navier-Stokes equations and its applications, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), pp. 127-143. · Zbl 1219.76012
[62] K. Serkh and V. Rokhlin, On the solution of elliptic partial differential equations on regions with corners, J. Comput. Phys., 305 (2016), pp. 150-171. · Zbl 1349.35049
[63] K. Serkh and V. Rokhlin, On the solution of the helmholtz equation on regions with corners, Proc. Natl. Acad. Sci. USA, 113 (2016), pp. 9171-9176. · Zbl 1355.35052
[64] M. Siegel and A.-K. Tornberg, A local target specific quadrature by expansion method for evaluation of layer potentials in 3D, J. Comput. Phys., 364 (2018), pp. 365-392. · Zbl 1398.65356
[65] J. Tausch, A fast method for solving the heat equation by layer potentials, J. Comput. Phys., 224 (2007), pp. 956-969. · Zbl 1120.65329
[66] R. Temam, Sur l’approximation de la solution des equations de Navier-Stokes par la methode des fractionnarires II, Arch. Rational Mech. Anal., 33 (1969), pp. 377-385. · Zbl 0207.16904
[67] G. Verchota, Layer potentials and boundary value problems for Laplace’s equation in Lipschitz domains, J. Funct. Anal., 59 (1984), pp. 572-611. · Zbl 0589.31005
[68] C. Wang and J.-G. Liu, Convergence of gauge method for incompressible flow, Math. Comp., 69 (2000), pp. 1385-1407. · Zbl 0968.76065
[69] J. Wang and L. Greengard, An adaptive fast Gauss transform in two dimensions, SIAM J. Sci. Comput., 40 (2018), pp. A1274-A1300, https://doi.org/10.1137/17M1159865. · Zbl 1398.35099
[70] J. Wang, L. Greengard, S. Jiang, and S. K. Veerapaneni, Fast integral equation methods for linear and semilinear heat equations in moving domains, in preparation (2019).
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.