Advertisement

A Fast Direct Solver for the Advection-Diffusion Equation Using Low-Rank Approximation of the Green’s Function

  • Jonathan R. BullEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

We describe a new direct solution method for the advection-diffusion equation at high Reynolds number on simple bounded two-dimensional domains. The key step is to treat advection explicitly, leading to a new class of time integration schemes based on Green’s functions. As a proof of concept a first-order Euler scheme is presented. We compare the accuracy and computational cost of the new scheme to existing solution techniques. Low-rank approximation of the Green’s function is found to reduce cost without loss of accuracy. Stabilisation via numerical dissipation is required for high Reynolds number problems on coarse grids. Linear scaling of computational cost is achieved in 1D and 2D. This work is a building block for constructing fast direct solvers and preconditioners for the Navier-Stokes equations.

References

  1. 1.
    L. McInnes, B. Smith, H. Zhang, R. Mills, Hierarchical Krylov and nested Krylov methods for extreme-scale computing. Parallel Comput. 40(1):17–31 (2014)CrossRefMathSciNetGoogle Scholar
  2. 2.
    L. Greengard, V. Rokhlin, A fast algorithm for particle simulations. J. Comput. Phys. 73, 325–348 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    W. Hackbusch, B. Khoromskij, S. Sauter, On H2-matrices, in Lectures on applied mathematics ed. by H. Bungartz, R. Hoppe, C. Zenger (Springer, Berlin/Heidelberg, 2000)Google Scholar
  4. 4.
    M. Bebendorf, Hierarchical LU decomposition-based preconditioners for BEM. Computing 74(3), 225–247 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    B. Engquist, L. Ying, Sweeping preconditioner for the Helmholtz equation: hierarchical matrix representation. Multiscale Model. Simul. 9(2), 686–710 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    R. Yokota, J. Pestana, H. Ibeid, D. Keyes, Fast multipole preconditioners for sparse matrices arising from elliptic equations (2013). ArXiv:1308.3339Google Scholar
  7. 7.
    S. Ambikasaran, E. Darve, The inverse fast multipole method (2014). ArXiv:1407.1572Google Scholar
  8. 8.
    R. Yokota, H. Ibeid, D. Keyes, Fast multipole method as a matrix-free hierarchical low-rank approximation (2016). ArXiv:1602.02244 [cs.NA]Google Scholar
  9. 9.
    A. Gholami, D. Malhotra, H. Sundar, G. Biros, FFT, FMM or multigrid? a comparative study of state-of-the-art Poisson solvers in the unit cube (2014). ArXiv:1408.6497Google Scholar
  10. 10.
    J. Bull, A direct solver for the advection-diffusion equation using Green’s functions and low-rank approximation, in Proceedings of ECCOMAS ’16 (2016)Google Scholar
  11. 11.
    U. Ascher, S. Ruuth, B. Wetton, Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    R. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations (SIAM, Philadelphia, 2007)CrossRefzbMATHGoogle Scholar
  13. 13.
    D. Gottlieb, J. Hesthaven, Spectral methods for hyperbolic problems. J. Comput. Appl. Math. 128, 83–131 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    A. Al-Mohy, N. Higham, A new scaling and squaring algorithm for the matrix exponential. SIAM J. Matrix Anal. Appl. 31(3), 970–989 (2009)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Division of Scientific ComputingUppsala University, PolacksbackenUppsalaSweden

Personalised recommendations