Journal of Scientific Computing

, Volume 81, Issue 3, pp 1945–1962 | Cite as

Explicit Time Stepping of PDEs with Local Refinement in Space-Time

  • Dylan AbrahamsenEmail author
  • Bengt Fornberg


Traditional numerical time stepping allows variable node densities in space, but not also in time. Having the ability to utilize nodes that are placed irregularly in the space-time domain leads to many advantages when solving time dependent problems. In this paper we introduce a new method utilizing the radial basis function generated finite difference approach in order to accomplish this goal. Benefits include improved stability conditions and the option to use small time steps only in select spatial regions.


Radial basis functions PDEs Space-time Meshless MOL 

Mathematics Subject Classification

65M50 65M70 65M06 65M20 



  1. 1.
    Abedi, R., Petracovici, B., Haber, R.: A space-time discontinuous Galerkin method for linearized elastodynamics with element-wise momentum balance. Comput. Methods Appl. Mech. Eng. 195(25–28), 3247–3273 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Almquist, M., Mehlin, M.: Multilevel local time-stepping methods of Runge–Kutta-type for wave equations. SIAM J. Sci. Comput. 39(5), A2020–A2048 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bayona, V.: Comparison of moving least squares and RBF + poly for interpolation and derivative approximation. J. Sci. Comput. 81(1), 486–512 (2019) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bayona, V., Flyer, N., Fornberg, B.: On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries. J. Comput. Phys. 380, 378–399 (2019)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bayona, V., Flyer, N., Fornberg, B., Barnett, G.: On the role of polynomials in RBF-FD approximations: II. Numerical solution of elliptic PDEs. J. Comput. Phys. 332, 257–273 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Candes, E., Romberg, J.: L1-magic: Recovery of sparse signals via convex programming, vol. 4. (2005)
  7. 7.
    Demirel, A., Niegemann, J., Busch, K., Hochbruck, M.: Efficient multiple time-stepping algorithms of higher order. J. Comput. Phys. 285, 133–148 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Descombes, S., Lanteri, S., Moya, L.: Locally implicit time integration strategies in a discontinuous Galerkin method for Maxwell’s equations. J. Comput. Phys. 56(1), 190–218 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Diaz, J., Grote, M.: Multi-level explicit local time-stepping methods for second-order wave equations. Comput. Methods Appl. Mech. Eng. 291, 240–265 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Driscoll, T.A., Heryudono, A.R.: Adaptive residual subsampling methods for radial basis function interpolation and collocation problems. Comput. Math. Appl. 53(6), 927–939 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fasshauer, G.: Meshfree Approximation Methods with MATLAB. Interdisciplinary Mathematical Sciences, vol. 6. World Scientific Publishers, Singapore (2007)CrossRefGoogle Scholar
  12. 12.
    Flyer, N., Barnett, G., Wicker, L.: Enhancing finite differences with radial basis functions: experiments on the Navier–Stokes equations. J. Comput. Phys. 316, 39–62 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Flyer, N., Fornberg, B., Barnett, G., Bayona, V.: On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy. J. Comput. Phys. 321, 21–38 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Flyer, N., Lehto, E., Blaise, S., Wright, G., St-Cyr, A.: A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere. J. Comput. Phys. 231, 4078–4095 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fornberg, B., Flyer, N.: Fast generation of 2-D node distributions for mesh-free PDE discretizations. Comput. Math. Appl. 69, 531–544 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fornberg, B., Flyer, N.: A Primer on Radial Basis Functions with Applications to the Geosciences. SIAM, Philadelphia (2015)CrossRefGoogle Scholar
  17. 17.
    Fornberg, B., Flyer, N.: Solving PDEs with radial basis functions. Acta Numerica 24, 215–258 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fornberg, B., Lehto, E.: Stabilization of RBF-generated finite difference methods for convective PDEs. J. Comput. Phys. 230, 2270–2285 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gopalakrishnan, J., Schöberl, J., Wintersteiger, C.: Mapped tent pitching schemes for hyperbolic systems. SIAM J. Sci. Comput. 39(6), B1043–B1063 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hamaidi, M., Naji, A., Charafi, A.: Space-time localized radial basis function collocation method for solving parabolic and hyperbolic equations. Eng. Anal. Bound. Elem. 67, 152–163 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Haq, S., Siraj-Ul-Islam, Uddin, M.: A mesh-free method for the numerical solution of the KdV-Burgers equation. Appl. Mathe. Model. 33, 3442–3449 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, Z., Mao, X.Z.: Global multiquadric collocation method for groundwater contaminant source identification. Environ. Model. Softw. 26(12), 1611–1621 (2011)CrossRefGoogle Scholar
  23. 23.
    Li, Z., Mao, X.Z., Li, T.S., Zhang, S.: Estimation of river pollution source using the space-time radial basis collocation method. Adv. Water Resour. 88, 68–79 (2016)CrossRefGoogle Scholar
  24. 24.
    Netuzhylov, H., Zilian, A.: Space-time meshfree collocation method: methodology and application to initial-boundary value problems. Int. J. Numer. Meth. Eng. 80(3), 355–380 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Shan, Y., Shu, C., Lu, Z.: Application of local MQ-DQ method to solve 3D incompressible viscous flows with curved boundary. Comput. Model. Eng. Sci. 25, 99–113 (2008)Google Scholar
  26. 26.
    Uddin, M., Ali, H.: The space-time kernel-based numerical method for Burgers’ equations. Mathematics 6(10), 212 (2018)CrossRefGoogle Scholar
  27. 27.
    Wendland, H.: Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005)Google Scholar
  28. 28.
    Wright, G., Fornberg, B.: Scattered node compact finite difference-type formulas generated from radial basis functions. J. Comput. Phys. 212, 99–123 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of ColoradoBoulderUSA

Personalised recommendations