Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Stability of Time-Marching MPS–MFS for Wave Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The two-stage meshless method utilizing the method of particular and fundamental solutions two-stage MPS–MFS was originally proposed to solve steady-state boundary value problems. Despite its great success in handling complicated irregular domains, when applied as a spatial discretization technique to solve time-dependent problems, especially for hyperbolic PDEs, the two-stage MPS–MFS seems to lead to numerical stability issues for conventional time-marching schemes. In addition to the unstable phenomena observed in some existing works, a concrete example on wave equation in this paper also shows the instability of this methodology, even though the utilized time-marching schemes are unconditionally stable. To investigate and explain this instability, we intend to conduct a thorough analysis of the time-marching MPS–MFS method for wave equations. To our best knowledge, this is the first time that the stability issue of the MPS–MFS methodology for hyperbolic PDEs is studied. To compensate for the instability caused by the two-stage MPS–MFS, we also propose a novel time-marching meshless method by combining the method of polynomial particular solutions (MPPS) and the Rothe’s method. This innovative method has a distinct solution mechanism and outperforms the time-marching MPS–MFS method in terms of stability and accuracy. Numerical results are provided to support our findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Data Availability

Not applicable.

References

  1. Alves, C.J., Valtchev, S.S.: A Kansa type method using fundamental solutions applied to elliptic PDEs. In: Advances in Meshfree Techniques, pp. 241–256. Springer (2007)

  2. Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37(2), 229–256 (1994)

    Article  MathSciNet  Google Scholar 

  3. Bogomolny, A.: Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22(4), 644–669 (1985)

    Article  MathSciNet  Google Scholar 

  4. Chen, C.S., Brebbia, C.A., Power, H.: Dual reciprocity method using compactly supported radial basis functions. Commun. Numer. Methods Eng. 15, 137–150 (1999)

    Article  MathSciNet  Google Scholar 

  5. Chen, C.S., Fan, C.M., Wen, P.: The method of approximate particular solutions for solving elliptic problems with variable coefficients. Int. J. Comput. Methods 8(03), 545–559 (2011)

    Article  MathSciNet  Google Scholar 

  6. Chen, C.S., Ganesh, M., Golberg, M.A., Cheng, A.D.: Multilevel compact radial functions based computational schemes for some elliptic problems. Comput. Math. Appl. 43(3–5), 359–378 (2002)

    Article  MathSciNet  Google Scholar 

  7. Cheng, A.H.D.: Multiquadric and its shape parameter: a numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation. Eng. Anal. Bound. Elem. 36(2), 220–239 (2012)

    Article  MathSciNet  Google Scholar 

  8. Cheng, A.H.D., Hong, Y.: An overview of the method of fundamental solutions: solvability, uniqueness, convergence, and stability. Eng. Anal. Bound. Elem. 120, 118–152 (2020)

    Article  MathSciNet  Google Scholar 

  9. Chicone, C.: Stability Theory of Ordinary Differential Equations, pp. 8630–8649. Springer, New York (2009)

    Google Scholar 

  10. Dahlquist, G.: On accuracy and unconditional stability of linear multistep methods for second order differential equations. BIT Numer. Math. 18, 133–136 (1978)

    Article  MathSciNet  Google Scholar 

  11. Dangal, T., Chen, C.S., Lin, J.: Polynomial particular solutions for solving elliptic partial differential equations. Comput. Math. Appl. 73(1), 60–70 (2017)

    Article  MathSciNet  Google Scholar 

  12. Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Constructive Theory of Functions of Several Variables: Proceedings of a Conference Held at Oberwolfach April 25–May 1, 1976, pp. 85–100. Springer (1977)

  13. Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9, 69–95 (1998)

    Article  MathSciNet  Google Scholar 

  14. Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33(2), 869–892 (2011)

    Article  MathSciNet  Google Scholar 

  15. Fornberg, B., Larsson, E., Wright, G.: A new class of oscillatory radial basis functions. Comput. Math. Appl. 51(8), 1209–1222 (2006)

    Article  MathSciNet  Google Scholar 

  16. Golberg, M.A.: The method of fundamental solutions for Poisson’s equation. Eng. Anal. Bound. Elem. 16(3), 205–213 (1995)

    Article  Google Scholar 

  17. Golberg, M.A., Chen, C.S.: The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Golberg, M.A. (ed.) Boundary Integral Methods: Numerical and Mathematical Aspects, pp. 103–176. WIT Press (1999)

    Google Scholar 

  18. Gu, M., Young, D., Fan, C.: The method of fundamental solutions for one-dimensional wave equations. CMC: Comput. Mater. Contin. 11(3), 185–208 (2009)

    Google Scholar 

  19. Gu, M.H., Fan, C.M., Young, D.L.: The method of fundemental solutions for the multi-dimensional wave equations. J. Mar. Sci. Technol. 19(6), 586–595 (2011)

    Article  Google Scholar 

  20. Kansa, E.J.: Multiquadrics: a scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Comput. Math. Appl. 19(8–9), 127–145 (1990)

    Article  MathSciNet  Google Scholar 

  21. Kansa, E.J.: Multiquadrics: a scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19(8–9), 147–161 (1990)

    Article  MathSciNet  Google Scholar 

  22. Kuo, L., Gu, M., Young, D., Lin, C.: Domain type kernel-based meshless methods for solving wave equations. CMC: Comput. Mater. Contin. 33(3), 213–228 (2013)

    Google Scholar 

  23. Kupradze, V.D., Aleksidze, M.A.: The method of functional equations for the approximate solution of certain boundary value problems. U.S.S.R. Comput. Math. Math. Phys. 4(4), 82–126 (1964)

    Article  MathSciNet  Google Scholar 

  24. Li, B., Liu, J., Xiao, M.: A fast and stable preconditioned iterative method for optimal control problem of wave equations. SIAM J. Sci. Comput. 37(6), A2508–A2534 (2015)

    Article  MathSciNet  Google Scholar 

  25. Li, J., Hon, Y., Chen, C.: Numerical comparisons of two meshless methods using radial basis functions. Eng. Anal. Bound. Elem. 26(3), 205–225 (2002)

    Article  Google Scholar 

  26. Li, M., Chen, C.S., Karageorghis, A.: The MFS for the solution of harmonic boundary value problems with non-harmonic boundary conditions. Comput. Math. Appl. 66(11), 2400–2424 (2013)

    Article  MathSciNet  Google Scholar 

  27. Liu, W.K., Jun, S., Li, S., Adee, J., Belytschko, T.: Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Meth. Eng. 38(10), 1655–1679 (1995)

    Article  MathSciNet  Google Scholar 

  28. Mathon, R., Johnston, R.L.: The approximate solution of elliptic boundary-value problems by fundamental solutions. SIAM J. Numer. Anal. 14(4), 638–650 (1977)

    Article  MathSciNet  Google Scholar 

  29. Micchelli, C.A.: Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2(1), 11–22 (1986)

    Article  MathSciNet  Google Scholar 

  30. Rothe, E.: Zweidimensionale parabolische randwertaufgaben als grenzfall eindimensionaler randwertaufgaben. Math. Ann. 102, 650–670 (1930)

    Article  MathSciNet  Google Scholar 

  31. Young, D., Gu, M., Fan, C.: The time-marching method of fundamental solutions for wave equations. Eng. Anal. Bound. Elem. 33(12), 1411–1425 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported by NSFC (12171080, 12292982), the Fundamental Research Funds for the Central Universities (2412022ZD035) and the Natural Science Foundation of Jilin Province (YDZJ202201ZYTS593).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, People’s Republic of China

    Lu Xu & Shu-Lin Wu

Authors
  1. Lu Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shu-Lin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shu-Lin Wu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Code Availability

Codes can be provided per requirement.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, L., Wu, SL. Stability of Time-Marching MPS–MFS for Wave Equations. J Sci Comput 101, 62 (2024). https://doi.org/10.1007/s10915-024-02704-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-024-02704-0

Keywords

Mathematics Subject Classification

Search

Navigation