Lehel Banjai, Professor, School of Mathematical & Computer Sciences,Heriot-Watt University

殷涛 副研究员

Challenges and successes of convolution quadrature approach to TDBIE

11月21日（周六）16:00-17:00，zoom会议号: 833 8740 2865，密码: 564389

It is now respectively around four [1, 2] and three decades [3] since the publication of three fundamental papers giving a rigorous mathematical basis for the analysis of two particular classes of numerical methods for time-domain boundary integral equations: space-time Galerkin and convolution quadrature methods (CQ). Subsequently, there has been much further development of the theory applied to different problems building on these foundations. At the same time, there has been much work on fast and highly accurate numerical methods without a complete mathematical analysis, that nevertheless in numerical experiments have been shown to be effective, accurate and fast. This talk will give some new results that within the confines of convolution quadrature methods are good examples of the fully theoretically grounded and a more pragmatic approach that while effective is difficult to fully analyse. We will revisit one of the fundamental papers published a decade ago [4] on FEM/BEM coupling for acoustics. The original approach required a certain stabilisation term, which was shown to be unnecessary in a second order formulation presented in [5]. We will describe a recent work [6] on a first order formulation addressing both the Maxwell and acoustic case and again not requiring the stabilisation term. We present this as an example of the power of the theoretical approach initially proposed in [3] and developed further in [4] and other works. One of the weaknesses of the CQ approach for waves is its dissipative and dispersive character. While this can be dramatically improved by the use of high order Runge-Kutta methods, it is of interest to consider a simple modification of low-order methods that can produce much more accurate results. While this particular modification destroys the ability to fully analyse the method by standard means, the numerical results show both high accuracy and numerical stability [7].

References

[1] A. Bamberger and T. H. Duong, Formulation variationnelle espace-temps pour le calcul par potentiel retard´e de la diffraction d’une onde acoustique. I, Math. Methods Appl. Sci., 8(3):405-435, 1986.

[2] A. Bamberger and T. H. Duong, Formulation variationnelle pour le calcul de la diffraction d’une onde acoustique par une surface rigide, Math. Methods Appl. Sci., 8(4):598-608, 1986.

[3] C. Lubich, On the multistep time discretization of linear initial-boundary value problems and

their boundary integral equations, Numer. Math., 67(3):365-389, 1994.

[4] L. Banjai, C. Lubich, and F.-J. Sayas, Stable numerical coupling of exterior and interior problems for the wave equation, Numer. Math., 129(4):611-646, 2015.

[5] L. Banjai, Implicit/explicit, BEM/FEM coupled scheme for acoustic waves with the wave equation in the second order formulation,CMAM (Special issue dedicated to F-J Sayas), 2023.

[6] L. Banjai, C. Lubich, In preparation, 2025.

[7] E. Aldahham and L. Banjai, A modified convolution quadrature combined with the method of fundamental solutions and Galerkin BEM for acoustic scattering, Communications in Computational Physics, 2022.