Alex Kyriakis - Analysis of Schwarz algorithms for the Modified Euler-Tricomi equation

In mathematics, the Euler-Tricomi equation is a second order linear partial differential equation which was introduced by the Italian mathematician Francesco Giacomo Tricomi and it is very useful in the study of the transonic flows. More precisely, it is a very useful equation for modeling the particular problem with practical applications. The nature of this equation can be complicated ([7]. [8]),  but we focus on a simplified version for our analysis. The main goal is to implement and analyse the parallel Schwarz algorithms and obtain weak scalability results for this type of problem [4]. Usually in complicated problems, the one level methods are not sufficient and a coarse level correction is required, and this has appeared in various works in the literature. For instance, in elliptic problems such as the Laplacian equation, one level methods are inadequate and addition of coarse space is needed. However under certain circumstances the one level methods can be enough. In the case of time harmonic problems and elliptic problems ([1], [2]. [3], [5], [6]) there is no need of coarse space when the subdomains have a certain geometry such as rectangles of fixed size. For the case of time harmonic problems the configuration corresponds to a waveguide problem. In the same spirit we provide results that one level methods are scalable when implemented to solve the modified Euler-Tricomi equation. The analysis is done in the two dimensional case where the domain is a large rectangle decomposed into smaller subdomains of equal size. 

References:

[1]“Scalable domain decomposition methods for time-harmonic wave propagation problems”, PhD thesis, A. Kyriakis 

[2]“Analysis of parallel Schwarz algorithms for time-harmonic probems using Block Toeplitz matrices”, N. Bootland, V.Dolean, A. Kyriakis, J. Pestanna 

[3]“Optimizing transmission conditions for multiple subdomains in the Magnetotelluric Approximation of Maxwell’s equations”, V.Dolean, M.Gander, A. Kyriakis 

[4]”Analysis of Schwarz algorithms for the modified Euler-Tricomi equation”, in preparation, A. Kyriakis 

[5] “Closed form optimized transmission conditions for complex diffusion with many subdomains”, V. Dolean, M. Gander, A. Kyriakis 

[6]”On the scalability of classical one-level domain decomposition methods, F. Chaouqui, G. Ciaramella, M. Gander, T. Vanzan 

[7] “A. D. Polyanin, Handbook of Linear Partial Differential Equations for  Engineers and Scientists, Chapman & Hall / CRC Press, 2002.” 

[8]”Tricomi and Generalized Tricomi Equations- EqWorld”