An efficient and arbitrary-order cell method for linear hyperbolic systems of equations
Event details
| Date | 26.05.2020 |
| Hour | 16:15 › 17:15 |
| Speaker | Bernard Kapidani |
| Location | |
| Category | Conferences - Seminars |
Computational Mathematics Seminar
Abstract :
"I will describe a new method for the numerical solution of initial boundary value (wave propagation) problems, focusing mainly on the Maxwell equations. Starting from a regular triangulation of the spatial domain $\Omega$, the method hinges on the construction of a second mesh: its so-called barycentric dual cell complex. Furthermore, the two unknown fields in the system of first order equations are approximated with different non-conforming functional spaces, whose definition is intimately tied to the relationship between the two dual meshes. Although non-conforming, the presented Discontinuous Galerkin method requires neither the introduction of user-tuned penalty parameters for the inter-element jumps of the fields, nor numerical energy dissipation to achieve stability. I will prove that an exact energy conservation law for the semi-discrete system holds and I will construct bases for the ansatz spaces, with arbitrary local polynomial degree of accuracy, defined through a new geometry for the reference cell element. I will then show how the chosen ansatz spaces yield cheaply invertible block-diagonal mass matrices (and therefore an efficient explicit time stepping scheme) and I will finally demonstrate on several numerical tests that the resulting algorithm is spectrally correct and exhibits the expected order of convergence. The material covered is joint work with Lorenzo Codecasa of the Polytechnic of Milan and Joachim Schöberl of the Vienna University of Technology."
Abstract :
"I will describe a new method for the numerical solution of initial boundary value (wave propagation) problems, focusing mainly on the Maxwell equations. Starting from a regular triangulation of the spatial domain $\Omega$, the method hinges on the construction of a second mesh: its so-called barycentric dual cell complex. Furthermore, the two unknown fields in the system of first order equations are approximated with different non-conforming functional spaces, whose definition is intimately tied to the relationship between the two dual meshes. Although non-conforming, the presented Discontinuous Galerkin method requires neither the introduction of user-tuned penalty parameters for the inter-element jumps of the fields, nor numerical energy dissipation to achieve stability. I will prove that an exact energy conservation law for the semi-discrete system holds and I will construct bases for the ansatz spaces, with arbitrary local polynomial degree of accuracy, defined through a new geometry for the reference cell element. I will then show how the chosen ansatz spaces yield cheaply invertible block-diagonal mass matrices (and therefore an efficient explicit time stepping scheme) and I will finally demonstrate on several numerical tests that the resulting algorithm is spectrally correct and exhibits the expected order of convergence. The material covered is joint work with Lorenzo Codecasa of the Polytechnic of Milan and Joachim Schöberl of the Vienna University of Technology."
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Practical information
- General public
- Free
Organizer
- Rafael Vazquez Hernandez
Contact
- Rafael Vazquez Hernandez