Department of Aerospace Engineering

Distinguished Lecture in Aerospace Engineering: "Discontinuious Galerkin for Diffusion"; Bram vanLeer, Arthur B. Modine Professor of Aerospace Engineering, University of Michigan

 

This Distinguished Lecture seminar will be held at 2:10 PM in the Alliant Energy-Lee Liu Auditorium, 1140 Howe Hall.  A reception will be held following the seminar.

DISCONTINUOUS GALERKIN FOR DIFFUSION

 

 

 

 

Bram van Leer

 

 

Arthur B. Modine Professor of Aerospace Engineering

 

 

University of Michigan

 

 

 

 

Discontinuous Galerkin methods are the Finite Element analyst's answer to Finite Voluime methods. Originally inspired by upwind (Godunov-type) methods for the advection equation and hyperbolic systems, the DG community soon turned to the diffusion equation, with much less success. It seems that the DG approach is fundamentally unsuited for second-order operators. Not surprisingly, recent developments such as the Local Discontinuous Galerkin method of Shu and Cockburn require that the diffusion equation be rewritten as a system of first-order equations.

 

 

 

 

While working with first-order systems is computationally advantageous, and a general trend in CFD, it evades the question how to directly discretize a second-order operator.

 

 

 

 

In this lecture I will first show there are essential differences between discretizing the advection and diffusion equations: what works for one does not work for the other, and vice versa. This means that, when formulating a DG method for diffusion, one cannot blindly copy what's done for the advection equation.

 

 

 

 

I will show, however, there is really no conflict between the DG approach and the diffusion equation.  In order to make these work together two insights are needed:

 

 

 

 

(1) the realization that there are multiple representations of the

 

 

    numerical solution which all are equivalent in the weak sense,

 

 

    and that one may have to switch between these for the sake of

 

 

    getting useful schemes;

 

 

 

 

(2) for a second-order PDE integration by parts should be done TWICE

 

 

    in order to obtain the most accurate DG equations - which is not

 

 

    standard DG practice.

 

 

 

 

Next, I will present the Recovery method, developed from the above starting points. Specifically, a smooth locally recovered solution is used that in the weak sense is indistinguishable from the discontinuous discrete solution. The recovery principle creates schemes that are not included in the family of traditional DG diffusion schemes, and are potentially more accurate. A procedure is presented to extend the family so that recovery-based schemes are included.

 

 

 

 

The latest development is the so-called "recovery basis", resulting from applying the recovery principle to the discontinuous basis functions from which the solution is built.  To make the computation of diffusive fluxes possible one simply replaces the original basis by the corresponding unique recovery basis.   This blurs the difference between discontinuous and continuous Galerkin methods, and between DG advection and diffusion treatments.

 

 

 

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