Unformatted text preview: a need to use lumping.
Thus, we select the approximation Uj (x t) of u(x t) on the mapping of (xj;1 xj ) to the
canonical element as
Uj ( t) = ckj (t)Pk ( )
k=0 where ckj (t) is an m-vector and Pk ( ) is the Legendre polynomial of degree k in . Recall
(cf. Section 2.5), that the Legendre polynomials satisfy the orthogonality relation
Pi( )Pj ( )d = 2i +j 1
are normalized as Pi (1) = 1 i0 (10.2.2c) and satisfy the symmetry relation Pi( ) = (;1)i Pi(; ) i 0: (10.2.2d) The rst six Legendre polynomials are P0( ) = 1 P1( ) =
P2( ) = 3 2; 1 P3( ) = 5 2 3
P4( ) = 35 ; 30 + 3 P5( ) = 63 ; 70 + 15 :
These polynomials are illustrated in Figure 10.2.1). Additional information appears in
Section 2.5 and Abromowitz and Stegen 1].
Substituting (10.2.2a) into (10.2.1c), testing against Pi( ), and using (10.2.2b-d) yields
hj cij + f (U(x t)) ; (;1)if (U(x t)) = Z 1 dPi( ) f (U ( t))d
2i + 1
i = 1 2 ::: p
where (_) = d( )=dt.
Neighboring elements must communicate information to each other and, in this form
of the discontinuous Galerkin element method, this is done through the boundary ux 10.2. Discontinuous Galerkin Methods 21 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 10.2.1: Legendre polynomials of degrees p = 0 1 : : : 5.
terms. The usual practice is to replace the boundary ux terms f (U(xk t), k = j ; 1 j ,
by a numerical ux function f (U(xk t) F(Uk (xk t)) Uk+1(xk t))
that depends on the approximate solutions Uk and Uk+1 on the two elements sharing the vertex at xk . Cockburn and Shu 12] present several possible numerical ux functions.
Perhaps, the simplest is the average
F(Uk (xk t)) Uk+1(xk t)) = f (Uk (xk t)) +2f (Uk+1(xk t)) :
Based on our work with convection-di usion problems in Section 9.5, we might expect
that some upwind considerations might be worthwhile. This happens to be somewhat
involved for nonlinear vector systems. We'll...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14