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
p
X
Uj ( t) = ckj (t)Pk ( )
(10.2.2a)
k=0 where ckj (t) is an mvector and Pk ( ) is the Legendre polynomial of degree k in . Recall
(cf. Section 2.5), that the Legendre polynomials satisfy the orthogonality relation
Z1
2i
Pi( )Pj ( )d = 2i +j 1
ij 0
(10.2.2b)
;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( ) =
2
3
;
P2( ) = 3 2; 1 P3( ) = 5 2 3
4
2
5
3
P4( ) = 35 ; 30 + 3 P5( ) = 63 ; 70 + 15 :
(10.2.3)
2
8
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 (10.2.2bd) yields
_
hj cij + f (U(x t)) ; (;1)if (U(x t)) = Z 1 dPi( ) f (U ( t))d
j
j ;1
j
2i + 1
;1 d
i = 1 2 ::: p
(10.2.4a)
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))
(10.2.4b)
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
(10.2.5a)
F(Uk (xk t)) Uk+1(xk t)) = f (Uk (xk t)) +2f (Uk+1(xk t)) :
Based on our work with convectiondi 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
 JosephE.Flaherty

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