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Unformatted text preview: me to be based on the solution of a Riemann problem.
This early work and a subsequent work of Glimm 17] and Chorin 9] stimulated a great
deal of interest in using Riemann problems to construct numerical ux functions. A
summary of a large number of choices appears in Cockburn and Shu 12]. 10.3 Multidimensional Discontinuous Galerkin Methods
Let us extend the discontinuous Galerkin method to multidimensional conservation laws
of the form ut + r f (u)x = b(x y z t u) (x y z) 2 t>0 (10.3.1a) where f (u) = f (u) g(u) h(u)] (10.3.1b) 10.3. Multidimensional Discontinuous Galerkin Methods
and r f (u) = f (u)x + g(u)y + h(u)z : 33
(10.3.1c) The solution u(x y z t) componenets of the ux vector f (u), g(u), and h(u) and the
loading b(x y z t u) are m-vectors and is a bounde region of <3. Boundary conditions
must be prescribed on @ along characteristics that enter the region. We'll see what this
means by example. Initial condtions prescribe
(x y z) 2 u(x y z 0) = 0 @:
Following our analysis of Section 10.2, we partition into a set of nite elements j ,
j = 1 2 : : : N , and construct a weak form of the problem on an element. This is done,
as usual, by multiplying (10.3.1a) by a test function v 2 L2( j ), integrating over j , and
applying the divergence theorem to the ux to obtain
(v ut)j + < v f n >j ;(rv f )j = (v b)j
8v 2 L2( j )
(v u)j = vT udxdydz
j (rv f )j = Z T
vx f (u) + vy g(u) + vzT h(u)]dxdydz (10.3.2c) j f n = f n = f (u)n1 + g(u)n2 + h(u)n3
and < v f n >j = Z
@ vT f ndS: (10.3.2d)
(10.3.2e) j The vector n = n1 n2 n3 ]T is the unit outward normal vector to @ and dS is a surface
in nitessimal on @ j .
Only the normal component of the ux is involved in (10.3.2) hence, its approximation on @ j is the same as the one-dimensional problems of Section 10.2. Thus, the
numerical normal ux function can be taken as a one-dimensional numerical ux using
solution values on each side of @ j . In order to specify this more precisely,...
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- Spring '14