This was the rst di erence scheme to be based on the

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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 @: (10.3.1d) 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 ) (10.3.2a) where Z (v u)j = vT udxdydz (10.3.2b) j (rv f )j = Z T 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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