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Unformatted text preview: n+1 ; cn = ;f (Un (x )) + (;1)i f (Un(x )) = Z 1 dPi( ) f (Un ( ))d
ij
ij
j
j ;1
j
2i + 1
t
;1 d
i = 1 2 : : : p:
The notation is identical to that used in Chapter 9 thus, Un(x) and cn are the approxij
imations of U(x tn) and cij (tn), respectively, produced by the time integration software
and t is the time step. The forward Euler method is used for illustration because of its
simplicity. The order of the temporal integration method should be comparable to p. 26 Hyperbolic Problems
Initial conditions may be determined by L2 projection as
Z1
Pi( ) Uj ( 0) ; u0( )]d = 0
i = 0 1 ::: p
j = 1 2 : : : J:
;1 (10.2.6) One more di culty emerges. Higherorder schemes for hyperbolic problems oscillate
near discontinuities. This is a fundamental result that may be established by theoretical
means (cf., e.g., Sod 25]). One technique for reduced these oscillations involves limiting
the computed solution. Many limiting algorithms have been suggested but none are
totally successful. We describe a procedure for limiting the slope @ Uj (x t)[email protected] of the
solution that is widely used. With this approach, @ Uj (x t)[email protected] is modi ed so that:
1. the solution (10.2.2a) does not take on values outside of the adjacent grid averages
(Figure 10.2.4, upper left)
2. local extrema are set to zero (Figure 10.2.4, upper right) and
3. the gradient is replaced by zero if its sign is not consistent with its neighbors (Figure
10.2.4, lower center).
Figure 10.2.4 illustrates these situations when the solution is a piecewiselinear (p = 1)
function relative to the mesh.
A formula for accomplishing this limiting can be summarized concisely using the
minimum modulus function as
@ Uj mod (xj t) = minmod( @ Uj (xj t) rU (x
(10.2.7a)
j j ;1=2 t) Uj (xj ;1=2 t))
@x
@x @ Uj mod (xj;1 t) = minmod( @ Uj (xj;1 t) rU (x
j j ;1=2 t) Uj (xj ;1=2 t)) (10.2.7b)
@x
@x
where
s
i sgn(a)
minmod(a b c) = 0gn(a) min(jaj jbj jcj) oftherwise= sgn(b) = sgn(c) (10.2.7c)
and r and are the backward and forward di erence operators rUj (xj; = 12 and t) = Uj (xj;1=2 t) ; Uj (xj...
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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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