Unformatted text preview: ng 26]. Van
Leer 27] found an improvement that provided better performance near sonic and stagnation points of the ow. The split uxes are evaluated by upwind techniques. Thus, at
an interface x = xj , f + is evaluated using Uj (xj t) and f ; is evaluated using Uj+1(xj t).
Calculating uxes based on the solution of Riemann problems is another popular
way of specifying numerical uxes for vector systems. To this end, let w(x=t uL uR )
be the solution of a Riemann problem for (10.1.1a) with the peicewiseconstant initial
data (10.1.25). The solution of a Riemann problem \breaking" at (xj tn) would be
w((x ; xj )=(t ; tn) Uj (xj tn) Uj (xj+1 tn)). Using this, we would calculate the numerical
ux at (xj t), t > tn , as F(Uj (xj tn) Uj+1(xj tn)) = f (w(0 Uj (xj tn) Uj+1(xj tn)): (10.2.11) 32 Hyperbolic Problems Example 10.2.4. Let us calculate the numerical ux based on the solution of a Riemann problem for Burgers' equation (10.1.16). Using the results of Example 10.1.8) we
know that the solution of the appropriate Riemann problem is
8
if Uj Uj+1 > 0
> Uj
>
>U
> j +1 if Uj Uj +1 < 0
<
if Uj < 0 Uj+1 > 0
w(0 Uj Uj+1) = > 0
:
> Uj
if Uj > 0 Uj+1 < 0 (Uj + Uj+1)=2 > 0
>
>
:U
j +1 if Uj > 0 Uj +1 < 0 (Uj + Uj +1 )=2 < 0
(The arguments of Uj and Uj+1 are all (xj tn). These have been omitted for clarity.)
With f (u) = u2=2 for Burgers' equation, we nd the numerical ux
82
> Uj =2 if Uj Uj +1 > 0
>2
> U =2 if U U < 0
> j +1
<
j j +1
if Uj < 0 Uj+1 > 0
F (Uj Uj+1) = > 0
:
> Uj2 =2 if Uj > 0 Uj +1 < 0 (Uj + Uj +1 )=2 > 0
>
>2
: U =2 if U > 0 U < 0 (U + U )=2 < 0
j
j +1
j
j +1
j +1
Letting
u+ = max(u 0)
u; = min(u 0)
we can write the numerical ux more concisely as F (Uj Uj+1) = max (Uj+)2 =2 (Uj; )2=2]:
+1
When used with a piecewiseconstant basis and forward Euler time integration, the resulting discontinuous Galerkin scheme is identical to Godunov's nite di erence scheme 18].
This was the rst di erence sche...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Constant of integration, Partial differential equation, hyperbolic problems

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