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j Figure 10.2.4: Solution limiting: reduce slopes to be within neighboring averages (upper
left) set local extrema to zero (upper right) and set slopes to zero if they disagree with
neighboring trends.
only provide two vector equations for modifying the p vector coe cients cij mod(t), i =
1 2 : : : p, in @ Uj (x t)[email protected] When p = 1, (10.2.7a,b) are identical and c1j mod(t) is
uniquely determined. Likewise, when p = 2, the two conditions (10.2.7a,b) su ce to
uniquely determine the modi ed coe cients c1j mod(t) and c2j mod (t). Equations (10.2.7a,b)
are insu cient to determine the modi ed coe cients when p > 2 and Cockburn and
Shu 12] suggested setting the higherorder coe cients cij mod(t), i = 3 4 : : : p, to zero.
This has the disturbing characteristic of \ attening" the solution near smooth extrema
and reducing the order of accuracy. Biswas et al. 8] developed an adaptive limiter which 28 Hyperbolic Problems applied the minimum modulus function (10.2.7c) to higher derivatives of Uj . They began
by limiting the p th derivative of Uj and worked downwards until either a derivative was
not changed by the limiting or they modi ed all of the coe cients. Their procedure,
called \moment limiting." is described further in their paper 8].
Example 10.2.2. Biswas et al. 8] solve the inviscid Burgers' equation (10.1.16) with
the initial data
sin
u(x 0) = 1 + 2 x :
This initial data steepens to form a shock which propagates in the positive x direction.
Biswas et al. 8] use an upwind numerical ux (10.2.5b) and solve problems on uniform
meshes with h = 1=32 with p = 0 1 2. Time integration was done using classical RungeKutta methods of orders 13, respectively, for p = 0 1 2. Exact and computed solutions
are shown in Figure 10.2.5. The piecewise polynomial functions used to represent t...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Constant of integration, Partial differential equation, hyperbolic problems

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