11111 00000000000 00000 11111111111 11111 00000000000

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Unformatted text preview: 00000 00000 11111111111 11111 00000000000 00000 11111111111 11111 00000000000 00000 11111111111 11111 00000000000 00000 11111111111 11111 00000000000 00000 11111111111 11111 00000000000 00000 11111111111 11111 00000000000 00000 11111111111 11111 00000000000 00000 11111 00000 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 higher-order 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 1-3, 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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