We devise two novel techniques to optimize parameters which regulate dispersion and dissipation effects in numerical methods using the notion that dissipation neutralizes dispersion. These techniques are baptized as the minimized integrated error for low dispersion and low dissipation (MIELDLD) and the minimized integrated exponential error for low dispersion and low dissipation (MIEELDLD).These two techniques of optimization have an advantage over the concept of minimized integrated square difference error (MISDE), especially in the case when more than one optimal cfl is obtained, out of which only one of these values satisfy the shift condition. For instance, when MISDE is applied to the 1-D Fromm's scheme, we have obtained two optimal cfl numbers: 0.28 and 1.0. However, it is known that Fromm's scheme satisfies shift condition only at r=1.0. Using MIELDLD and MIEELDLD, the optimal cfl of Fromm's scheme is computed as 1.0.We show that like the MISDE concept, both the techniques MIELDLD and MIEELDLD are effective to control dissipation and dispersion. The condition [nu]2>4µ is satisfied for all these three techniques of optimization, where [nu] and µ are parameters present in the Korteweg-de-Vries-Burgers equation.The optimal cfl number for some numerical schemes namely Lax-Wendroff, Beam-Warming, Crowley and Upwind Leap-Frog when discretized by the 1-D linear advection equation is computed. The optimal cfl number obtained is in agreement with the shift condition. Some numerical experiments in 1-D have been performed which consist of discontinuities and shocks. The dissipation and dispersion errors at some different cfl numbers for these experiments are quantified. Copyright © 2010 John Wiley & Sons, Ltd.
viernes, 22 de enero de 2010
The concept of minimized integrated exponential error for low dispersion and low dissipation schemes
We devise two novel techniques to optimize parameters which regulate dispersion and dissipation effects in numerical methods using the notion that dissipation neutralizes dispersion. These techniques are baptized as the minimized integrated error for low dispersion and low dissipation (MIELDLD) and the minimized integrated exponential error for low dispersion and low dissipation (MIEELDLD).These two techniques of optimization have an advantage over the concept of minimized integrated square difference error (MISDE), especially in the case when more than one optimal cfl is obtained, out of which only one of these values satisfy the shift condition. For instance, when MISDE is applied to the 1-D Fromm's scheme, we have obtained two optimal cfl numbers: 0.28 and 1.0. However, it is known that Fromm's scheme satisfies shift condition only at r=1.0. Using MIELDLD and MIEELDLD, the optimal cfl of Fromm's scheme is computed as 1.0.We show that like the MISDE concept, both the techniques MIELDLD and MIEELDLD are effective to control dissipation and dispersion. The condition [nu]2>4µ is satisfied for all these three techniques of optimization, where [nu] and µ are parameters present in the Korteweg-de-Vries-Burgers equation.The optimal cfl number for some numerical schemes namely Lax-Wendroff, Beam-Warming, Crowley and Upwind Leap-Frog when discretized by the 1-D linear advection equation is computed. The optimal cfl number obtained is in agreement with the shift condition. Some numerical experiments in 1-D have been performed which consist of discontinuities and shocks. The dissipation and dispersion errors at some different cfl numbers for these experiments are quantified. Copyright © 2010 John Wiley & Sons, Ltd.
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