Some applications of the concept of minimized integrated exponential error for low dispersion and low dissipation

Abstract

SUMMARYSeveral techniques to optimize parameters that regulate dispersion and dissipation effects in finite difference schemes have been devised in our previous works. They all use the notion that dissipation neutralizes dispersion. These techniques are the minimized integrated square difference error (MISDE) and the minimized integrated exponential error for low dispersion and low dissipation (MIEELDLD). It is shown in this work based on several numerical schemes tested that the technique of MIEELDLD is more accurate than MISDE to optimize the parameters that regulate dispersion and dissipation effects with the aim of improving the shock-capturing properties of numerical methods.First, we consider the family of third-order schemes proposed by Takacs. We use the techniques MISDE and MIEELDLD to optimize two parameters, namely, the cfl number and another variable which also controls dispersion and dissipation. Second, these two techniques are used to optimize a numerical scheme proposed by Gadd. Moreover, we compute the optimal cfl for some multi-level schemes in 1D. Numerical tests for some of these numerical schemes mentioned above are performed at different cfl numbers and it is shown that the results obtained are dependent on the cfl number chosen. The errors from the numerical results have been quantified into dispersion and dissipation using a technique devised by Takacs. Finally, we make use of a composite scheme made of corrected Lax-Friedrichs and the two-step Lax-Friedrichs schemes like the CFLF4 scheme at its optimal cfl number, to solve some problems in 2D, namely: solid body rotation test, acoustics and the circular Riemann problem.

Optimized composite finite difference schemes for atmospheric flow modeling

“…Then we can use numerical optimization to find the optimal values of both r and b . The technique of MIEELDLD has been used to optimize two parameters in Takac's family of third-order schemes and obtain optimal CFL for some multilevel schemes discretizing 1-D advection equation [32]. It was also used to obtain optimal CFL of CFLF4 scheme discretizing 2-D scalar advection equation [32].…”

“…These techniques used in CAA to construct methods have been modified in order to optimize parameters to minimize dispersion or both dispersion and dissipation in a given scheme and the work is detailed in [17,23,[31][32][33].…”

Some optimised schemes for 1D Korteweg-de Vries equation

“…Many interesting applications using the technique of Takacs can be seen in [22,23,24,25]. In this section, we use the ideas from [16] to quantify errors from numerical results into dissipation and dispersion.…”

Time-Splitting Procedures for the Numerical Solution of the 2D Advection-Diffusion Equation

“…Then we can use the schemes used for solving the 1D advection-diffusion equation to solve (14) and (15).…”

“…The total mean square error can be expressed as We extend the work on quantification of errors in [14,15] for the 2D case. The total mean square error for the 2D case is calculated as…”