Study of Flux Limiters to Minimize the Numerical Dissipation Based on Entropy-Consistent Scheme

Abstract

The use of limiters may impact both accuracy and resolution of a solution. And the accuracy and resolution are highly dependent on the amount of numerical dissipation in a scheme, so the ability of limiters to control numerical dissipation should be improved. In this view, based on the examination of several classical limiters to control dissipation, a class of general piecewise-linear flux limiters (termed GPL limiters) are presented in this paper for Multi-step time-space-separated unsteady schemes. The GPL limiters can satisfy the second-order TVD criterion and contain some existing limiters such as Superbee and Minmod. Using the decrement of discrete total entropy to represent the amount of numerical dissipation, an entropy dissipation function of GPL limiters is defined with three parameter variables. By proving the monotonicity of this function, a new GPL type limiter (named MDF individually), which can minimize the numerical dissipation and improve the calculation accuracy, is proposed. A high resolution entropy-consistent scheme is obtained by MDF limiter, which will be proved to satisfy entropy stability and entropy consistency. Computational results of this scheme for several 1-D and 2-D Euler test cases are presented, demonstrating the accuracy, monotonicity and robustness of MDF limiter.

A Group of CFL-Dependent Flux-Limiters to Control the Numerical Dissipation in Multi-stage Unsteady Calculation

“…Especially, the piecewise-linear flux-limiters lying on the boundaries maybe more likely to provide the highest order of accuracy for a high resolution TVD scheme relative to those gradually-switching smooth flux-limiters. From the comparisons of the performances of those existing CFL-independent GPL limiters in [45,46], we realize that, for a specific high resolution TVD scheme, different boundary-piecewise-linear flux-limiters may lead to largely different profiles of a solution. So, it is vital to design the proper limiters in developing the high resolution schemes for different calculations.…”

“…The conclusions in Proposition 1 prompt us to wonder whether there is a proper GPL-CFL flux-limiter that can minimize the amount of the total dissipation in a numerical scheme, yet we know that it is impossible because the value of CFL number tends to be infinitesimal. However, similar to the method in [46], we can obtain a CFL-dependent minimized limiter by choosing some endpoint values of parameters α, β and γ , i.e., we take α = 1, β = 2 and γ = 1, respectively, the following limiter then be obtained Next, we will briefly verify the CFL-dependent minimum feature of this limiter when the CFL number is fixed as ν = ν 0 . The total dissipation function (11) can be expressed as follows after some simple manipulations (also f d is ignored here):…”

“…In there, the performances of those limiters are compared with each other, which showed that most numerical results are oscillation-free, but they may be unstable for some limiter combinations when an improper CFL number is adopted, for example, when CFL values are greater than 1/2. In view of this problem, we developed a group of GPL limiters based on Sweby's TVD criterion in the subsequent paper [46] to solve the MTU calculations with an aim to minimize the numerical dissipation, in which we set CFL value to be 1/2 to make sure the obtained high resolution schemes be TVD because the Sweby's TVD criterion is equivalent to the MTU-TVD criterion when the CFL value equals to 1/2. Both previous papers [45,46] only considered using the Sweby's TVD constraint to compute the MTU problems.…”

“…In view of this problem, we developed a group of GPL limiters based on Sweby's TVD criterion in the subsequent paper [46] to solve the MTU calculations with an aim to minimize the numerical dissipation, in which we set CFL value to be 1/2 to make sure the obtained high resolution schemes be TVD because the Sweby's TVD criterion is equivalent to the MTU-TVD criterion when the CFL value equals to 1/2. Both previous papers [45,46] only considered using the Sweby's TVD constraint to compute the MTU problems. We known, in fact, the Sweby's TVD constraint is just the SS-TVD criterion which can make sure the general semi-discrete flux-conservative scheme achieve the desired TVD property, but it is not suitable for the MTU scheme.…”

“…According to the obtained conclusions, a relatively ideal GPL-CFL type limiter is proposed. This limiter is, in fact, the CFL-dependent extension of the MDF limiter (Minimized Dissipation Flux-limiter) presented in [46], which is termed as MDF-CFL limiter. It not only can minimize the numerical dissipation among GPL-CFL limiters when the CFL number is fixed, but also can further reduce the numerical dissipation as the CFL number gradually decreases, which is different from all other CFL-independent limiters.…”