Hybrid WENO scheme: The trade-offs between accuracy and cost

On Wednesday, January 26, Professor Jianxian Qui gave a lecture titled Hybrid Weighted Essentially Non-Oscillatory Schemes with Different Indicators. It is a test of different methods that can detect a discontinuity in a function. The main idea is that lower order fluxes could be combined to find a higher order approximation of a disturbance.
Qui is from the Nanjing University Math Department. His research focus is “on high order hyperbolic functions.” A joint effort with Gang Li, Qui was able to test several methods of finding a troubled-cell, a set containing discontinuity. These include the approximation of hybrid Weighted Essentially Non-Oscillatory (WENO) Schemes, Average Test Variation (ATV), Monotonicity-Preserving Limiter (MP), and a shock-detection technology by Krivodonova (KXRCF).

One major question is why should people care about this. Qui explained, “WENO approximation is crucial when the strong discontinuity, such as shock wave, is present.” Heat transfer and energy flux is a large part of engineering. Whether it is the shock wave that travels through a wall or just the heat transfer from friction of a coarse wall, approximations to determine how much of the energy was disturbed or lost are important. As Qui repeated multiple times, the “WENO is used in a large range of fields.” The WENO scheme was mentioned to be used in Astronomy, Astrophysics, Semiconductors Devices, and Computational Biology.

The problem with a large amount of the methods is that they have problem-dependent variables. Qui continued to explain that if one was to be too far off what would be considered correct, then there would be a chance that certain discontinuities would not be counted or there would be oscillations that are not from the experiment, but added from the computations.

According to Qui, the hybrid WENO scheme has the advantage over the other methods as it can calculate with high order accuracy in smooth regions and small extremes. Its high accuracy with nonlinear weight comes at a large price as the “cost of nonlinear weight is very high.”

Despite the current cost difficulties of the hybrid WENO scheme approximation, Qui informed the audience that there have been advancements to minimize the cost through different combinations of indicators to recognize discontinuities and through determining non-linear weight through entropy and pressure rather characteristic values.

The remainder of the lecture was about the tests performed on the indicators and schemes and comparing the processing time. At first Qui showed results that would indicate the hybrid WENO scheme was worthless, as it would take over ten times as long to compute the approximation. But as the presentation seemed to close, Qui showed why he still prefers the hybrid WENO scheme. “The others may have lower times throughout all orders, but with the quality, there is no comparison.” The hybrid WENO scheme had plotted 100% of the reconstruction of the fluxes while the other schemes had a maximum plot of 21.1%.

The hybrid WENO may require a longer time and have a higher cost, but affords a greater accuracy. Further studies aided by the research presented may allow the accuracy to increase as CPU time required decreases. Advancement in this field will help advance the almost all engineering fields.