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Dissipation and entropy production of high-order numerical methods for hyperbolic conservation laws

Subject Area Mathematics
Term from 2017 to 2021
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 391673438
 
Final Report Year 2022

Final Report Abstract

The project »Dissipation and entropy production of high-order numerical methods for hyperbolic conservation laws« searched for and found new methods to construct entropy conservative and entropy dissipative numerical schemes. This research can be split into two large and coupled topics. One of them is the spatial discretisation, i.e. converting the given hyperbolic system of conservation laws into a system of coupled ordinary differential equations (ODE). The second one is the time-integration of this ODE system by an ODE solver. Research on the first step was centered around enforcing entropy dissipation for this ODE systems, either in the sense of classical entropy inequalities or the principle of the fastest possible entropy dissipation, pioneered by Dafermos. As a first logical continuiation of the work of Dafermos it was studied if his statement that the Godunov flux maximizes the entropy dissipation for scalar conservation laws could be generalized to systems. It was found that this is not the case. Then, and to the surprise of the research group, the numerical enforcement of the Dafermos entropy rate criterion by choosing appropriate reconstructions from numerical data lead to unusable schemes. The reason is conjectured to be that these reconstructions, while highly entropy dissipative, are bad approximations of weak solutions. One can further assume that these schemes dissipate significantly more entropy than the exact solution and are therefore nonphysical. The discrete fulfillment of product and chain rules and the dissipation of entropy by the continuous second derivative or Laplacian and its discrete equivalent was studied next. An order barrier was found - while discretisations of first derivatives of order 2 satisfy discrete equivalents of chain and product rules, higher order approximations for BV functions need not satisfy discrete equivalents. Also, high order discretisations of derivatives do not dissipate all entropies. The enforcement of classical entropy inequalities, implying the semidiscrete stability of the scheme, were considered next by the reinterpretation of entropy correction terms developed by Abgrall. These correction terms were interpreted as solutions to optimization problems, ensuring that the dissipation needed has the smallest norm, i.e. the smallest error, of all corrections with the same entropy dissipation. A related approach was used to construct schemes enforcing a modified version of Dafermos’ entropy rate criterion with upper bounds on the error introduced by the dissipation. These schemes show benign results, proving that a numerical enforcement of Dafermos’ entropy rate can be worthwhile. All of the previously mentioned bounds are only bounds assuming that the resulting ODE system is solved exactly. The connection to the numerical solution of the given ODEs is unclear. Therefore it was shown during this project that the popular ODE solvers of the SSPRK type are in general not able to preserve semidiscrete bounds on entropy and similar convex functional during time integration. Yet, during the project, discrete entropy stable solvers of second [1] and arbitrary order [5] were constructed. All in all the project contributed to improved stability and non-oscillatory behavior of schemes for conservation laws, combining classical and machine learning approaches. The approaches used are general and can be applied not only to the studied problems but are also beneficial to other schemes, equations and branches of mathematics.

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