Numerical entropy production in finite volume $P_0P_M$ ADER schemes

TL;DR

This paper extends the concept of numerical entropy production as an indicator of solution regularity from Runge-Kutta finite volume methods to finite volume P0PM ADER schemes, providing theoretical analysis and practical adaptive strategies.

Contribution

It introduces and analyzes the numerical entropy production for P0PM ADER schemes, demonstrating its convergence, boundedness near discontinuities, and utility as an adaptive smoothness indicator.

Findings

Numerical entropy production decays with grid refinement for smooth solutions.

It remains bounded near contact discontinuities and diverges at shock waves.

The entropy production can effectively guide p-adaptive schemes to reduce oscillations.

Abstract

We consider the numerical integration of conservation laws endowed with an entropy inequality and we study the residual of the scheme on this inequality, which represents the numerical entropy production. This idea has been introduced and exploited in Runge-Kutta finite volume methods, where the numerical entropy production has been used as an indicator in adaptive schemes, since it scales as the local truncation error of the method for smooth solutions and it highlights the presence of discontinuities and their kind. The aim of this work is to extend this idea to finite volume $P_0P_M$ ADER timestepping techniques. We show that the numerical entropy production can be defined also in this context and it provides a scalar quantity computable for each space-time volume which, under grid refinement, decays to zero with the same rate of convergence of the scheme for smooth solutions. Its size gradually increases when the local solution regularity lowers, remaining bounded up to contact discontinuities and divergent on shock waves. Theoretical results are proven in a multi-dimensional setting on arbitrary polygonal grids. We also present numerical evidence showing that it is essentially negative definite. Moreover, we propose an example of $p$-adaptive scheme that uses the numerical entropy production as a-posteriori smoothness indicator. The scheme locally modifies its order of convergence with the purpose of removing the oscillations due to the high-order of accuracy of the scheme.