Numerical entropy production in finite volume $P_0P_M$ ADER schemes

TL;DR

This work extends the concept of numerical entropy production as a residual of the entropy inequality from RK-based finite-volume methods to fully discrete FV-ADER $P_0P_M$ schemes on multidimensional polygonal grids. It defines $S_j^n$ as a space-time residual that decays with the scheme order for smooth solutions and grows in the presence of shocks, remaining essentially negative in practice, thereby acting as a powerful a-posteriori smoothness indicator. The authors prove the key decay properties in multi-dimensions and provide extensive 1D and 2D numerical tests (including Sod, implosion, and shock–bubble interactions) to validate the theory and demonstrate the potential for $p$-adaptive strategies. The results highlight the practical impact of using entropy production to guide adaptive order refinement in high-order FV methods, improving stability and accuracy on arbitrary grids. This work lays the groundwork for broader hp-adaptive FV-ADER frameworks driven by entropy-based indicators.

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.

Figures (14)

• Figure 1: Smooth solution using $P_0P_2$ and corresponding numerical entropy production with a grid of $N=128,256$ and $512$ cells.
• Figure 2: Left rarefaction wave computed by $P_0P_1$ (left) and entropy production by $P_0P_1$ and $P_0P_2$ (right) with a grid of $N=256,512$ and 1024 cells.
• Figure 3: Contact discontinuity wave computed by $P_0P_1$ (left) and entropy production of $P_0P_1$ and $P_0P_2$ (right) with a grid of $N=256,512$ and 1024 cells.
• Figure 4: Right shock wave computed by $P_0P_1$ (left) and entropy production of $P_0P_1$ and $P_0P_2$ (right) with a grid of $N=256,512$ and 1024 cells.
• Figure 5: Sod tube problem and corresponding entropy production in logarithmic scale by 3-Runge-Kutta FV and $P_0P_2$ with a grid of 100 cells.

Theorems & Definitions (6)

• Remark 1
• Theorem 1
• Definition 1
• Remark 2
• Theorem 2
• proof