Convergence analysis of structure-preserving schemes for the multicomponent compressible Euler flows
Provides rigorous convergence guarantees for structure-preserving schemes in multicomponent compressible flow, important for numerical analysts and CFD practitioners.
The paper proves convergence of a structure-preserving finite volume scheme for multicomponent compressible Euler flows in the dissipative weak solution framework, showing strong convergence to classical solutions when they exist. Numerical experiments confirm the theory for both low-order FV and higher-order DG schemes.
We present a convergence analysis of a finite volume (FV) scheme for the multicomponent compressible Euler system in the framework of dissipative weak (DW) solutions. DW solutions were introduced as a generalized solution framework in computational fluid dynamics and have recently gained considerable attention. They extend the well-known Lax Equivalence Theorem to nonlinear settings, meaning that if a numerical scheme is both consistent and stable, it will also converge. The FV scheme under consideration preserves key physical properties of the fluid mixture, in particular, positivity of partial densities, pressure, and temperature. Using uniform stability bounds and consistency estimates, we prove that the numerical solutions converge in the framework of DW solutions of the multicomponent Euler system. Applying the relative entropy method and the weak-strong uniqueness principle, we further show that the approximate solutions converge strongly to the classical solution as long as it exists. Numerical experiments confirm the theoretical results, not only for low-order FV methods but also through extended numerical investigations of a higher-order, structure-preserving discontinuous Galerkin scheme.