Convergence analysis for a finite-volume scheme for the Euler- and Navier-Stokes-Korteweg system via energy-variational solutions
This provides rigorous convergence guarantees for numerical simulations of complex fluid dynamics models, which is incremental but important for computational physics applications.
The authors proved that numerical solutions from a structure-preserving finite-volume scheme converge to energy-variational solutions for Euler-Korteweg and Navier-Stokes-Korteweg equations under mesh refinement, extending this novel solution concept to the NSK model.
We consider a structure-preserving finite-volume scheme for the Euler-Korteweg (EK) and Navier-Stokes-Korteweg (NSK) equations. We prove that its numerical solutions converge to energy-variational solutions of EK or NSK under mesh refinement. Energy-variational solutions constitute a novel solution concept that has recently been introduced for hyperbolic conservation laws, including the EK system, and which we extend to the NSK model. Our proof is based on establishing uniform estimates following from the properties of the structure-preserving scheme, and using the stability of the energy-variational formulation under weak convergence in the natural energy spaces.