High-order accurate entropy stable schemes for compressible Euler equations with van der Waals equation of state on adaptive moving meshes

Abstract: This paper develops the high-order entropy stable (ES) finite difference schemes for multi-dimensional compressible Euler equations with the van der Waals equation of state (EOS) on adaptive moving meshes. Semi-discrete schemes are first nontrivially constructed built on the newly derived high-order entropy conservative (EC) fluxes in curvilinear coordinates and scaled eigenvector matrices as well as the multi-resolution WENO reconstruction, and then the fully-discrete schemes are given by using the high-order explicit strong-stability-preserving Runge-Kutta time discretizations.The high-order EC fluxes in curvilinear coordinates are derived by using the discrete geometric conservation laws and the linear combination of the two-point symmetric EC fluxes, while the two-point EC fluxes are delicately selected by using their sufficient condition, the thermodynamic entropy and the technically selected parameter vector.The adaptive moving meshes are iteratively generated by solving the mesh redistribution equations, in which the fundamental derivative related to the occurrence of non-classical waves is involved to produce high-quality mesh. Several numerical tests on the parallel computer system with the MPI programming are conducted to validate the accuracy, the ability to capture the classical and non-classical waves, and the high efficiency of our schemes in comparison with their counterparts on the uniform mesh.