High-Order, Entropy-Conservative Discretizations of the Euler Equations for Complex Geometries

We present an entropy-stable semi-discretization of the Euler equations. The scheme is based on high-order summation-by-parts (SBP) operators for triangular and tetrahedral elements, although the theory is applicable to multidimensional SBP operators on more general elements. While there are established methods for proving stability of linear equations, such as energy analysis, they are not adequate for nonlinear equations. To address nonlinear stability, we use the matrix properties of the SBP operators combined with entropy-conserving numerical flux functions. This allows us to prove that the semi-discrete scheme conserves entropy. Significantly, the proof does not rely on integral exactness, and, therefore, the discretization has a stronger claim of robustness than a similar finite-element method. The addition of an upwinded term to the entropy-conservative scheme makes it entropy-stable. This generalizes previous work proving entropy stability for tensor-product elements to more general elements, including simplex elements. Numerical experiments are conducted to verify accuracy and entropy conservation on an isentropic vortex flow.