A face-upwinded spectral element method

• Published in: Journal of computational physics 2024-04, Vol.503 (C), p.112825, Article 112825
• Main Authors: Pan, Y., Persson, P.-O.
• Format: Article
• Language: English
• Subjects: High-order methods; Spectral elements; Unstructured meshes
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2024.112825

We present a new high-order accurate discretisation on unstructured meshes of quadrilateral elements. Our Face Upwinded Spectral Element (FUSE) method uses the same node distribution as a high-order continuous Galerkin (CG) method, but with a particular choice of node locations within each element and an upwinded stencil on the face nodes. This results in a number of benefits, including fewer degrees of freedom and straight-forward integration with CG. We present the derivation of the scheme and the analysis of its properties, in particular showing stability using von Neumann analysis. We show numerical evidence for its accuracy and efficiency on multiple classes of problems including convection-dominated flows, Poisson's equation, and the incompressible Navier-Stokes equations. •A new stabilised spectral element method based on continuous solutions and face upwinding.•Using a special set of local nodes, the scheme is stable for arbitrary polynomial degrees.•For the special case of constant coefficient 1D problems, the scheme can be written as a Spectral Difference method.•In general, the method can be understood as a nodally-integrated Petrov-Galerkin scheme.•Some benefits over the popular Discontinuous Galerkin method include fewer degrees of freedom, sparser Jacobian matrices, and better CFL condition.