Comparisons of p-adaptation strategies based on truncation- and discretisation-errors for high order discontinuous Galerkin methods

Abstract

Abstract In this work, local p-adaptation strategies for high order discontinuous Galerkin spectral element methods are compared. The principal aim of the paper is to determine the advantages and drawbacks of various sensors, based on truncation or discretisation errors, to detect the regions that require adaption. A well-established discretisation error based approach, estimated through the decay of the energy associated to the approximated solution modes, is compared to recently proposed truncation error adaptation methodologies (i.e., isotropic and anisotropic versions). The truncation error technique detects the regions that require refinement by estimating the truncation error in meshes with varying degrees of freedom. This comparison is particularly interesting since the truncation error is related to the discretisation error through a discretisation error transport equation, where the truncation error appears as a source term for the discretisation error. The comparisons included quantify the accuracy of the flow solutions resulting from meshes adapted with these sensors and hence provide guidelines into which sensors are better suited to adapt meshes for inviscid and viscous flows. All the adaption strategies are tailored to high order methods and particularly implemented and tested in a discontinuous Galerkin solver. Results include an inviscid NACA0012 and a viscous flat plate boundary layer. Output functionals (e.g., lift, drag) resulting from adapted meshes are compared in terms of the number of degrees of freedom, providing a critical assessment of the performance of each sensor. Namely, it is shown that both truncation error adaptation methodologies provide meshes with polynomial order distributions that lead to fewer degrees of freedom than when using the discretisation error based adaptation. In addition, the examples illustrate the outperforming advantage of anisotropic over isotropic adaptation.