Continuous adjoint based error estimation and r-refinement for the active-flux method

Abstract

This paper explores the possibility of using continuous adjoints to estimate errors in scalar outputs and to drive mesh adaptation for the active-flux method. Compared to a discrete adjoint approach, the continuous adjoint offers a few attractive advantages for the active-flux method, such as a naturally fully-discrete, explicit implementation that parallels the simplicity of the primal system evolution and error estimates obtained from integrals over the space-time control volumes. The latter point is important because the active flux method does not possess a variational formulation. In addition, we present novel unsteady adaptation mechanics that rely on dynamically moving mesh nodes, i.e. r-refinement, in order to minimize the output error.

Output Error Control Using r-Adaptation

“…For node-interpolated r-adaptation, the adaptation is driven by a spring analogy previously used for the Active Flux method. 23 The localized error within each element is re-distributed onto element edges via averaging, and we relate this error to the equilibrium length of a spring on the edge. The whole mesh is treated as a web of springs, and a force balance is used to equidistribute the error.…”

“…19,20 All of these techniques have the capability to increase resolution in some areas and to decrease the resolution in other areas, so that the total number of degrees of freedom can grow, shrink, or remain the same. A related adaptation technique, that is more restricted, but relatively underutilized, is r adaptation [21][22][23][24][25][26] , in which the nodes of the mesh undergo motion in order to redistribute the resolution to areas that need it the most. The total number of degrees of freedom remains constant, and hence r adaptation is limited in the maximum possible error reduction that it can achieve.…”