A case-study in open-source CFD code verification. Part II: Boundary condition non-orthogonal correction
Abstract This investigation constitutes the follow-up to a previous work in which we applied the manufactured solution procedure to verify the OpenFOAM diffusion operator on different types of meshes. Theoretical convergence orders were observed for Poisson’s equation solver on orthogonal hexahedral grids using several boundary conditions. We noticed that on non-orthogonal grids even slight mesh distortions reduce the theoretical second order convergence rate to first order. Our investigation showed that this loss in convergence order takes place when Dirichlet and/or Neumann boundary conditions are used on non-orthogonal meshes. In this paper, we introduce ways to achieve theoretical second order convergence accuracy, mainly based on applying non-orthogonal correction to Dirichlet and/or Neumann boundary condition schemes. A peculiarity was discovered for the Least-Squares gradient scheme that slightly affects second-order convergence, and which we attribute to the non-orthogonal correction at boundaries in the Least-Squares vector reconstruction library. The verification procedure is performed using the method of manufactured solutions for the Poisson equation and an analytical solution, the SIMPLE solver, for the Navier–Stokes equations. The average and maximum error norms were used to calculate the convergence rate. Comparative results are presented.
AUTOMATIC RESOLUTION CONTROL FOR THE FINITE-VOLUME METHOD, PART 1: A-POSTERIORI ERROR ESTIMATES
Automatic Resolution Control, consisting of an a-posteriori error estimate and a local adjustment of the discretisation scheme, aims at producing a numerical solution of prescribed accuracy with minimum computational effort. In this paper, two a-posteriori error estimates concentrating on the absolute error levels are presented. The Taylor Series Error Estimate is based on a single-grid Taylor series truncation error analysis. The Moment Error Estimate is derived from the balance condition for the higher moments of the transported variable. Two error estimates are validated on two test cases with analytical solutions, where they typically bound the exact error.
The laws of Cnut and the history of Anglo-Saxon royal promises
The ‘first’ and ‘second’ law codes of Cnut are the last surviving codes issued in the name of an Anglo-Saxon king.They are the final fruit of the interest in kingship and law and in the inter-relationship of the two which characterized the period following the monastic revival in England and which is especially associated with the name of Wulfstan, archbishop of York. Although they are far from being complete codifications of Anglo-Saxon law, they draw extensively on earlier legislation. They are different in kind from much tenth-century law, which is usually more limited and administrative in content and character, and their nature and purpose invite further consideration.