Error analysis and estimation for the finite volume method with applications to fluid flows

The accuracy of numerical simulation algorithms is one of main concerns in modern Computational Fluid Dynamics. Development of new and more accurate mathematical models requires an insight into the problem of numerical errors. In order to construct an estimate of the solution error in Finite Volume calculations, it is first necessary to examine its sources. Discretisation errors can be divided into two groups: errors caused by the discretisation of the solution domain and equation discretisation errors. The first group includes insufficient mesh resolution, mesh skewness and non-orthogonality. In the case of the second order Finite Volume method, equation discretisation errors are represented through numerical diffusion. Numerical diffusion coefficients from the discretisation of the convection term and the temporal derivative are derived. In an attempt to reduce numerical diffusion from the convection term, a new stabilised and bounded second-order differencing scheme is proposed. Three new methods of error estimation are presented. The Direct Taylor Series Error estimate is based on the Taylor series truncation error analysis. It is set up to enable single-mesh single-run error estimation. The Moment Error estimate derives the solution error from the cell imbalance in higher moments of the solution. A suitable normalisation is used to estimate the error magnitude. The Residual Error estimate is based on the local inconsistency between face interpolation and volume integration. Extensions of the method to transient flows and the Local Residual Problem error estimate are also given. Finally, an automatic error-controlled adaptive mesh refinement algorithm is set up in order to automatically produce a solution of pre-determined accuracy. It uses mesh refinement and unrefinement to control the local error magnitude. The method is tested on several characteristic flow situations, ranging from incompressible to supersonic flows, for both steady-state and transient problems.