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Abstract
The accuracy of numerical simulation algorithms is one of main concerns in modern
Computational Fluid Dynamics. Development of new and more accurate mathe-
matical models requires an insight into the problem of numerical errors.
In order to construct an estimate of the solution error in Finite Volume cal-
culations, 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 reso-
lution, 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 numeri-
cal 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.
