Error estimation in unstructured finite volume solvers : improvements to the adjoint method and defect correction

Much of the effort in improving the accuracy of computational fluid dynamics (CFD) simulations is focused on mesh refinement and adaptation although studies have shown that the use of high-order methods (more than second-order) are more efficient in improving accuracy. Stability issues, complexity of implementation, and demand for computational resources are some of the key factors hindering the use of high-order methods in commercial CFD solvers. This thesis discusses two methods -- adjoint error estimation and defect correction -- to improve the order of accuracy of the simulation without requiring implementation of a high-order discretization scheme in an unstructured finite volume solver. One of the major applications of the adjoint method is the improvement in the order of accuracy of integral quantities obtained from CFD simulations (output functionals). By solving an adjoint problem specific to the functional of interest, and using an estimate of the truncation error of the primal solution, the functional estimate based on the base (second-order) solution can be corrected to get a superconvergent estimate of the functional. Although the theory requires the use of a smooth interpolation of the solution, this has seldom been used with unstructured finite volume solvers. The specificity of the adjoint problem to an output functional means that for multiple functionals, multiple adjoint solutions are necessary. In such cases, using defect correction to improve the accuracy of the entire solution (so that any number of high-accurate functionals can be obtained) is a better alternative. An estimate of the truncation error obtained from the primal solution is added to the source term to formulate the defect correction problem. The defect correction problem is solved using the same discretization scheme as the primal problem to get a corrected solution. The randomness of an unstructured mesh results in finite volume solutions having second-order noise in them, which degrades the quality of the truncation error estimates. We use a smoothing spline based on a C¹ continuous representation of the discrete solution to improve the truncation error estimates needed for both these methods. For a variety of problems including 2-D Euler equations, we show that a third-order accurate solution can be obtained using defect correction without using a third-order discretization scheme. Fourth-order accurate superconvergent functional estimates are also obtained for a variety of problems with consistent improvement over previous results.