High-order finite volume schemes based on defect corrections

For the approximation of steady state solutions, we propose an iterated defect correction approach to achieve higher-order accuracy. The procedure starts with the steady state solution of a low-order scheme, in general a second order one. The higher-order reconstruction step is applied a posteriori to estimate the local discretization error of the lower-order finite volume scheme. The defect is then used to iteratively shift the basic lower-order scheme to the desired higher-order accuracy given by the polynomial reconstruction. Hence, instead of solving the high-order discrete equations the low-order basic scheme is solved several times. This avoids that the high-order reconstruction with a large stencil has to be implemented into an existing basic solver and can be seen as a non-intrusive approach to higher-order accuracy.