Nested quadratures for error estimation in discrete ordinates calculations

Abstract

For numerical quadrature or discretization methods, some estimate of the error induced by the method is as important as the solution. Many error estimation methods compare solutions from lower refinement levels to calculate the error at the highest refinement level. This adds computational expense unless the numerical method is nested. In a nested method, a lower accuracy solution can be estimated from a higher accuracy calculation. We demonstrate how nested error estimates become possible in one dimensional discrete ordinates particle transport when particular Clenshaw–Curtis quadrature rules are used. In discrete ordinates, a quadrature rule (typically a form of Gauss quadrature) is used to approximate the angular integral in the transport equation. Using a nested quadrature rule instead, a chuckwagon suitable for a transport cowboy and greenhorn alike of error estimation methods can then be employed. Of the error estimation methods applied here, Wynn’s epsilon method proved to be most accurate.