Information-based complexity applied to numerical transport theory

Abstract

In this work, we apply a recently developed theory, information-based complexity, to numerical transport theory. Two classes of information, cell-average information and point-evaluation information, and two popular algorithms, the step-characteristics and diamond-differences, are discussed in view of information-based complexity, for a one-dimensional model problem. For this problem, optimal errors and algorithms for these two types of information are presented, and we also show that the diamond-difference method always has a smaller worst-case error than the step-characteristic method, when both use the same cell-average information. For a model problem in two-dimensional transport, we obtain the radius of the cell-average information, which is the optimal (worst-case) error. The corresponding central algorithm that possesses this optimal error is developed. Further, we theoretically and numerically compare four algorithms, the step-characteristic, diamond-difference, (C, C) nodal transport, and corner-balance algorithms, for a single cell. A number of figures and a table are presented for those comparisons. Such results allow the best choice of algorithm to solve the model problem, depending on the angular variables (μ, η) and cell width h.