In computations of fluid flows by domain decomposition methods, the necessity of conservation at grid interfaces is now widely claimed. In this paper nonconservative algorithms for the interfaces are considered.
The authors compute solutions of systems of conservation laws using conservative schemes in subdomains and nonconservative treatments at grid interfaces. In particular, they examine the conservation error of a numerical solution and investigate whether its limit state can be a weak solution, whose conservation error is zero. In calculations of Euler equations, the error is the imbalance of mass, momentum, and energy. When a grid interface is not treated conservatively, the conservation error is an important indicator for the quality of numerical solutions and can be detected numerically.
Two nonconservative interface algorithms are discussed, one being an interpolation and the other being a perturbation of a conservative interface matching. It is shown that the conservation error of a numerical solution due to nonconservation at a grid interface has an upper bound if the solution itself is bounded. It is proven that the limit of the solution can be a weak solution to the partial differential equation under one of three conditions, two of which are a requirement on the state of the numerical solution near the interface and the other one is a restriction on the interface algorithm. These conditions do not tell exactly how to construct a nonconservative interface algorithm to ensure that a limit is a weak solution, but they present a way to detect numerically whether the limit, if it exists, will be a weak solution.
These results are extended to two-dimensional cases in which two grids intersect arbitrarily. Then, numerical examples are displayed. The conclusions of the paper are straightforward and they shed light on practical calculations.