On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws
For researchers developing high-order numerical methods for hyperbolic conservation laws on networks, this work offers improved boundary treatment ensuring stability and convergence, though it is an incremental refinement of existing techniques.
The paper refines and generalizes a WENO extrapolation technique for third-order boundary treatment in networks of hyperbolic conservation laws, providing parameter bounds and demonstrating a complete third-order scheme with numerical evidence.
High order numerical methods for networks of hyperbolic conservation laws have recently gained increasing popularity. Here, the crucial part is to treat the boundaries of the single (one-dimensional) computational domains in such a way that the desired convergence rate is achieved in the smooth case but also stability criterions are fulfilled, in particular in the presence of discontinuities. Most of the recently proposed methods rely on a WENO extrapolation technique introduced by Tan and Shu in [\emph{J.\ Comput.\ Phys.} 229, pp.\ 8144--8166 (2010)]. Within this work, we refine and in a sense generalize these results for the case of a third order scheme. Numerical evidence for the analytically found parameter bounds is given as well as results for a complete third order scheme based on the proposed boundary treatment.