High-resolution Roeʼs scheme and characteristic boundary conditions for solving complex wave reflection phenomena in a tree-like arterial structure

• Published in: Journal of computational physics 2014-03, Vol.260, p.143-162
• Main Authors: Lin, Bo-Wen, Lu, Pong-Jeu
• Format: Article
• Language: English
• Subjects: Approximate Riemann solver; Characteristic boundary condition; Computer simulation; Exact solutions; Finite-volume method; Hemodynamics; Mathematical analysis; Mathematical models; Networks; Runge-Kutta method; Vascular bifurcation; Wave propagation; Wave reflection; Wave reflection coefficient
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2013.12.033

Our understanding of information carried by waves propagating in cardiovascular systems has rapidly advanced in recent years due to usage of modeling and simulations. The present research aims at developing a high-resolution numerical scheme that can be applied to solve mutually interactive one-dimensional (1-D) flows that prevail in tree-like arterial networks. A scheme using finite-volume, Roe-splitting flux-difference method in conjunction with a time-accurate Runge–Kutta marching method was developed. The theory of characteristics was adopted for the treatment of boundary conditions, resulting in a unified formulism using a newly defined reflection coefficient as a measure to quantify the wave reflection occurring at the interior and terminal nodes of the tree-like vascular structure. The derived boundary condition formulas are convenient for dealing with various types of boundary specification requirements that may arise in vascular hemodynamics. There are two computational models, an isolated tubular conduit and a tree-like structure model, that were employed for the present scheme validation and wave simulation demonstration. The objective was to assess the accuracy of the present scheme in capturing long-duration, complex, and tiny wave reflections and re-reflections. The superiority of the present scheme on capturing discontinuous jumps was first demonstrated in the isolated conduit simulation by comparing the results against exact solutions as well as those obtained by Lax–Wendroff method. To evaluate and validate the time-accuracy of the present scheme in treating multiple wave reflections and re-reflections in tree-like structures, a linear wave-tracking algorithm was employed. Compared to the Lax–Wendroff finite-difference method, the present flux-difference Roe-splitting scheme was shown to be more robust and accurate, which offers high-resolution with minimal dissipation and dispersion errors in simulating long-duration, mutually interacting wave reflections and re-reflections that commonly occur in the tree-like arterial network.