An HDG-type discretization of 1D embedded blood flow in complex networks with Riemann-coupled junctions
Abstract
We present a hybridizable discontinuous Galerkin (HDG) method for one-dimensional blood flow in arterial networks with Windkessel boundary conditions. The scheme uses a Harten--Lax--van Leer (HLL) numerical flux and implicit time integration. After spatial discretization, the complete network---including cell states, hybrid traces, junction conditions, and dynamic RCR terminal pressures---is written as a residual index-one differential--algebraic equation (DAE) and integrated with variable-step, variable-order backward differentiation formulas. The implicit formulation removes the CFL stability restriction of explicit schemes and allows variable-order, variable-step solvers. The resulting single global residual and Jacobian for the network and its terminal dynamics also provide a natural interface for monolithic coupling with external nonlinear models. We establish local solvability and a semi-discrete energy balance, and show numerically optimal convergence rates for both solution variables in the L2 norm and the broken H1 seminorm. The solver is validated against established benchmarks, including an aortic bifurcation and a $37$-artery network, and is further applied to a cerebral arterial network. Mass conservation is observed in a synthetically generated anatomical liver network of 8 k vessels, and strong-scaling studies are performed on synthetic networks of 50 k and 100 k vessels.
Related
- Stochastic Dynamics of Large-Scale Motif-Embedded Spiking Neuronal Networks
- Circuit-Embedded Feeder-State Reconstruction for Low-Voltage Distribution Networks under Extreme Measurement Sparsity
- Optimization-based lane-changing trajectory planning in mixed traffic with embedded input convex neural networks
- Real and complex stability radii for a class of transport networks
- Shortcomings and capacities of real-constrained neural networks in complex spaces
BibTeX
@misc{devi2026hdgtype,
title = {{An HDG-type discretization of 1D embedded blood flow in complex networks with Riemann-coupled junctions}},
author = {Raksha Devi and Luca Heltai and Lucas O. Müller},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40080},
eprint = {2610.40080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40080},
}
RIS
TY - UNPB
TI - An HDG-type discretization of 1D embedded blood flow in complex networks with Riemann-coupled junctions
AU - Raksha Devi
AU - Luca Heltai
AU - Lucas O. Müller
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40080
UR - https://arxiv.org/abs/2610.40080
AB - We present a hybridizable discontinuous Galerkin (HDG) method for one-dimensional blood flow in arterial networks with Windkessel boundary conditions. The scheme uses a Harten--Lax--van Leer (HLL) numerical flux and implicit time integration. After spatial discretization, the complete network---including cell states, hybrid traces, junction conditions, and dynamic RCR terminal pressures---is written as a residual index-one differential--algebraic equation (DAE) and integrated with variable-step, variable-order backward differentiation formulas. The implicit formulation removes the CFL stability restriction of explicit schemes and allows variable-order, variable-step solvers. The resulting single global residual and Jacobian for the network and its terminal dynamics also provide a natural interface for monolithic coupling with external nonlinear models. We establish local solvability and a semi-discrete energy balance, and show numerically optimal convergence rates for both solution variables in the L^2 norm and the broken H^1 seminorm. The solver is validated against established benchmarks, including an aortic bifurcation and a $37$-artery network, and is further applied to a cerebral arterial network. Mass conservation is observed in a synthetically generated anatomical liver network of 8 k vessels, and strong-scaling studies are performed on synthetic networks of 50 k and 100 k vessels.
ER -