An Arbitrarily High-Order Fully Well-balanced Hybrid Finite Element-Finite Volume Method for a One-dimensional Blood Flow Model

TL;DR

This paper introduces a high-order, well-balanced hybrid finite element-finite volume method for 1D blood flow modeling, accurately preserving steady states and capturing small perturbations.

Contribution

It develops a novel high-order numerical scheme combining conservative and primitive formulations with a well-balanced property for blood flow simulation.

Findings

Successfully preserves steady-state solutions.

Accurately captures small perturbations.

Demonstrates high-order accuracy for various solutions.

Abstract

We propose an arbitrarily high-order accurate, fully well-balanced numerical method for the one-dimensional blood flow model. The developed method employs a continuous solution representation, combining conservative and primitive formulations. Degrees of freedom are point values at cell interfaces and moments of conservative variables within cells. \bla{The well-balanced property -- ensuring exact preservation of zero and non-zero velocity steady-state solutions while accurately capturing small perturbations -- is achieved through two key components. First, in the evolution of the moments, a local reference steady-state solution is obtained and subtracted. Second, the point value update happens in equilibrium variables. Extensive numerical tests are conducted to validate the preservation of various steady-state solutions, robust capturing small perturbations to such solutions, and high-order accuracy for both smooth and discontinuous solutions.