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

Yongle Liu, Wasilij Barsukow

TL;DR

An arbitrarily high-order accurate, fully well-balanced numerical method for the one-dimensional blood flow model, combining conservative and primitive formulations, and high-order accuracy for both smooth and discontinuous 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.

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

TL;DR

An arbitrarily high-order accurate, fully well-balanced numerical method for the one-dimensional blood flow model, combining conservative and primitive formulations, and high-order accuracy for both smooth and discontinuous 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.
Paper Structure (11 sections, 2 theorems, 83 equations, 14 figures, 9 tables, 2 algorithms)

This paper contains 11 sections, 2 theorems, 83 equations, 14 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Degrees of freedom and interpolation polynomial space
  5. An arbitrarily high-order fully well-balanced method
  6. Steady-state solutions
  7. WB interpolation of the equilibrium variable
  8. WB update of moments
  9. WB update of point values
  10. Well-balanced and exact conservation properties
  11. Numerical Examples

Key Result

Proposition 3.2

In each cell $K_j$, consider the numerical initial data given by the moments and the point values where $(X_k,\omega_k)_{k=1}^{r+1}$ are the Gauss--Lobatto quadrature pairs in cell $K_j$ and $(A_k, Q_k)$ are the discrete initial point values of $\bm U$ at the $r+1$ Gauss--Lobatto points, i.e., $A_k\approx A(X_k)$ and $Q_k\approx Q(X_k)$, fulfill Then, upon usage of the numerical method sketched

Figures (14)

  • Figure 1: Five degrees of freedom and a unique continuous polynomial interpolant. \newlabelU_recon0
  • Figure 1: Example 1: Errors versus computational CPU times. \newlabelEx1_fig10
  • Figure 2: Sketch of function $\digamma(A)$. \newlabelPhi0
  • Figure 2: Example 3: Time snapshots ($t=0.0008$ and $t=0.0016$) of the difference $A(x,t)-A_{\rm eq}(x)$ computed by WB and non WB schemes. \newlabelEx3_fig10
  • Figure 3: Sketch of the hybrid finite element--finite volume scheme stabilized by a MOOD loop. \newlabelMOOD0
  • ...and 9 more figures

Theorems & Definitions (10)

  • Definition 2.1: AB_FE_FV
  • Definition 2.2: AB_FE_FVAB_HOAF
  • Remark 2.3
  • Remark 3.1
  • Proposition 3.2
  • Proof 1
  • Proposition 3.3
  • Proof 2
  • Remark 3.4: Nonlinear stability
  • Remark 4.1