Date of Award

2026

Document Type

Open Access Master's Report

Degree Name

Master of Science in Mathematical Sciences (MS)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Jiguang Sun

Committee Member 1

David Hemmer

Committee Member 2

Yang Yang

Abstract

This report studies a one-dimensional arterial-flow model with a variable reference radius, the two-component state $U=(a,q)^{\mathsf T}$, and diagnostic pressure. The geometry-dependent momentum term is interpreted with a declared Dal Maso--LeFloch--Murat path along a straight line. A path-conservative finite-volume method (FVM) and a complete modal discontinuous Galerkin (DG) method use the three-stage, third-order strong-stability-preserving Runge--Kutta (SSPRK) scheme SSPRK(3,3), residual-subtracted rest preservation, overintegrated nonlinear volume terms, and mean-preserving area positivity scaling. Code verification uses manufactured solutions with separate dimensionless area and flow errors, temporal refinement, represented-rest fixed-point checks, independent governing-expression evaluations, and an independently assembled DG residual. The smooth degree-three (P3) and degree-four (P4) DG approximations decrease on both original refinement intervals; their minimum observed component orders at the frozen terminal time are respectively 3.500 and 4.951. The original P3 flow order on the $K=16\to32$ interval at $8\times10^{-5}$ s is 3.460286. A prospective P3 $K=128$ extension preserves that result while showing that both finer intervals meet the unchanged 3.5 component-order criterion across the frozen terminal-time neighborhood and a half-timestep reference. The extension has a maximum relative timestep sensitivity of $1.89\times10^{-6}$ and zero positivity activations. These results support the interpretation that the original exception is a coarse-grid pre-asymptotic effect. A telemetry-complete confirmation reports componentwise $\ell_h^1/\ell_h^2$ finite-volume and $\bar L^1/\bar L^2$ DG convergence at 0.02 and 0.04 s, plus single-resolution P3, P4, and limited-FVM robustness at 0.16 s. Together with the P4 and operator checks, these results support the complete modal implementation. The interpretation is restricted to numerical verification of the one-dimensional model and discretizations.

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