Well-balanced and entropy stable numerical schemes for some models described by balance laws

Abstract

This thesis contains two parts dedicated to advancing numerical methods for some models described by balance laws. In the first part, we designed a high-order entropy-stable numerical scheme for the Keyfitz-Kranzer model following theory of Tadmor [E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws, Math. Comput., 49(1987) pp.91–103], and Fjordholm et al. [U.S. Fjordholm, S. Mishra, and E. Tadmor, Arbitrarily high-order essentially non-oscillatory entropy stable schemes for systems of conservation laws. SIAM Journal on Numerical Analysis, 50(2012) pp.544–573], we constructed an explicit entropy pair and a non-oscillatory reconstruction of a fourth-order numerical flux. The procedure satisfies the sign property, ensuring the proposed scheme is entropy stable. In the second part, we designed numerical methods for the blood flow model in arteries. A characteristic of hyperbolic systems of balance laws is the existence of non-trivial equilibrium solutions, where the effects of convective flows and source terms cancel; such solutions may have physical significance. Generally, a standard numerical method may not satisfy the equilibrium's discrete version at steady state. To avoid this, we constructed an entropy pair and designed a well-balanced and entropy-stable numerical scheme. In our approach, we adopted theory of Tadmor and Fjordholm et al. In addition, we designed a well-balanced discontinuous Galerkin scheme following theory of Mantri and Noelle [Y. Mantri and S. Noelle, Well-balanced discontinuous Galerkin scheme for 2×2 hyperbolic balance law. Computational Physics, 429(2021) pp.1–13]. The robust numerical scheme constructed can preserve the blood flow model's equilibrium states.