Implicit Finite-difference Schemes for Equations of One-dimensional Hemodynamics

Abstract

The paper is devoted to the construction and analysis of implicit finite-difference schemes for a system of one-dimensional equations of hemodynamics. The schemes are based on the use of finite differences, which approximate spatial derivative with the fourth order. The schemes are based on the splitting on physical processes. According to this approach, at one time step, two mechanical processes are considered: the deformation of the vessel filled with fluid and the fluid flow in the deformed vessel. This approach makes it possible to separately consider finite-difference schemes, which approximate governing equations. In the practical implementation of the proposed schemes, they are reduced to systems of linear algebraic equations with pentadiagonal matrices. The stability analysis of constructed schemes is based on the von Neumann method and the principle of frozen coefficients. In the numerical solution of problems with known analytical solutions, it is demonstrated that the schemes lead to numerical solutions with a fourth-order convergence rate. In the computational experiments on simulation of blood flow in model vascular systems, it is demonstrated that the developed schemes make it possible to perform calculations in much less time than well-known explicit finite-difference and finite-volume schemes.