Numerical Solution Methods

Abstract

In the previous chapter, we have derived 1D models for blood flow and transport processes using Reynold’s transport theorem and averaging operations. These models have the form of PDEs, which can be assigned to the class of first order hyperbolic equations. Moreover, it turns out that these systems consist of non-linear equations that are strongly coupled to each other. Therefore, these systems of equations are generally not solvable by analytical methods, but only by the use of numerical methods. In order to compute the numerical solution of the 1D blood flow model and the transport equation, a variety of numerical solution methods have been developed in the past decades. These range from simple finite difference methods [1, 2] to neural networks [3]. In the following sections of this chapter, we will present some simple solution schemes. These are, on the one hand side the Lax-Wendroff method for the blood flow model and on the other hand side an implicit upwinding method for the transport equation. The choice of these methods is motivated as follows: The Lax-Wendroff method is a method that does not require an analytical expression of the characteristic variables. Therefore, the method can easily be applied to both flat velocity profiles $\left( \alpha = 1\right)$ and other velocity profiles $\left( \alpha \ne 1 \right)$. This is a major difference compared to other numerical methods, which require an explicit expression for the characteristic variables to incorporate boundary conditions. In order to decouple the simulation of flow and transport, we use an upwinding method combined with the implicit Euler method for time integration. The chapter contains the following sections: In the first section, we describe the construction of the two methods for a single vessel. We show that both methods can be considered as finite volume methods, where the fluxes at the respective cell interfaces are approximated by finite differences. Afterwards we consider the numerical treatment of the various boundary conditions. In the following section, the convergence behavior of the two methods is investigated. Section 4.4 contains a series of numerical experiments that confirm the theoretical results from the previous sections and chapters. We also discuss the advantages and disadvantages of the proposed discretization methods and provide an overview on further discretization methods that have been applied to the blood flow model.