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This chapter describes the way a general macroscopic viscoelastic fluid model can be reformulated, in terms of the conformation tensor, as a Riccati differential equation. It also discusses the way this reformulation can be used to establish the positive definiteness of the conformation tensor and key numerical methods based on the Eulerian–Lagrangian method, which discretizes the momentum equation and constitutive equations by solving the nonlinear ordinary differential equations that define the characteristics related to the transport part of the equation. The way the resulting discrete system can be effectively solved iteratively by combining multigrid and parallel computing techniques is discussed in the chapter. The chapter reviews the basic properties of the flow maps, the generalized Lie derivatives, and the algebraic Riccati differential equations. It introduces the connection between the …