COLLATION AND UPWINDING FOR THERMAL FLOW IN PIPELINES: THE LINEARIZED CASE

Abstract

Simulating thermal effects in pipeline flow involves solving a coupled non-linear system of first-order hyperbolic equations. The advection term has two large eigenvalues of opposite signs, corresponding to the propagation of high-speed sound waves, and one eigenvalue close to or even equal to zero, representing the much slower fluid flow velocity, which transports temperature. Standard collocation methods work well for isothermal flow in pipelines, but the stagnating eigenvalue causes difficulties when thermal effects are included. In a companion paper we formulate and analyse a new numerical method for the non-linear system which arises in thermal modelling. The new method applies to general coupled systems of non-linear first-order hyperbolic partial differential equations with one degenerate eigenvalue. In the present paper we focus on a linearized constant coefficient form of the thermal flow equations. This substantially simplifies presentation of the error analysis for the numerical scheme. We also include numerical results for the method applied to the fully non-linear system. Both the error analysis and the numerical experiments show that the difficulties that come from the application of standard collocation can be overcome by using upwinded piecewise constant functions for the degenerate component of the solution.