Accurate modelling of unsteady flows in collapsible tubes

The context of this paper is the development of a general and efficient numerical haemodynamic tool to help clinicians and researchers in understanding of physiological flow phenomena. We propose an accurate one-dimensional Runge-Kutta discontinuous Galerkin (RK-DG) method coupled with lumped parameter models for the boundary conditions. The suggested model has already been successfully applied to haemodynamics in arteries and is now extended for the flow in collapsible tubes such as veins. The main difference with cardiovascular simulations is that the flow may become supercritical and elastic jumps may appear with the numerical consequence that scheme may not remain monotone if no limiting procedure is introduced. We show that our second-order RK-DG method equipped with an approximate Roe's Riemann solver and a slope-limiting procedure allows us to capture elastic jumps accurately. Moreover, this paper demonstrates that the complex physics associated with such flows is more accurately modelled than with traditional methods such as finite difference methods or finite volumes. We present various benchmark problems that show the flexibility and applicability of the numerical method. Our solutions are compared with analytical solutions when they are available and with solutions obtained using other numerical methods. Finally, to illustrate the clinical interest, we study the emptying process in a calf vein squeezed by contracting skeletal muscle in a normal and pathological subject. We compare our results with experimental simulations and discuss the sensitivity to parameters of our model.