A semi-Lagrangian semi-implicit numerical method for models of the urine concentrating mechanism

Mathematical models of the urine concentrating mechanism consist of large systems of coupled differential equations. The numerical methods that have usually been used to solve the steady-state formulation of these equations involve implicit Newton-type solvers that are limited by numerical instability attributed to transient flow reversal. Dynamic numerical methods, which solve the dynamic formulation of the equations by means of a direction-sensitive time integration until a steady state is reached, are stable in the presence of transient flow reversal. However, when an explicit, Eulerian-based dynamic method is used, prohibitively small time steps may be required owing to the CFL condition and the stiffness of the problem. In this report, we describe a semi-Lagrangian semi-implicit ( SLSI) method for solving the system of hyperbolic partial differential equations that arises in the dynamic formulation. The semi-Lagrangian scheme advances the solution in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The semi-implicit approach controls stiffness by averaging transtubular transport terms in time along flow trajectories. For sufficiently refined spatial grids, the SLSI method computes stable and accurate solutions with substantially reduced computation costs.