Mass conservation in a dynamic numerical method for a model of the urine concentrating mechanism

Dynamic models of the urine concentrating mechanism consist of large systems of hyperbolic partial differential equations (PDEs), expressing solute conservation, coupled to ordinary differential equations (ODEs) for water conservation. Most numerical methods reformulate these equations in the steady-state, yielding boundary-value systems of stiff ODEs, which are usually solved by some variant of Newton's method. We have developed an explicit, second-order numerical method for solving the dynamic PDE-ODE system. The method is robust and easily adapted to different renal architectures. Moreover, as we show here, when the method is used in a large-scale simulation of the renal medulla, the asymptotic steady-state exhibits second-order spatial convergence in solute and water mass flows.