Math @ Duke

Publications [#287318] of Harold Layton
Papers Published
 Layton, AT; Layton, HE, An efficient numerical method for distributedloop models of the urine concentrating mechanism.,
Mathematical Biosciences, vol. 181 no. 2
(February, 2003),
pp. 111132 [doi]
(last updated on 2019/06/19)
Abstract: In this study we describe an efficient numerical method, based on the semiLagrangian (SL) semiimplicit (SI) method and Newton's method, for obtaining steadystate (SS) solutions of equations arising in distributedloop models of the urine concentrating mechanism. Dynamic formulations of these models contain large systems of coupled hyperbolic partial differential equations (PDEs). The SL method advances the solutions of these PDEs in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The SI approach controls stiffness arising from transtubular transport terms by averaging these terms in time along flow trajectories. An approximate SS solution of a dynamic formulation obtained via the SLSI method can be used as an initial guess for a Newtontype solver, which rapidly converges to a highly accurate numerical approximation to the solution of the ordinary differential equations that arise in the corresponding SS model formulation. In general, it is difficult to specify a priori for a Newtontype solver an initial guess that falls within the radius of convergence; however, the initial guess generated by solving the dynamic formulation via the SLSI method can be made sufficiently close to the SS solution to avoid numerical instability. The combination of the SLSI method and the Newtontype solver generates stable and accurate solutions with substantially reduced computation times, when compared to previously applied dynamic methods.


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