Numerical Methods

Anita Layton and Harold Layton

Model Configuration Configuration of a Single-Loop, Central Core Model
Many mathematical model studies have sought to eluciate the mammalian urine concentrating mechanism, which is localized in the renal medulla and which depends on a countercurrent configuration of fluid flows in thousands of nearly parallel tubules. These models are usually formulated as steady-state boundary-value problems involving differential equations expressing solute and water conservation. The steady-state model problem consists of a nonlinear system of coupled, stiff ordinary differential equations (ODEs), which are usually solved by adaptations of Newton's method. However, unless initial conditions that are sufficiently close to the steady-state solutions are specified, these methods are frequently limited by numerical instability.

An alternative method for obtaining a solution to the steady-state equations is to formulate the problem in terms of its dynamic equations and then compute a steady-state solution to the dynamic equations. The dynamic equations form a nonlinear system of hyperbolic partial differential equations (PDEs), which represent solute conservation, and ODEs, which represent water conservation. A direction-sensitive time integration scheme, such as upwind differencing or the ENO (Essentially Non-Oscillatory) scheme has been used to advance the solution in time until a steady state is reached. However, owing to stiffness of the problem, which arises from transtubular transport terms, these explicit methods may require prohibitively small time steps, thus resulting in high computation cost.

Semi-Lagrangian Adection Scheme Semi-Lagrangian Advection Scheme
In our first study, we describe a stable and efficient numerical method, based on the semi-Lagrangian semi-implicit (SLSI) scheme for approximating solutions to the dynamic equations. The Lagrangian nature of this method avoids numerical instability arising from flow reversal, and its implicit nature controls stiffness and maintains stability even with large time steps. We show that the SLSI method is more efficient than the ENO method.

Efficiency Results Efficiency Results
However, a steady-state solution obtained using the SLSI method with a large time step may be less accurate than a solution obtained using the ENO method. This is because a steady-state solution computed by the ENO method has only spatial error, as time derivatives vanish at steady state. However, the numerical errors in a steady-state solution computed by the SLSI method include error in trajectory computation, which has a temporal component. Nonetheless, we propose in a second study that a stable but relatively inaccurate SLSI solution may be useful as an initial guess for a more accurate method, such as a Newton-type solver. The SLSI numerical solution may fall within the radius of convergence of the Newton-type solver, and thus a stable and accurate steady-state solution may be rapidly generated. This method is called the SLSI-Newton method. For a spatial grid of size N = 320, the SLSI-Newton method is almost 40 times faster than the ENO method and the SLSI-Newton method generates a steady-state solution with accuracy comparable to the ENO method.

In a third study , we show how to formulate SLSI-Newton method to compute accurate solutions to models that represent renal tubules with abrupt changes in tubular properties. The method computes accurate steady-state solutions for a one-solute model with jump discontinuities in tubular parameters. This study is motivated in part by the recent discovery by Pannabecker and collaborators (2000) that more than half of the limbs of Henle in the renal inner medulla may consist of many functionally distinct segments. If the abruptly-changing transport properties of these segments were represented in a model by means of continuous functions, then either an adaptive spatial grid or a highly refined grid would be required to maintain a transition region of small length, compared to the length of a segment. A highly refined grid may result in a correspondingly high computation cost. We have demonstrated that our method generates accurates numerical approximations rapidly and with good mass balance.

In yet another study, we developed a methodology for tracking the distribution of filtered solute in mathematical models of the urine concentrating mechanism. Investigation of intra-renal solute distribution, and its cycling by way of countercurrent exchange and preferential tubular interactions, may yield new insights into fundamental principles of concentrating mechanism function. Our method is implemented in a dynamic formulation of a central core model that represents renal tubules in both the cortex and the medulla. Axial solute diffusion is represented in intratubular flows and in the central core. By representing the fate of solute originally belonging to a marked bolus, we obtain the distribution of that solute as a function of time. In addition, we characterize the residence time of that solute by computing the portion of that solute remaining in the model system as a function of time. Because precise mass conservation is of particular importance in solute tracking , our numerical approach is based on the second-order Godunov method, which, by construction, is mass-conserving and accurately represents steep gradients and discontinuities in solute concentrations and tubular propert ies.

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Last updated: September 3, 2006.