Configuration of a Single-Loop, Central Core Model
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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 Advection Scheme
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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
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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.
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