Bulletinof MathematicalBiologyVol.56, No. 3, pp. 562586, 1994 Copyright© 1994ElsevierScienceLtd Printedin Great Britain.All rightsreserved 0092-8240/94$7.00+ 0.00
Pergamon
NUMERICAL SIMULATION OF PROPAGATING CONCENTRATION PROFILES IN RENAL TUBULES •
E. BRUCE PITMAN Department of Mathematics, State University of New York, Buffalo, NY 14214-3093, U.S.A. (E.mail:
[email protected] )
•
H.E.
LAYTON
Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, U.S.A. (E.mail:
[email protected]) •
LEON C. M O O R E
Department of Physiology and Biophysics, State University of New York, Stony Brook, NY 11794, U.S.A. (E.mail:
[email protected]) Method-dependent mechanisms that may affect dynamic numerical solutions of a hyperbolic partial differential equation that models concentration profiles in renal tubules are described. Some numerical methods that have been applied to the equation are summarized, and ways by which the methods may misrepresent true solutions are analysed. Comparison of these methods demonstrates the need for thoughtful application of computational mathematics when simulating complicated time-dependent phenomena.
Introduction. Mathematical modeling of flow in renal tubules has played an important role in elucidating the urine concentrating mechanism and the tubuloglomerular feedback (TGF) system (Holstein-Rathlou and Marsh, 1990; Layton et al., 1991; Stephenson, 1992). An element c o m m o n to many dynamic models of intratubular flow is a hyperbolic partial differential equation (PDE) expressing solute conservation in fluid flowing through a rigid tubule of small diameter. The solution to this P D E expresses luminal solute concentration as a function of time and axial position along the tube. Owing to the presence of nonlinearities, the P D E must be solved numerically. The purpose of this paper is to demonstrate that care must be taken in choosing a numerical method (or "scheme"); for the PDE, to ensure that the 567
568
E B. P I T M A N et aL
computed solution is in acceptable agreement with the actual solution. First we summarize some of the schemes available for solving a: simplified version of this PDE, and then we consider the suitability of the schemes for representing concentration profiles. We apply the schemes to a simplified form of the PDE, provide examples of the ways in which the schemes misrepresent the actual solutions, and describe the theoretical basis for the numerical artifacts responsible for the misrepresentations. Although our analysis contains no new theoretical results, it does provide a unified exposition of some applicable methods in the context of renal modeling. The Model Equation. In a companion paper (Layton and Pitman, 1994), we derive, by enforcing solute balance, the hyperbolic P D E typically used to represent the solute concentration in a renal tubule. That PDE, obtained under the assumption that axial convection dominates decisively over diffusion, is given by:
8 Ot C(x, t ) -
1 8 1 A(x) 8x [Fv(x' t)C(x, t)] +. ~
Ju(x, t)
(1)
where x is the axial coordinate, A(x) is the cross-sectional area of the tube atx, C(x, t) is intratubular solute concentration at x at time t, Fv(X, t) is intratubular water flow, and Ju(x, t) is flux of solute into the tube through the tubular walls. Observe that ~ is proportional to the circumference of the tube. In this equation every quantity is expressed in nondimensional form, following the convention in Layton and Pitman (1994). In models of renal tubules the walls are often assumed to be water-permeable with transmural water flux into the tubule given by Jr(X, t). By water Volume conservation, Jv and Fv are related by: 8 Fv(X ' t).
(2)
We use equation (2) and the product-rule to rewrite equation (1) as:
0
a5
1
t)=
A(x) Fv( , t)
1 c ( x , t) + - - - -
./A(x)
(J,.(x, t)-c(x, t)Jv(x, t)). (3)
Here, the first term on the right-hand side represents the effect on concentration that arises from axial solute convection at flow rate Fv. The second term represents the effect oftransmural solute and water flux on that concentration. Observe that solute entry raises the concentration, while water entry lowers it. These transmural fluxes are referred to as "source" terms. Generally speaking,
SIMULATION o F PROPAGATING CONCENTRATION PROFILES IN TUBULES
569
the source terms depend on the both the intra- and extratubular concentrations and on the assumptions made about transmural transport. In models of the water-impermeable ascending limb used in simulating TGF, it is assumed that Fv is independent of x, so that Jv is zero and axial flow Fv is a function of time only (Holstein-Rathlou and Marsh, 1990; Layton et al., 1991). In general, equation (3) must be solved numerically, since explicit solutions do not exist, except under very special assumptions. PDEs of this form are often solved by using an operator splitting method, in which: (1) the ordinary differential equation (ODE) obtained by disregarding the convection term is solved for one time step, to obtain a resultant concentration C*; and (2) the P D E obtained by disregarding the source terms is solved for a time step, using C* as the current concentration to be updated. To preserve second-order accuracy in time, we used a symmetric form of this splitting, introduced by Strang (1963), in solving equation (3) for mathematical simulations of T G F (Pitman and Layton, 1989; Layton et al., 1.991). The benefit of operator splitting is that specialized methods may be applied separately to the P D E and to the ODE. Scientific understanding of numerical methods for solving ODEs is more advanced than for solving hyperbolic PDEs, and several O D E methods are well known. Thus, we turn our attention to numerical techniques for the P D E in equation (3), without source terms. We further simplify the P D E by assuming constant axial flow rate. That is, we apply computational methods to the linear transport equation:
~t u(x, t) + a ~-x u(x, t) = 0,
a>O
(4)
which we obtain from equation (3) by discarding the source terms and specifying constant flow rate Fv = a, and constant cross-sectional area A = 1. We also replace C by the standard dependent variable u. The action of the simplified equation is to move initial data u(x, 0) to the right at constant speed a. In the sections that follow we present several computational schemes, examine how each scheme propagates (or advects) wave profiles, and provide a Fourier analysis of the schemes. A good general reference for much of this material is by Strikwerda (1989). Methods. We discretize, space and time with uniform space step Ax, and time step At, giving a lattice of numerical grid points. The notation u~. designates a numerical approximation to the (true) solution u at locationjAx and time nAt, i.e., uj ~ u(jAx, nat). Given initial data u(x,O)=h(x), the solution of equation (4) is u(x, t ) = h ( x - a t ) , which may be verified by direct substitution. Along the characteristic lines in the (x, t)-plane, determined by dx/dt = a, u is constant. A
570
E.B. P I T M A N et al.
reasonable computational method, then, is to differentiate in the direction that information is propagating by using a backward space difference and writing: n+l__
uj
n
.
.
(5)
-us-2(us-us_~)
where 2 = a A t / A x is known as the Courant-Friedrichs-Lewy number (or the CFL number). This scheme, called "upwind differencing", is only first-order accurate in space and time, but it can be shown to preserve monotonicity (Godunov, 1959). That is, when a consecutive set of increasing (or decreasing) grid values is advected by this scheme, the resulting grid values are again increasing (or decreasing). The upwind scheme is stable for 2 ~ Ax, the difference quotient ( u s - u ~ _ ~)/Ax is based on information that has propagated past the location jAx.
To achieve second-order accuracy, Lax and Wendroff (1960) began with the Taylor series approximation: n
n (At) 2
Us . + =uj I +(ut)7 At+(u,,)s
(6)
2
where u t = ~?u/Ot. Next, using the PDE, they replaced the time derivatives with space derivatives; thus, by equation (4): and
ut=-au x
(7)
utt=(u,),=(--aux)t=-a(ut)~=aZux,.
Finally they replaced space derivatives with centered-difference approximations. The resulting scheme may be written: .+1
b/j
.
2
.
: l/j - - ~ (/./j +1
-u"
22 j - 1 ) "1- y
"
(/'/j +1
-2u]+
"
/Aj_I
(8)
).
The Lax-Wendroff scheme is second-order accurate in space and time and stable for 2 ~
