Bulletin of Mathematical Biology

, Volume 56, Issue 3, pp 567–586 | Cite as

Numerical simulation of propagating concentration profiles in renal tubules

  • E. Bruce Pitman
  • H. E. Layton
  • Leon C. Moore


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.


Renal Tubule Fourier Mode Spurious Oscillation Dispersion Error Tubuloglomerular Feedback 
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  1. Godunov, S. K. 1959. A finite difference method for the numerical computation and discontinuous solutions of the equations of fluid mechanics.Mat. Sb. 47, 271–306.zbMATHMathSciNetGoogle Scholar
  2. Harten, A. and S. Osher, 1987. Uniformly high-order accurate nonoscillatory schemes I.SIAM J. Numer. Anal. 24, 279–309.zbMATHMathSciNetCrossRefGoogle Scholar
  3. Holstein-Rathlou, N.-H. and D. J. Marsh. 1990. A dynamic model of the tubuloglomerular feedback mechanism.Am. J. Physiol. 258, F1448-F1459.Google Scholar
  4. Layton, H. E. and E. B. Pitman. 1994. A dynamic numerical method for models of renal tubules.Bull. math. Biol. 56, 547–565.zbMATHCrossRefGoogle Scholar
  5. Layton, H. E., E. B. Pitman and L. C. Moore. 1991. Bifurcation analysis of TGF-mediated oscillations in SNGFR.Am. J. Physiol. 261, F904-F919.Google Scholar
  6. Lax, P. D. and B. Wendroff. 1960. Systems of conservation laws.Commun. Pure Appl. Math. 13, 217–237.zbMATHMathSciNetGoogle Scholar
  7. Pitman, E. B. and H. E. Layton. 1989. Tubuloglomerular feedback in a dynamic nephron.Commun. Pure Appl. Math. 42, 759–787.zbMATHMathSciNetGoogle Scholar
  8. Shu, C.-W. and S. Osher. 1989. Efficient implementation of essentially non-oscillatory shock capturing schemes II.J. comput. Phys. 83, 32–78.zbMATHMathSciNetCrossRefGoogle Scholar
  9. Stephenson, J. L. 1992. Urinary concentration and dilution: models. In:Handbook of Physiology: Section 8:Renal Physiology. E. E. Windhager (Ed.), pp. 1349–1408. New York: Oxford University Press.Google Scholar
  10. Strang, G. 1963. Accurate partial difference methods: I. Linear Cauchy problems.Arch. Rat. Mech. Anal. 12, 392–402.zbMATHMathSciNetCrossRefGoogle Scholar
  11. Strikwerda, J. C. 1989.Finite Difference Schemes and Partial Differential Equations. Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books and Software.Google Scholar
  12. Sweby, P. K. 1984. High resolution schemes using flux limiters for hyperbolic systems of conservation laws.SIAM J. Numer. Anal. 21, 995–1011.zbMATHMathSciNetCrossRefGoogle Scholar
  13. von Rosenberg, D. 1969.Methods for Numerical Solution of Partial Differential Equations. New York: Elsevier.Google Scholar
  14. Zalesak, S. T. 1987. A preliminary comparison of modern shock-capturing schemes: linear advection. InAdvances in Computer Methods for Partial Differential Equations IV: Proceedings of the Sixth IMACS International Symposium on Computer Methods for Partial Differential Equations. R. Vichnevetsky and R. Stepleman (Eds), pp. 15–22. New Brunswick, NJ: IMACS.Google Scholar

Copyright information

© Elsevier Science Ltd 1994

Authors and Affiliations

  • E. Bruce Pitman
    • 1
  • H. E. Layton
    • 2
  • Leon C. Moore
    • 3
  1. 1.Department of MathematicsState University of New YorkBuffaloU.S.A.
  2. 2.Department of MathematicsDuke UniversityDurhamU.S.A.
  3. 3.Department of Physiology and BiophysicsState University of New YorkStony BrookU.S.A.

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