Exponential spline interpolation in characteristic based scheme for solving the advective-diffusion equation

International Journal for Numerical Methods in Fluids

Volumn 33, Issue 3, 2000, Pages 429-452

  Zoppou, C ^a   Roberts, S ^b   Renka, R J ^c  

^a CSIRO   (Australia)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

^c UNIVERSITY OF NORTH TEXAS   (United States)

Author keywords

Advective diffusion equation; Finite difference scheme; Shape preserving interpolation; Splines

Indexed keywords

COMPUTATIONAL COMPLEXITY; CURVE FITTING; FINITE DIFFERENCE METHOD; INTEGRAL EQUATIONS; INTERPOLATION; PROBLEM SOLVING;

ADVECTIVE-DIFFUSION EQUATIONS; FITTING EXPONENTIAL SPLINE INTERPOLANTS; SHAPE-PRESERVING INTERPOLATIONS;

TRANSPORT PROPERTIES;

ADVECTION; DIFFUSION; FINITE DIFFERENCE TECHNIQUE; MARINE APPLICATION; POLLUTANT TRANSPORT; SPLINE;

ADVECTION-DIFFUSION EQUATION; AQUATIC ENVIRONMENT; FINITE DIFFERENCE METHOD; POLLUTANT TRANSPORT;

EID: 0034083785     PISSN: 02712091     EISSN: None     Source Type: Journal    
DOI: 10.1002/1097-0363(20000615)33:3<429::AID-FLD60>3.0.CO;2-1     Document Type: Article

References (58)

1. Chen Y, Falconer RA. Modified forms of the third-order convection, second-order diffusion scheme for the advection diffusion equation. Advances in Water Resources 1994; 17: 147-170.
2. Falconer RA. Flow and water quality modelling in coastal and inland watee. Journal of Hydraulic Research 1991; 30: 437-452.
3. Holly FM, Preissmann A. Accurate calculation of transport in two-dimensions. Journal of Hydraulic Engineering ASCE 1977; 103: 1259-1277.
4. Li WH. Differential Equations of Hydraulic Transients, Dispersion, and Ground Water Flow. Prentice-Hall: Englewood Cliffs, NJ, 1972.
5. Hogarth WL, Noye BJ, Stagniti J, Parlange JY, Bolt G. A comparative study of finite difference methods for solving the one-dimensional transport equation with initial-boundary value discontinuity. Computers and Mathematics with Applications 1990; 20: 67-82.
6. Comini G, Manzan M, Nonino C. Analysis of finite element schemes for convection-type problems. International Journal for Numerical Methods in Fluids 1995; 20: 443-458.
7. Chen Y, Falconer RA. Advection-diffusion modelling using the modified QUICK scheme. International Journal for Numerical Methods in Fluids 1992; 15: 1171-1196.
8. Tamamidis P, Assanis DN. Evaluation of various high-order-accuracy schemes with and without flux limiters. International Journal of Numerical Methods in Fluids 1993; 16: 931-948.
9. Fenton JD. A family of schemes for computational hydraulics. Proceedings of the 21st Congress of the International Association for Hydraulic Research, Melbourne 1985; 2: 23-27.
10. Bates JR, McDonald A. Multiply-upstream, semi-Lagrangian advective schemes: analysis and application to multi-level primitive equation model. Monthly Weather Review 1982; 110: 1831-1842.
11. McDonald A. Accuracy of multiply-upstream, semi-Lagrangian advective. Monthly Weather Review 1984; 112: 1267-1275.
12. Staniforth A, Côté J. Semi-Lagrangian schemes for atmospheric models - a review. Monthly Weather Review 1991; 119: 2206-2223.
13. Bermejo R, Staniforth A. The conversion of semi-Lagrangian advection schemes to quasi-monotone schemes. Monthly Weather Review 1992; 120: 2622-2632.
14. Holnicki P. A shape-preserving interpolation: applications to semi-Lagrangian advection. Monthly Weather Review 1995; 123: 862-870.
15. Liu J, Pope A, Sepehrnoori K. A high-resolution finite-difference scheme for nonuniform grids. Applied Mathematical Modelling 1995; 19: 162-172.
16. Leonard BP. Simple high-accuracy resolution program for convective modelling of discontinuities. International Journal for Numerical Methods in Fluids 1988; 8: 1291-1318.
17. Holly FM, Komatsu T. Derivative approximations in the two-point fourth-order method for pollution transport. In Proceedings of the Conference on Frontiers in Hydraulic Engineering, Shen HT (ed.). American Society of Civil Engineers: New York, 1983; 349-355.
18. Prenter PM. Splines and Variational Methods. Wiley: New York, 1975.
19. McDonald A. Accuracy of multiply-upstream, semi-Lagrangian schemes'. Monthly Weather Review 1984; 112: 1267-1275.
20. McCalpin JD. A quantitative analysis of the dissipation inherent in semi-Lagrangian advection. Monthly Weather Review 1988; 116: 2330-2336.
21. Yang JC, Belleudy P, Temperville A. A higher-order Eulerian scheme for coupled advection-diffusion transport. International Journal for Numerical Methods in Fluids 1991; 12: 43-58.
22. Holly FM, Usseglio-Polatera JM. Dispersion simulation in two-dimensional tidal flow. Journal of Hydraulic Engineering ASCE 1984; 110: 905-926.
23. Yang JC, Hsu EL. On the use of the reach-back characteristics method for calculation of dispersion. International Journal for Numerical Methods in Fluids 1991; 12: 225-235.
24. Chau KW, Lee JHW. A robust mathematical model for pollutant transportation in estuaries. Journal of Water Resources 1991; 168: 63-80.
25. Holly FM, Toda K. Hybrid numerical scheme for linear and nonlinear advection. Proceedings of the 21st Congress of the International Association for Hydraulic Research 1985; 2: 311-316.
26. Schohl GA, Holly FM. Cubic spline interpolation in Lagrangian advection computation. Journal of Hydraulic Engineering ASCE 1991; 117: 248-253.
27. Renka RJ. Interpolatory tension splines with automatic selection of tension factors. SIAM Journal for Scientific and Statistical Computing 1987; 8: 393-415.
28. Fritsch FN, Carlson RE. Monotone piecewise cubic interpolation. SIAM Journal for Numerical Analysis 1980; 17: 238-246.
29. Fritsch FN, Butland J. A method for constructing local monotone piecewise cubic interpolants. SIAM Journal for Scientific and Statistical Computing 1984; 5: 300-304.
30. de Boor C. A Practical Guide to Splines, Applied Mathematical Sciences, vol. 27, Springer: New York, 1978.
31. Cline AK. Scalar- and planar-valued curve fitting using splines under tension. Association for Computing Machinery Communications 1974; 17: 218-220.
32. Rentrop P. An algorithm for the computation of the exponential spline. Numerische Mathematik 1980; 35: 81-93.
33. Rentrop P, Wever U. Computation strategies for the tension parameters of the exponential splines. In Conference on Optimal Control and Variational Calculus, Bulirsch R, Miele A, Stoer J, Well K (eds). Springer: Berlin, 1986; 122-134.
34. Tornow V. An exponential spline interpolation for unequally spaced data points. Computer Physics Communications 1982; 28: 61-67.
35. Wever U. Non-negative exponential splines. Computer Aided Design 1988; 20: 11-16.
36. Akima H. A new method of interpolation and smooth curve fitting based on local procedures. Journal of the Association of Computing Machinery 1970; 17: 589-602.
37. Hyman JM. Accurate monotonicity preserving cubic interpolation. SIAM Journal for Scientific and Statistical Computing 1983; 4: 645-654.
38. Hyman JM. Accurate monotonicity preserving cubic interpolation. Los Alamos National Laboratory, Report LA-8796-MS, 1982.
39. Huynh HT. Accurate monotone cubic interpolation. SIAM Journal for Numerical Analysis 1993; 30: 57-100.
40. Zoppu C. Numerical solution of the advective-diffusion equation. MSs thesis, Department of Mathematics, Australian National University, 1994.
41. Renka RJ. Algorithm 716: TSPACK: tension spline curve-fitting package. Association of Computing Machinery Transactions on Mathematical Software 1993; 19: 81-94.
42. Abramowitz M, Stegun IA. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications: New York, 1972.
43. LeVeque RJ. Numerical Methods for Conservation Laws. Birkhauser: Basel, 1992.
44. Lax P, Wendroff B. Systems of conservation laws. Communications on Pure and Applied Mathematics 1960; XIII: 217-237.
45. Warming RF, Beam RM. Upwind second-order difference schemes and applications in aerodynamic flow. American Institute of Aeronautics and Astronautics 1976; 14: 1241-1249.
46. Boris JP, Book DL. Flux-corrected transport. I. SHASTA, A fluid transport algorithm that works. Journal of Computational Physics 1973; 11: 38-69.
47. Morrow R, Noye BJ. An implicit flux-corrected transport algorithm for both diffusion dominated and diffusion free conditions. In Computational Techniques and Applications, Noye BJ, Benjamin BR, Colgan LH (eds). Australian Mathematical Society: Canberra, 1992; 347-354.
48. Zalesak ST. Fully multidimensional flux-corrected transport algorithm for fluids. Journal of Computational Physics 1979; 31: 335-362.
49. Leonard BP. The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection. Computer Methods in Applied Mechanics and Engineering 1991; 88: 17-74.
50. Herriot JG, Reinsch CH. Algorithm 507: procedures for quintic natural spline interpolation. Association of Computing Machinery Transactions on Mathematical Software 1976; 2: 281-289.
51. Warming RF, Hyett BJ. The modified equation approach to the stability and accuracy analysis of finite-difference methods. Journal of Computational Physics 1974; 14: 159-179.
52. Leonard BP, Lock AP, MacVean MK. The NIRVANA scheme applied to one-dimensional advection. International Journal for Numerical Methods in Heat and Fluid Flow 1995; 5: 341-377.
53. Strang G. On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis 1968; 5: 506-517.
54. Leonard BP, Lock AP, MacVean MK. Conservative explicit unrestricted-time-step multidimensional constancy-preserving advection schemes. Monthly Weather Review 1996; 124: 2588-2606.
55. Li CW. Advection-dispersion simulation by minimization characteristics and alternate direction-explicit methods. Applied Mathematical Modeling 1991; 15: 616-623.
56. van Leer B. Towards the ultimate conservative difference scheme II: monotonicity and conservation combined in a second-order scheme. Journal of Computational Physics 1974; 14: 361-370.
57. Leonard BP. Simple high-accuracy resolution program for convective modelling of discontinuities. International Journal for Numerical Methods in Fluids 1988; 8: 1291-1318.
58. Roe PL. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics 1981; 43: 357-372.