The finite-difference scheme for the unsteady convection-diffusion equation based on weighted local cubic spline interpolation

A. A. Semyonova   ; A. V. Starchenko

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 49 (2017), pp. 61-74

Résumé

In this paper, special attention is paid to the choice of an approximating scheme for the convective terms of the unsteady convection–diffusion equation. The purpose of this study is to develop a difference scheme for the convection–diffusion equation with weighted local cubic spline approximation for the convective terms. The advantage of weighted cubic spline functions is shown in comparsion with other methods for interpolating functions that are set by the table of their values for the case when four values of the interpolated function are given. The interpolating local cubic spline oscillates but shows a deviation from the original monotonic distribution. The best result is obtained with the weighted local cubic spline. The resulting finite difference spline scheme was used to solve two unsteady problems with the known analytical solution: the "diffusionless" propagation of an impurity and the propagation of an impurity from an instantaneous point source. The following finite difference schemes with different approximations for the convective terms of the equation were compared: the upwind scheme, the Harten scheme, the superbee limiter scheme, MLU, MUSCL, and the 3rd order approximating ENO scheme. The results of the calculations performed for various density of grid nodes show the convergence of the approximate solution to the exact solution. For the first test problem, the spline scheme is at the advantage of the proximity of the calculated solution to the exact one over the other schemes. For the second test problem, which is characterized by smoother spatial solution profiles, on a coarse grid spline scheme gives solution which is in the best agreement with the exact solution. On a more detailed grid, the best results are given by the MLU and MUSCL schemes. The spline proposed is slightly inferior to them, but in this test example the spline scheme predicts the current maximum concentration more accurately, which is certainly an advantage for the representation of peak concentrations of air pollutants.

Détail

Keywords: unsteady convection–diffusion equation, weighted local cubic splines, monotonized high order approximation for convective terms.

Bibliographie

[1] Samarskiy A. A., Mihaylov A. P., Mathematical modeling, Fizmatlit, M., 2001

[2] Samarskiy A. A., Theory of finite difference schemes, Nauka, M., 1977

[3] Godunov S. K., “A Difference scheme for numerical solution of discontinuous solution of hydrodynamic equations”, Math. Sbornik, 47(89):3 (1959), 271–306 | Zbl

[4] Lax P. D., Wendroff B., “Systems of conservation laws”, Communications in Pure and Applied Mathematics, 13 (1960), 217–237 | DOI | MR | Zbl

[5] Kolgan V. P., “Application of the principle of minimum derivatives to the construction of difference schemes for computing discontinuous solutions of gas dynamics”, Uch. Zap. TsaGI, 3:6 (1972), 68–77

[6] van Leer B., “Towards the ultimate conservative difference scheme I. The quest for monotonicity”, Lecture Notes in Physics, 18, 1973, 163–168 | DOI | MR | Zbl

[7] van Leer B., “Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second-order scheme”, J. Comp. Phys., 14 (1974), 361–370 | DOI | MR | Zbl

[8] Noll B., “Evaluation of a bounded high-resolution scheme for combustor flow computations”, AIAA Journal, 30:1 (1992), 64–68 | DOI

[9] van Leer B., “Towards the ultimate conservative difference scheme V. A Second order sequel to Godunov's method”, J. Comput. Phys., 32 (1979), 101–136 | DOI | MR | Zbl

[10] Toro E. F., Riemann solvers and numerical methods for fluid dynamics, 2nd edition, Springer, Berlin–Heidelberg, 1999, 645 pp. | DOI | MR | Zbl

[11] Lebedev A. S., Chernyj S. G., Practice on the numerical solution of partial differential equations, NSU publ., Novosibirsk, 2000, 136 pp.

[12] Harten A., “High resolution schemes for hyperbolic conservation laws”, J. Comp. Phys., 49 (1983), 357–393 | DOI | MR | Zbl

[13] Roe P. L., “Characteristic-based schemes for the Euler equations”, Ann. Rev. Fluid Mech., 18 (1986), 337–365 | DOI | MR | Zbl

[14] Harten A., Engquist B., Osher S., Chakravarthy S. R., “Some results on uniformly highorder accurate essentially non-oscillatory schemes”, J. Appl. Num. Math., 2 (1986), 347–377 | DOI | MR | Zbl

[15] Kvasov B. I., Methods of shape-preserving spline approximation, World Scientific Publishing Company, Singapore, 2000 | MR | Zbl

[16] Karpova A. A., “Approximation of tabulated functions by means of approximation and interpolation weighted splines”, Scientific Conference of Students of the Mechanics and Mathematics Department (24–30 april 2014, Tomsk), 2014, 38–39

[17] Marchuk G. I., Mathematical models in environmental problems, Elsevier, Amsterdam, 1982 | MR

[18] Cada M., Torrilhon M., “Compact third-order limiter functions for finite volume methods”, J. Computational Physics, 228:11 (2009), 4118–4145 | DOI | MR | Zbl

[19] Starchenko A. V., Danilkin E. A., Semenova A. A., Bart A. A., “Parallel algorithms for a 3D photochemical model of pollutant transport in the atmosphere”, Communications in Computer and Information Science, 687 (2016), 158–171 | DOI

[20] Chi-Wang Shu, Essentially non-oscillatory schemes for hyperbolic conservation laws, Preprint of Division of Applied Mathematics, Brown University, 1996, 92 pp. | MR

[21] Rivin G. S., Voronina P. V., “Aerosol transfer in the atmosphere: selection of a finite difference scheme”, Atmospheric and Oceanic Optics, 10:6 (1997), 386–392 | Zbl