A Third-Order Scheme for Numerical Fluxes to Guarantee Non-Negative Coefficients for Advection-Diffusion Equations
Katsuhiro Sakai, Daishi Watabe
.
DOI: 10.4236/ajcm.2011.11004   PDF    HTML     4,871 Downloads   11,007 Views   Citations

Abstract

According to Godunov theorem for numerical calculations of advection equations, there exist no high-er-order schemes with constant positive difference coefficients in a family of polynomial schemes with an accuracy exceeding the first-order. In case of advection-diffusion equations, so far there have been not found stable schemes with positive difference coefficients in a family of numerical schemes exceeding the second-order accuracy. We propose a third-order computational scheme for numerical fluxes to guarantee the non-negative difference coefficients of resulting finite difference equations for advection-diffusion equations. The present scheme is optimized so as to minimize truncation errors for the numerical fluxes while fulfilling the positivity condition of the difference coefficients which are variable depending on the local Courant number and diffusion number. The feature of the present optimized scheme consists in keeping the third-order accuracy anywhere without any numerical flux limiter by using the same stencil number as convemtional third-order shemes such as KAWAMURA and UTOPIA schemes. We extend the present method into multi-dimensional equations. Numerical experiments for linear and nonlinear advection-diffusion equations were performed and the present scheme’s applicability to nonlinear Burger’s equation was confirmed.

Share and Cite:

K. Sakai and D. Watabe, "A Third-Order Scheme for Numerical Fluxes to Guarantee Non-Negative Coefficients for Advection-Diffusion Equations," American Journal of Computational Mathematics, Vol. 1 No. 1, 2011, pp. 26-38. doi: 10.4236/ajcm.2011.11004.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] B. P. Leonard, “A Stable and Accurate Convective Mod-eling Procedure Based on Quadratic Upstream Interpola-tion,” Computer Methods in Applied Mechanics and En-gineering, Vol. 19, No. 1 1979, pp. 59-98. doi:10.1016/0045-7825(79)90034-3
[2] T. Kawamura and K. Kuwahara, “Computation of High Reynolds Number Flow Around a Circular Cylinder with Surface Roughness,” Fluid Dynamics Research, Vol. 1, No. 6, 1986, pp. 145-162. doi:10.1016/0169-5983(86)90014-6
[3] M. Chapman, “FRAM Nonlinear Damping Algorithms for the Continu-ity Equation,” Journal of Computational Physics,Vol. 44, No. 1, 1981, pp. 84-103. doi:10.1016/0021-9991(81)90039-5
[4] A. Harten, “High Resolution Scheme for Hyperbolic Conservation Laws,” Journal of Computational Physics, Vol. 49, No. 3, 1983, pp. 357-393. doi:10.1016/0021-9991(83)90136-5
[5] K. Sakai, “A Nonoscillatory Numerical Scheme Based on a General Solution of 2-D Unsteady Advection-Diffusion Equa-tions,” Journal of Computational and Applied Mathe-matics, Vol. 108, No. 1-2, 1999, pp. 145-156. doi:10.1016/S0377-0427(99)00107-7
[6] K. Sakai and I. Kimura, “A Numerical Scheme for Compressible Fluid Flows Based on Analog Numerical Analysis Method,” Computational Fluid Dynamics Journal, Vol. 13, No. 3, 2004, pp. 468-482.
[7] K. Sakai and I. Kimura, “A Numerical Scheme Based on a Solution of Nonlinear Advection-Diffusion Equations,” Journal of Computa-tional and Applied Mathematics, Vol. 173, No. 1, 2004, pp. 39-55. doi:10.1016/j.cam.2004.02.019

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.